Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Dynamical error suppression for quantum information
(USC Thesis Other)
Dynamical error suppression for quantum information
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
DYNAMICAL ERROR SUPPRESSION FOR QUANTUM INFORMATION PROCESSING by Kaveh Khodjasteh Lakelayeh A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2007 Copyright 2007 Kaveh Khodjasteh Lakelayeh Epigraph What you call passion is not spiritual force, but friction between the soul and the outside world. Where passion dominates, that does not signify the presence of greater desire and ambition, but rather the misdirection of these qualities toward an isolated and false goal, with a consequent tension and sultriness in the atmosphere. Those who direct the maximum force of their desires toward the center, toward true being, toward perfection, seem quieter than the passionate souls because the flame of their fervor cannot always be seen. In argument, for example, they will not shout and wave their arms. But I assure you, they are nevertheless burning with subdued fires. - Herman Hesse, Das Glasperlenspiel [The Glass Bead Game] ii Dedication To my loving parents, beloved companions, compassionate friends, and generous teachers. iii Acknowledgments My deepest thanks and respect first go to my parents as they gave me support, encour- agement and much more during my course of study. Words will simply fail in their case. I also wish to thank my thesis advisor, Professor Daniel A. Lidar whose support, guidance, and friendship were essential in my progress as a student. I am also especially grateful to him for providing a perfect research environment and the freedom of pursuing my scientific interests. It will be impossible to thank all of my teachers by mentioning their names here. All of my achievements during my studies and research are directly influenced by them in some way or another. I would like to especially thank Professors John E. Sipe, Charles Dyer, and Tony Key who were exceptionally inspiring during my studies at the Univer- sity of Toronto. I would also like to go back a few years and thank Professors Hesam Arfaei, Farhad Ardalan, Siavash Shahshahani, and Mohammad Khorrami for scientific inspirations that drove me along the path of theoretical physics. I also would like to thank Kasra Rafi and Ramin Takloo-Bighash (now professors) who during their under- graduate studies took exception to the ordinary and showed me the beauty and utility of higher mathematics at our high school club. I would like to thank Mr. Kazemi who taught me how to rigorously count and prove and Mr. Hashemi who thought I was a good student despite my grades. iv It will also be impossible to name all of my supportive, kind, and generous friends to thank them. Arnavaz B. Danesh has a special place as she helped and coped along as I managed through my studies and led me in a way to find myself. I should warmly thank Yashar, Wolfgang, and Niayesh who motivated me in science and beyond in profound ways. I should also thank Lian-ao Wu, Mark Byrd, Sara Schneider, Alioscia Hamma, Masoud Mohseni, Alireza Shabani, and Joseph Geraci who as colleagues bore with my crazy ideas and requests and helped me far more than is typical of fellow travelers. I would also like to thank Bahareh, Ryan, Pedram, and Shahin for their generosity and support which came in unexpected ways. For all of them, I wish the best. I thank Professor John E. Sipe, along with Professor Aephraim M. Steinberg for their support and guidance as the former members of my graduate advisory committee at the University of Toronto. I would also like to thank the members of my current committee: Professor Werner D¨ appen, Professor Grae Jia Lu, Professor Todd Brun, Professor Igor Davetak, Professor Paolo Zanardi, and Professor Stephan Haas. I need to thank Professor Haas especially, as he has tremendously helped with my transfer to the University of Southern California and made it possible to focus on my research. I should also thank all those who facilitated my life and enhanced my experience as a university student. I am especially grateful to Krystyna Biel and Marianne Khurana for their selfless help and various wise and creative solutions that they came up with for all of my seemingly impossible administrative problems. I should also thank Frank Bures, Frank Niertit, and Daniel Gruner for handling and coping with my almost catastrophic computer incidents. Finally my deep thanks and gratitude to Nelia for sharing with me an endless source of inspiration and for her vast kindness. v Table of Contents Epigraph ii Dedication iii Acknowledgments iv List of Figures ix Abstract x Chapter 1: A Biased (Re)View of Quantum Information Processing 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 System, Measurement and Control . . . . . . . . . . . . . . . . . . . . 2 1.3 Quantum Computers and Quantum Algorithms . . . . . . . . . . . . . 6 1.4 Decoherence and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Realization of Quantum Information Processing . . . . . . . . . . . . . 9 1.6 The Road Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Control, Universality, Channels, Errors, and Error Correction 12 2.1 Control and Universality . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Pathology of Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Geometric Characterization of the Bath . . . . . . . . . . . . . . . . . 15 2.4 Quantum Channels and Markovian Dynamics . . . . . . . . . . . . . . 16 2.5 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.1 Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Decoherence free subspaces . . . . . . . . . . . . . . . . . . . 26 2.5.3 Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6.1 Ion Traps as Qubits . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6.2 Nuclear Spins as Qubits . . . . . . . . . . . . . . . . . . . . . 28 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 vi Chapter 3: Dynamical Decoupling 30 3.1 Effective Hamiltonian Theory . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Dynamical Decoupling: First Order Canonical Setting . . . . . . . . . . 31 3.3 Universal Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . 36 3.4 Decoupling of the Spin-Boson Model . . . . . . . . . . . . . . . . . . 38 3.5 Dynamical Decoupling as Means of Quantum Error Correction . . . . . 41 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 4: Fault-Tolerant QC in the Presence of Spontaneous Emission 47 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Detected Jump-Corrected Codes . . . . . . . . . . . . . . . . . . . . . 49 4.2.1 Example: Encoded Logic for 4-Qubit DJC Code . . . . . . . . 51 4.2.2 Generalization: DJC Code Encoding Several Qubits . . . . . . 53 4.3 Fault Tolerant Measurement and Recovery . . . . . . . . . . . . . . . . 54 4.3.1 State Preparation and Read-Out . . . . . . . . . . . . . . . . . 56 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 5: Efficient Error Correction and QC in the presence S.E. 58 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Eliminating the Conditional Evolution using BB pulses . . . . . . . . . 61 5.3 Correcting Spontaneous Emission Jumps with a QECC . . . . . . . . . 63 5.4 Fault Tolerant Preparation, Measurement, and Recovery . . . . . . . . . 66 5.5 Fault Tolerant Computation . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5.1 Case 1: Natural{Z i ,X i ,X i X j } . . . . . . . . . . . . . . . . . 71 5.5.2 Case 2:{Z i ,X i ,XY Model} . . . . . . . . . . . . . . . . . . . 71 5.5.3 Case 3:{Z i ,X i , Heisenberg interaction} . . . . . . . . . . . . 72 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 6: Fault-Tolerant Quantum Dynamical Decoupling 74 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2 The noisy quantum control problem . . . . . . . . . . . . . . . . . . . 75 6.2.1 Periodic DD . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2.2 Concatenated DD . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.3 Numerical Results for Spin-Bath Models . . . . . . . . . . . . . . . . . 78 6.4 Imperfect decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.4.1 Convergence of CDD in the limit of zero-width pulses . . . . . 82 6.4.2 Finite width pulses . . . . . . . . . . . . . . . . . . . . . . . . 84 6.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 84 Chapter 7: Efficiency Analysis of Dynamical Decoupling 85 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Universal Dynamical Decoupling for a Qubit . . . . . . . . . . . . . . 88 vii 7.3 Analysis of the Universal Decoupling Pulse Sequence . . . . . . . . . . 91 7.3.1 Ideal Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.3.2 Magnus Expansion . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3.3 Pulses of Finite Width . . . . . . . . . . . . . . . . . . . . . . 97 7.4 Decoupling Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.4.1 Error Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.4.2 Periodic Decoupling . . . . . . . . . . . . . . . . . . . . . . . 102 7.4.3 Concatenated Decoupling . . . . . . . . . . . . . . . . . . . . 104 7.4.4 Finite Width . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.4.5 Example: Decoupling in Spin Quantum Dots . . . . . . . . . . 111 7.5 Higher Order (Trotter-Suzuki) Universal Decoupling . . . . . . . . . . 116 Table 1: Comparison of error phase for various decoupling schemes . . 118 7.6 Decoupling with Very Narrow Pulses Cannot Increase Error Norms . . 119 7.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.8 Addendum: Validity of the Magnus Expansion . . . . . . . . . . . . . . 124 7.9 Addendum: Finite Pulse Width Analysis for CDD . . . . . . . . . . . . 128 7.10 Addendum: Concatenation of the CPMG pulse sequence . . . . . . . . 130 Chapter 8: Self-Correcting Single Qubit Unitary Operations 132 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.2 Formulation and Characterization of Errors . . . . . . . . . . . . . . . 134 8.3 Single Qubit Operations . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3.1 Error Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3.2 First-order perfect self-correcting qubit operations . . . . . . . 139 8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.4.1 Higher order sequences . . . . . . . . . . . . . . . . . . . . . . 141 8.4.2 Improvement at What Cost? . . . . . . . . . . . . . . . . . . . 142 8.4.3 Pulse Shape Considerations . . . . . . . . . . . . . . . . . . . 143 8.4.4 Using Composite Pulse Techniques . . . . . . . . . . . . . . . 144 8.5 Addendum: Calculation of higher order correction terms . . . . . . . . 145 Chapter 9: Speculations and Outlook 148 9.1 Higher Order Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.2 Dynamical Decoupling of Extended Systems . . . . . . . . . . . . . . 150 9.3 Combining Error Correcting Codes and DD for QC . . . . . . . . . . . 151 9.4 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References 154 [Table of Contents] viii List of Figures 6.1 Performance of CDD and PDD in presence of pulse errors . . . . . . . 79 6.2 Performance of CDD and PDD as a function of the interaction strengths 79 6.3 Performance of CDD and PDD as a function of the concatenation level . 80 6.4 Renormalization of Hamiltonian in Dynamical Decoupling . . . . . . . 81 7.1 Comparison of Performance of PDD and CDD . . . . . . . . . . . . . 114 7.2 Comparison of Performance of PDD and CDD for non-ideal sequences 115 ix Abstract Quantum information theory is based on the premise of manipulating quantum systems. Decoherence and noisy control directly limit this manipulation. Quantum error correc- tion theory aims to understand the sources of errors in manipulation of quantum systems and to remedy the problems caused by the errors in an efficient manner. In this thesis I focus on error correction mechanisms that are based on a realistic and physical picture of the interactions of the quantum system with the environment. In chapters 1, 2, and 3, I provide a brief introduction to quantum information processing, quantum error correc- tion, and dynamical decoupling. In chapters 4 and 5, I consider error correction of a set of qubits in the presence of spontaneous emission as the main source of errors. These results have been published in [KL02] and [KL03]. The quantum trajectories picture is used for describing the error processes. Two error correction schemes are provided in this scenario and are both built on simple quantum error detecting codes for detecting quantum jump errors. The qubit number overhead in this encoding is reduced in the first method [KL02] by exploiting the symmetry of the conditional dynamics that can be used to create a decoherence free subspace. In the second method [KL03], the condi- tional dynamics is canceled by applying parallel population swapping operations on the qubits. For both methods, I describe means of integrating the proposed error correction schemes with various proposals to achieve fault tolerant quantum computation. Chap- ters 6 and 7 are based on dynamical decoupling: a method for removal of undesired x interaction terms from a Hamiltonian evolution by application of fixed unitary quantum operators. These results have been published in [KL05a] and [KL06]. I describe general concatenated pulse sequences that are constructed recursively from simple dynamical decoupling pulse sequences. I show that using the concatenated dynamical decoupling sequences is (i) significantly more efficient than repeating traditional sequences and (ii) these sequences are more robust with respect to natural control errors [KL05a]. A com- prehensive leading order analysis of dynamical decoupling efficiency is provided in the process [KL06]. In chapter 8 (not yet published), I describe the construction of self- correcting pulse sequences for a single qubit. xi Chapter 1: A Biased (Re)View of Quantum Information Processing 1.1 Motivation In the modern world, information (or lack of ignorance for the negativist) is not a mys- tical concept anymore. Information can be quantified and processed and in turn be transferred. This comes in sharp contrast with the way quantum mechanics (and theory of relativity to some extent) has altered our conception of physical systems as essen- tially unavailable as a whole, chameleonic in features, and impossible to replicate. . . and yet [Lan96] information is physical. The apparent paradoxical nature of information, as ultimately a quantum phenomenon, can be sharpened by making a distinction between the quantum and the classical. Information as something absolute that can be possessed is fundamentally a classical object. Quantum information [NC00] on the other hand can only be measured into classical information and is encoded in the logical states of quantum systems. The processing of quantum and classical information by controlling quantum systems or simply quantum computation [Fey86](QC) promises more process- ing power and speed than conventional computing in terms of classical information. This is appreciated by the current efforts in the study and utilization of quantum systems and the exciting interdisciplinary study of quantum algorithms and quantum information processing schemes. Many quantum algorithms and communication protocols offer concrete advantages over their classical counterparts but unfortunately the quantum nature of these algo- rithms is checked by our means of control, isolation, and ultimately understanding, resulting in errors. The assumption of an isolated and controllable quantum system, 1 simply breaks down due to these errors. Errors leave us then with a classical statisti- cal mixture and the quantum advantages are lost. These phenomena, more specifically known as decoherence [Zur91] plague all physical realizations of quantum information processing and has led to the study of quantum error suppression theory where different methods or proposals for quantum information processing in the presence of differ- ent types and regimes of errors are explored. Some of these proposals are reviewed, expanded, and elaborated in this thesis. In this introductory chapter, I present a short review of the principles of quantum information processing and decoherence. This review is by no means complete, free of misrepresentation, nor self-contained and some of the concepts and methods will be covered in more detail again (and again). The biased structure of this review roughly reflects the way I have been exposed to the subject during my graduate studies and how they support the body of the thesis. 1.2 System, Measurement and Control States of a quantum system in isolation are described by the rays of a Hilbert space. The dimension of this Hilbert spaced is practically dictated by our understanding and realization of the quantum system under consideration. For example a two-level system such as a spin-1/2 is represented by a 2 dimensional Hilbert space. Further structure can be formally introduced into any Hilbert space by identifying its subsystems and subspaces. The notion of subsystems is particularly important in our understanding and control of quantum systems. Mathematically speaking, if a Hilbert spaceH is written asH = H 1 ⊗H 2 ,H 1 andH 2 are referred to as its subsystems. However, this becomes more than a formal construct, when the subsystems are physically meaningful/useful. Two 2 spin- 1 2 systems or qubits can be treated as subsystem of a bigger 4-dimensional Hilbert space but as long as they cannot be addressed individually, this representation is only superficial. Practically, operators acting on the subsystems available to us dictate our subsystem decomposition [ZLL04]. Furthermore, in many cases the representations of a symmetry group naturally dictate a subspace/subsystem decomposition for a Hilbert space. The evolution of the states of a quantum system is dictated by its Hamiltonian. A constant Hamiltonian provides an autonomous evolution but a time-varying Hamiltonian H(t) can be used for navigating the evolution. Formally the unitary evolution operator or the propagator corresponding to this Hamiltonian is written as U(t 1 ,t 0 ) =T + h exp(−i Z t 1 t 0 H(t)dt) i (1.1) whereT + denotes the time-ordering required for solving the Schrdinger equation when the driving Hamiltonian is not constant. In typical settings, one expects an isolated quantum system to have a constant Hamil- tonian while a time-varying Hamiltonian is intrinsically the result of the introduction of a second interacting system. A simple example is when the system is observed from the reference point of the second (controlling) quantum system. In this case, the interaction picture provides a simple time-varying Hamiltonian. More explicitly, suppose the sys- temH is composed of two subsystemsH S (controlled) andH C (controlling) and the Hamiltonian for the overall system is written as H =H S ⊗I C +H int (1.2) 3 whereI C denotes the identity operator acting onH C andH int does not commute with H S . After transforming into the interaction picture ofH int , the controlled system evolves according to the following time-dependent Hamiltonian starting att = 0 [Sak85]: H i (t) =e itH int H S e −itH int (1.3) While the interaction picture offers a natural introduction of time-dependent Hamiltoni- ans, we often treat ourselves to controllable system Hamiltonians whereH(t) is some- times designated arbitrarily. A fundamental discussion of this topic is beyond the scope of this thesis. Besides unitary evolution, we are also interested in measurements [WZ83]. Mea- surements, similar to quantum evolutions involve interaction with a measuring appa- ratus, itself another quantum system. Measurements can be implemented/understood as unitary evolutions on the larger quantum system of the measured system plus the measuring system, followed by forgetting about the measuring system [NC00]. The above paragraphs emphasize the fact that simple control and measurements of an individual quantum system, although postulated in quantum mechanics in a simple manner, are in reality based on the notion of external systems and may not be straight- forward. The introduction of external systems is also responsible for the notion of open quantum systems. Consider a quantum system described byH S interacting with the environment (or the bath 1 ), another quantum system described byH B over which we typically have no control or measurement power. Since the combined system is out of 1 The term bath often appears in a thermodynamical context where the environment is at a fixed tem- perature 4 reach, the unitary evolution of the combined system, manifests itself inH S with possi- bly non-unitary dynamics. This also requires open quantum systems to be described by density matrices. Starting with the overall Hamiltonian given by H =H S ⊗I B +I S ⊗H B +H SB (1.4) whereI S (I B ) is the identity operator acting on the system (the environment) andH SB denotes the interaction terms, any desired extra system can be added toH B but in prac- tice only states that contribute to the combined dynamics via H SB are included. The open system dynamics can be described by a quantum operationE given by E(ρ i ) = Tr B [Uρ i U † ] (1.5) whereU is the unitary propagator for the combined system governed byH andρ i refers to the initial combined density matrix for the system and the environment. In the study of open quantum systems, the statistical properties of the environment allow us to perform simplifying approximation on the dynamics. An important case is Markovian dynamics where the time-evolution of the density matrix is generated instan- taneously and does not depend on the history of the evolution in an explicit manner. In this case we can write dρ(t) dt =L[ρ(t)] (1.6) whereL is a function defined on system density matrices. Linearity of quantum mechan- ical evolution implies further that L is a linear function, a superoperator as it is an operator acting on operators. Much of the standard theory of open quantum systems relies on the Markovian approximation. This theory is somewhat unsuitable for deal- ing with many important quantum systems in quantum information processing and in 5 such non-Markovian settings one is often forced to consider the full Hamiltonian evo- lution. Understanding and characterization of the open systems dynamics of quantum systems is essential in the suppression of errors. As we shall see various error correction techniques rely on different representations of errors. 1.3 Quantum Computers and Quantum Algorithms A quantum computer is a quantum system that can be manipulated from a certain ini- tial condition to the point that measurement results on it correspond to the solution of a given computational problem. In contrast, a classical computer utilizes only classical measurement results at each stage of the computation and does not rely on the intermit- tent quantum states. The superiority of quantum computers to classical computers is still under debate although there are clear examples of quantum algorithms that outperform their classical counterparts in certain important computational problems. The famous examples are the Shor Algorithm for quantum Fourier transform and specifically for integer factorization and the Grover Search Algorithm [Sho94, Gro96]. Quantum algorithms [NC00] are usually described as a preparation, followed by a sequence of unitary evolutions (sometimes called quantum gates) and measurements, ending typically in a set of final readout measurements, where the solution is obtained either deterministically or with a significant probability. A striking feature of all efficient quantum algorithms is the appearance of entanglement. In vague and simple words, entanglement is a measure of how far the state of a quantum system is from being a ten- sor product of subsystem states. In the language of quantum algorithms entanglement is produced by application of multi-qubit gates which are typically harder to implement than single qubit gates. This difficulty is only the tip of the iceberg of problems of tran- sition from a quantum algorithm to a working quantum computer. In reality, one can 6 switch and control the Hamiltonian for a spatially extensive quantum system (such as a 2-dimensional array of spins) to a limited degree of precision and range both in time and space. The question of realization of a quantum algorithm on this system, natu- rally translates into the problem of approximating an arbitrary unitary operator on this system using the available controls such that the number of steps involved in the approx- imation does not offset the efficiency of the quantum algorithm. This is often referred to as the question of quantum universality. The celebrated Solovay-Kitaev theorem is the basic starting point for the assertion that given a limited set of unitary operations on two qubits, it is possible to generate an arbitrary two-qubit unitary operation with a given error and the steps involved will be asymptotically a polynomial of the inverse of the approximation error [DN05]. Despite encouraging universality results, even if the quantum gates required in a quantum algorithm are readily available, quantum comput- ers face the greater existential challenge of decoherence that we consider in the next section. 1.4 Decoherence and Errors In almost all applications, one idealizes a quantum system that is simply stable or con- trolled in a designated manner. Any deviation from this idealization in reality corre- sponds to an error. Errors and seemingly random deviations are present in all technolo- gies but quantum systems can be affected by errors in a more profound way. Ascetically interpreting quantum mechanics as a theory in which the basis of observation can give rise to superpositions of states as opposed to statistical mixture, we find out that depend- ing on the environment, these superpositions can quickly decay into statistical mixtures that are oblivious to the basis of observation and mathematically speaking, the density matrix of the system becomes diagonal [Zur91]. Decoherence makes quantum classical. 7 One way of quantifying these effects is to look at purity defined as Tr(ρ 2 ). Purity is a good measure of decoherence when we are not concerned with the subsystems as it directly measures the loss of the off-diagonal elements. When there are subsystems in the system such as a system of many qubits, a quantum fidelity measure is often used to quantify the effect of decoherence and other errors. We also speak of fidelities of opera- tions when we are comparing a strategy for realization of a particular quantum operation to that of an ideal operation. While decoherence can be thought of as a systematic error (due to the essentially fixed but unknown environmental interactions), random errors are also an important hurdle for realization of quantum information processing devices. Such errors can exist in various stages of a particular quantum protocol such as preparation, or the navigation of controllable parameters. It turns out that the combined effect of random errors and decoherence can become even more problematic. Before errors can be tackled, they should first be characterized and their sources identified. The shear multiplicity of the sources of errors in simple quantum systems makes this a difficult undertaking. Traditionally, tomography techniques can be used to characterize decoherence and error rates in a quantum memory but they involve many steps and are sometimes as hard as a quantum computing task [NC00, DP01]. A useful quantum information processing proposal should not only describe a strategy for deal- ing with possible errors but also should address our typical lack of knowledge and the possibly wide range of error time-scales present in a system. 8 1.5 Realization of Quantum Information Processing The fact that quantum mechanics is a fundamental theory of physics and is expected to describe all physical phenomena 2 at some level of complexity, comes in sharp con- trast with the fact that very few physical systems can be manipulated in a “quantum manner”. The effects of decoherence and the smallness of the Planck’s constant almost immediately destroy the visibility of superpositions in the realm of the tangible. Hence, all possibilities of quantum information processing are still in the microscopic realm (See [BPM + 97] for an important example of quantum teleportation). Technological advances such as the laser, electronics, and miniaturization have nonetheless made this realm much more accessible in the 21st century and presently many quantum systems can be manipulated to such degrees that we may realistically speak of physical quantum information processing systems despite all the challenge ahead. Quantum information processing proposals are as diverse as the quantum phenom- ena that can be examined in the lab. One one side of the spectrum, nuclear magnetic resonance [CLK + 00] and spin-based quantum computing [BEL00] relies on the manip- ulation and selectivity of spins of particles by external magnetic fields. On the other side of the spectrum, linear optics proposals [KLM01] encode qubits in photonic modes and interact them via post-measurement selections. At the time of writing, none of these proposals have developed into a working quan- tum computer. Requirements such as isolation, ease of manipulation, and measurements are hard to satisfy in a physical system and simply put, quantum technology is still by and large undeveloped [DiV00]. On the other hand many constraints in the hardware of a quantum computer can be eased by passive design considerations and improvements in the software. Quantum error correction [Ste99, CPM + 98], is the prominent example 2 Except gravity of course, as of now. 9 of using active/passive design to combat the undesired effects of errors and decoher- ence in a quantum computer. In fact many quantum computing proposals actively use such design ideas to define the qubits and encode technologically available switchings into quantum operations. The logical encoding of qubits and operations in the general sense of the word, forms the core of all viable quantum computing proposals. Error correction can be designed both at low and high levels. In high level error correction one often assumes a semi stable abstract notion of qubits and quantum gates and designs a correction strategy that utilizes algorithmic techniques. In low level error correction, one considers a candidate physical quantum system, where one cannot enjoy the luxu- ries of abstraction and the error correction should come from more fundamental control and detection principles. In this thesis the main focus will be on such low level error correction methods although we employ ideas from high level error correction at many points. 1.6 The Road Ahead Even if quantum computers is never built or their power is on par with classical com- puters, the science of quantum information seems to lead a life of its own, inspiring other areas of physics, mathematics, and computer science along the way. The theory of error correction in particular is deeply related to important questions such as quantum phase transitions and quantum control. Much about decoherence itselfcan be learned by studying and applying error correction. The end result is not just an isolated branch of physics or a mundane engineering task but a very exciting amalgam of deep math- ematical ideas on symmetry, interesting and revealing physical insights into evolution and characterization of fundamental systems such as spins and photons, and ingenious designs and protocols that bring these into a working and useful technical paradigm. As 10 for error correction, it is far from exaggeration to assert that any new quantum phase of matter that becomes accessible is immediately studied for its control and decoherence properties as a candidate for a quantum information processing device and every new scheme or proposal for manipulation of quantum systems is immediately considered for application on the available quantum systems. Suppressing quantum errors is exactly what makes us suitable to consider these questions and much more. 11 Chapter 2: Control, Universality, Channels, Errors, and Error Correction 2.1 Control and Universality The premise of control assumes that there exists a reliable “controlling system” that can be manipulated to change the state of another, less tangible “controlled system”. In the context of quantum systems, even the premise might not exactly hold. To control another system, the controlling system needs to interact with it, usually resulting in the entanglement of the controlled and the controlling system. This typically results in uncertainty in the state of the controlling system and affects the controlled system in turn. In practical settings one tends to simply ignore these effects and assume a “classical control” which simply allows us to modify some interaction Hamiltonian over time. This can also be extended to include measurements as well. Typically for a quantum system, our control over the unitary evolutions or measurements is constrained. For example, in the traditional NMR setting (Sec. 2.6.2), only rotations around an axis in theXY plane are directly available [NC00]: U φ [θ] = exp[−iθ(cosφσ x +sinφσ y )] These constraints limit the extent of control of the quantum system as both the set of possible “implementable” unitary operations and the set of reachable rays in the Hilbert 12 space of the system can be non-trivial subsets. In general, if a set of control specifi- cations generate a dense subset of all possible unitary operations on a Hilbert space, they are universal. More specifically, suppose the quantum system to be controlled consists of n qubits. A set of control specifications (such as quantum gates, etc.) are universal if they can approximate with arbitrary precision, anySU(2 n ) unitary opera- tion on the qubits [DBE95, DiV95, DNBT02a]. The notion of universality can be made more practical by requiring the control sequence to be efficient: Any unitary operation can be approximate with a precisionǫ P(n) where P(n) is a function of the number of steps (or time, or generically cost)n. The control sequence is efficient ifP(n) grows slower or as a polynomial inn. Efficient constructions of arbitrary operations in terms of the available control mechanisms is perhaps the most important gradient for build- ing a quantum computer. It turns out that a very limited set of Hamiltonians or gates can be used efficiently for quantum universality. This is based on the Solovay-Kitaev algorithm [DN05]. From a historic point of view, this positive, simple, and yet funda- mental result has encouraged directly the phenomenal growth of research in quantum information science. 2.2 Pathology of Control Let us return to the simple single spin NMR example and consider what can go wrong with the rotation of the spin state of a nucleus using a Magnetic field. Obviously every- thing will fail if there is no quantum system to be controlled. Less catastrophically, the spin system might not stay within the control field or it might change in nature due to some ideally absent interaction with another system. Furthermore, there might be other degrees of freedom in the system that could be “awakened” by our control and thus require a practical redefinition of the controlled quantum system. These existential 13 challenges, can be defined in terms of interactions and Hamiltonians. Direct control errors can also be characterized in terms of the control Hamiltonian. A minimal setting for describing the pathology of Hamiltonian control of a quantum system with a Hilbert spaceH S can be given as H =H S+B +H ctrl+B whereH S+B refers to a fixed collective Hamiltonian for the system and a surrounding “bath” or environment, collectively forming a Hilbert spaceH SB . The control part, H ctrl+B refers to a part that can be controlled. In case of a control that does not non- trivially couple to the environment (purely controllable), this Hamiltonian is simplified asH ctrl (t)⊗I B where I B is the identity operator on the pure-bath Hilbert space,H B . If system is also isolated from the environment then H S+B = H S ⊗I B +I S ⊗H B . Such an isolated and purely controllable system is an ideal candidate for quantum infor- mation processing. In reality this is never the case and all quantum systems under con- trol are open quantum systems that cannot be described only by the evolution of pure states. Open quantum systems [BP02], as we shall see in Section 2.4 are sometimes best described by the linear evolution of the density matrices, or quantum channels. Although the introduction of a surrounding environment system is the natural way of introducing rogue degrees of freedom into a quantum control settings, in some cases there might be no obvious limit on which degrees of freedom should be included into the system-bath setting. In the context of propagation of (undesired, in our case) cor- relations, the difference between the undesired effects of coupling between of the spin of a proton and a nearby nucleus and that of its coupling to the Planet Mars is a matter of time scales. We refer to the general setting of the system and the bath for describ- ing an open quantum system, as the system-bath dogma. An important result that can be used to rationalize the system-bath dogma and effectively characterize the size of 14 the bath using the speed of propagation of correlations is given by a corollary to the Lieb-Robinson bound [NS06]. 2.3 Geometric Characterization of the Bath In this section we use the language and the setting of [BHV06] for a short review. Con- sider a multi-part system that comprises several subsystems (degrees of freedom) with a Hilbert spaceH =H 1 ⊗···⊗H n . Let us assume that this system evolves under an inter- action HamiltonianH = P i 1 ,···,i k H i 1 ···i k (t) such that eachH i 1 ···i k (t) term act as identity on at leastn−k subsystems andk is fixed. Specifically, assume that all mutual interac- tions among the subsystems are 2-body and thusk = 2. Let us also define a topological concept of distance by defining a graph whose vertices correspond to the subsystem and each two vertices are to be connected iff there exists a term in the Hamiltonian that cou- ples them. Leth denote the maximum of the strength of all coupling terms defined as h := maxkH i 1 ···i k k 1 . Now suppose two operatorsO A and O B are defined on regions A andB on this graph and the shortest path betweenA andB is given byL. IfO A (t) denotes operatorO A in the Heisenberg picture at timet, we have: [O A (t),O B ]≤cNkO A kkO B kexp[− L−v|t| ξ ] (2.1) whereN is the number of vertices in the smallest ofA andB andc,v, andξ only depend onh and the maximum degree of the graph. The commutator[O A (t),O B ] measures the way the parts A and B affect each other. These effects decay exponentially to zero outside a “light cone” defined by the speed of propagationv which is a function ofh. A corollary of this result states the following [Osb06]: Suppose the connected graph 1 The operator norm used here, similar to the ones used elsewhere in this thesis is a unitarily invariant norm such askAk:= max hψ|ψi=1 |hψ|A|ψi| 15 defined above describes the interaction of a quantum system with the various degrees of freedom of the bath. Let the Hamiltonian describing the whole system and the bath be given as H =H 0 + X l H B l where H B l refers to all Hamiltonian terms that are at a topological distance l [in the sense just defined] from the system. Also suppose the speed of propagation v as it appears in Eq.(2.1) is given. Then at timet, all dynamical properties of the system can be approximated as generated by a truncated HamiltonianH tr (t) given by H tr (t) =H 0 + X l<vt H B l with an errorǫ =O(e vt−l ). This reduction of the Hilbert space [in the same spirit as the cluster decomposition principle in quantum field theory [Wei95]] is essential when the system-bath dogma is used to theoretically describe the interaction of a quantum system with the environment. This approximation can affect some of the results of Chapters 6 and 7 considerably in case of spatially extended environments. 2.4 Quantum Channels and Markovian Dynamics There are many instances where the Hamiltonian description of the system and the bath, while fundamental, is of no use in describing the dynamics of an open quantum system. In particular, in many cases the bath is not well defined and one might require an a posteriori description of how the system on its own evolves over time. The implicit tracing out of the bath and other uncertainties in the evolution require us to use the density matrix of the system to describe its evolution. In general, such a mapping would be linear, trace-preserving, and positive. Normally one also needs complete positivity 16 which requires the state of the system to remain positive under the map, even if it is initially considered along with another system, which is later traced out [SMR61]. For completely positive trace preserving maps or quantum channels there exists a relatively simple description. SupposeE : C s 7→ C s denotes a mapping of density matrices of the system (forming aC ⋆ -algebraC s ). Then it is a quantum channel (trace preserving, completely positive, and linear) iff it can be written in the Kraus form [BP02] E(ρ) = X i E i ρE † i (2.2) where E i are operators acting on the system such that P i E † i E i = I S . Probabilistic evolutions of the system can easily be described by quantum channels. In this chapter where not ambiguous we use operators X,Y and Z to denote the Pauli operators on a qubit. We use the basis states|0i and|1i to represent the eigenstates of the PauliZ operator in the increasing order of the eigenvalues. Let us consider a bit flip or polarizing channel on a qubit describing random flipping of the qubit in theZ basis is given by: B(ρ,t) = [1−p(t)]ρ+p(t)XρX where we haveE 1 = [1−p(t)] 1/2 I andE 2 = p(t) 1/2 X. In this simple example,p(t) describes the probability of finding the qubit in the flipped state at timet. An important class of dynamical quantum channels are those with the Markov property: Consider a quantum channelE(t 2 ,t 1 ) that describes the evolution of the system from timet 1 tot 2 with the following semi-group property: E(t 2 ,t 1 )E(t 1 ,t 0 ) =E(t 2 ,t 0 ) 17 The semi-group property allows for a so called Master equation, whereE(t 2 ,t 1 ) sym- bolically refers to the operator generating the solution of the following differential equa- tion forρ(t) at timet 2 with initial condition set at timet 1 : dρ(t) dt =Lρ (2.3) whereL denotes a linear superoperator that acts on ρ.Symbolically we can write E(t 2 ,t 1 ) = exp(L(t 2 −t 1 )). The superoperatorL can be decomposed into unitary and dissipative parts. The unitary part is identified by a Hamiltonian dynamics of the system while the dissipative part is often called the Lindbladian superoperator [Lin76]. Lρ = Unitary z }| { −i[H s ,ρ]− X n,m h n,m ρL m L n +L m L n ρ−2L n ρL m +h.c. | {z } Dissipative (2.4) Equations (2.3–2.4) are referred to as the Lindblad equations for Markovian open quan- tum systems. If h nm as a matrix can be diagonalized, we can simplify the dissipative part and write the Lindblad equation in the following compact form: ˙ ρ(t) =−i[H s ,ρ]+ X m 2L m ρL † m −{L † m L m ,ρ} . (2.5) We will revisit the Lindblad equation again in the coming chapters. The Lindblad form of the evolution of open quantum systems is especially suitable for describing errors due to the interaction with the environment. In Eq.(2.5), if H S is under our control, the dissipative term refers to errors which are not under control and can cause decoherence. In quantum error correction theory the operators L m are simply the error operators affecting the system [KL97a]. One can always find an error basis to expand all errors in terms of the error basis operators. One may also use a 18 general quantum operation for this purpose but the Lindblad form clearly distinguishes the unitary (probably not an error source 2 .) and the dissipative (probably an error source) parts of the dynamics. It turns out that the theory of quantum error correcting codes can be built based on both descriptions although the quantum channel picture is more general. 2.5 Quantum Error Correction The discovery of quantum error correcting codes marked the beginning of a new hope for quantum information processing [Sho95]. The fact that we can use control over the quantum systems not just for steering them in a desired manner but also to correct their evolution against unknown errors, is nontrivial and is rooted in the counterintu- itive features of quantum mechanics. It is worth mentioning that the error correction of classical information is fundamentally much more accessible due to the absence of the non-cloning theorem [WZ82] that states we cannot make copies of an unknown quantum state on a different system. In a deterministic classical setting such cloning is possible and even in probabilistic classical settings cloning is possible with an arbitrary small error. Cloning brings in an almost free source of redundancy that can immediately be put to use for error correction. In quantum mechanics such a redundancy has to be constructed using entanglement. Yet, as we shall see exploiting redundancies is not the only quantum error correction strategy. Different methods of error corrections have been constructed for dealing with various errors and error regimes. In a simplistic manner we go over three different (yet similar in character or notation) error correction strategies. All these strategies will later be used in the developed ideas of this thesis. 2 We are ignoring the so called Lamb shift induced by the Lindblad operators which is a Hermitian correction to the original Hamiltonian of the system [Zan98] 19 2.5.1 Quantum Codes While it is not possible to clone quantum states, it is possible to encode them in a higher dimensional Hilbert space [KL97a]. This allows the information content of the states to be “spread out” so that if there are not too many errors present, the information can be retrieved later. Before we review this idea in more detail, let us clarify the concept of errors as it applies to the standard theory of quantum codes. The errors are assumed to be random quantum operations interspersed along the quantum circuit. The evolution of the system can thus be visualized as a series of consecutive quantum channels, some of which contain control elements (quantum gates or measurements) and some will leave certain sub parts unchanged (No Operation), and some will be errors. The error channels might be non-trace-preserving as well. Although the simplicity of this model has been controversial [AHH + 02], it is generally accepted as a basic framework upon which the theory of quantum codes has been built. In the quantum operations picture one may also include the concept of errors correlated in space (quantum operations that affect more than a single part or qubit), errors correlated in time (quantum operations that tend to occur based on the previous ones) although they are not directly corrected in the canonical theory of error correction. An important exception are systematic coherent errors that need to be addressed separately in a low level manner [KLZ98a]. Consider a quantum system that is prone to errors such as a qubit affected by the polarizing channel. We refer to this quantum system as the single physical qubit. Now let us replace the system with another systemρ phys which is composed of copies of the Hilbert space of the original qubit,H phys , spanned by the basis{|ψ i i}. This replacement must be accessible by controllable quantum operations. Physically, the new system is formed by adding new qubits which are prepared in a pure state. The ”encoding” oper- ation of preparingρ phys is, represented by the quantum channelC. A simple example is bringing two similar physical qubits alongside the original ones and performing some 20 quantum operations on them. Of course, by enlarging the Hilbert space in this sense, we will face new errors affecting the new qubits. Assuming that the encoded system still undergoes a completely positive map, we can still describe the effect of the errors as a quantum channel E phys (ρ phys ) = X i E i ρ phys E † i . We will also require a method of undoing the effect of the errors on the encoded system. This will be performed by another quantum channel, the recovery operationR. In the presence of encoding and recovery the overall transformation of the physical states is given by the following combination of mappings: ρ phys 7→R◦E phys ◦C(ρ phys ). If successful as an error correction strategy, for trace preserving error channels we should haveR◦E phys ◦C(ρ lgc ) = ρ phys . For non-trace preserving maps the equal sign can be replaced with a proportionality sign. The quantum error correcting criteria states a simple sufficient and necessary condition [KL97a]for a quantum code to be successful against the error generatorsE i : D ψ i E † k E l ψ j E =C kl δ ij , (2.6) whereC kl is a Hermitian matrix. Intuitively, one can interpret Eq.(2.6) as follows: Con- sider the effect of two errorsE k on the basis states|ψ i i (also known as the codewords). The operatorsE k will also naturally include an operator proportional to identity which is not an error but needs to be included, as we shall see. If the action of each of the error operators completely transforms the basis states into a new basis that is connected to the original basis in a reversible way (e.g. orthogonal in case of non-degenerate codes 21 whereC kl is diagonal), they can be distinguished from each other and from “no error” in a way that does not return any information about the superposition content of the original state. An important corollary to the quantum error correction criteria is that a code that corrects a set of errors{E i } will also correct any linear combination of them. Note that the error correcting criteria only determines which errors can be corrected. The exact performance of an error correcting code will require comparing the action of the quantum channel representing the error correction of a given error model with that of the original error channel. For this purpose one uses a measure of fidelity. For density matricesρ andσ, one such a fidelity measureF , is defined as [NC00] F(ρ,σ) := Tr h p ρ 1/2 σρ 1/2 i . Now, suppose one devises a code that corrects errorsE i α of type{α} that are bound to happen on single qubits indexed withi, with a certain independent probabilityp over the error correcting period. Even in this picture, multiple errors can also occur but with a smaller probability, such asp 2 for two errors. If a quantum code that can only correct single errorsE α i , it typically will not be able to correct errorsE α i E β j that are bound to happen with a probabilityp 2 . Heuristically this leads to the following fidelity between the initial un-erred stateρ and the corrected stateσ: F(ρ,σ) = 1−O(p 2 ) where the minus sign is due to the fact that fidelity is always smaller than 1. It is possible to construct encodings that protect againstt (or fewer) qubit errors occurring on n physical qubits encodingk logical qubits. These codes are named [n,k,2t+1] codes 22 after their classical counterparts in classical coding theory. The quantum Hamming bound [Got96] sets a limit on the number of errors that can be corrected: t X j=0 n j 3 j 2 k ≤ 2 n . It follows that the smallest encoding that encodes a single logical qubit and protects it against single qubit errors has 5 physical qubits. With degenerate quantum codes in which the errors cannot be identified individually but can be corrected nonetheless, it is possible to achieve more efficient encoding [SS07]. Simple quantum codes that correct a small set of errors can be extended recursively by concatenation to correct more errors. One theoretically starts with a small quantum correcting code that has a fidelity of1−O(p 2 ) for correcting single errors of probabilityp usingn physical qubits. The resulting encoded single qubit can be used as the building block physical qubit of another copy of the same code. This will correct the logical single qubit errors in the previous code with a fidelity of 1−O (p 2 ) 2 andn 2 physical qubits will be used. Repeating this process inl levels results in a code that usesN =n l physical qubits and has a fidelity of 1−O p 2 l = 1−O(p N 1/logn ). Where the lack of fidelity is scaled down by a power of the order N 1/logn using N physical qubits. While not exact, this heuristic suggests that concatenated codes are efficient in dealing with errors if a large enough supply of qubits is available [RDM02]. The performance of the concatenated codes is directly related to the quantum error correction thresholds where errors with a strength below a certain level can be practically overcome with perfect fidelities in an efficient manner (requiring a sub-exponential number of qubits) [KLZ98b, Ste03]. An important requirement of quantum error correction theory is fault tolerance [ABO97, Got98, Pre99]: the ability to successfully combine error correction with the 23 course of the quantum operations in quantum algorithms. Construction of quantum fault tolerant circuits requires the circuit elements to be applied to logical qubits. As such the circuit elements need to be redesigned and they should be compatible with the error detection and recovery procedures. Stabilizer quantum codes [Got97] can be used for this purpose in an elegant fashion. These codes are a special class of error correcting codes onn physical qubits (with a Hilbert spaceH n ) that are constructed in conjunction with the products of the Pauli operators (P n , a multiplicative group modulo the phase) as the error generators. One starts with the stabilizer subgroupS ofP n that act on a sub- spaceH lgc as identity and can in principle be any abelian subgroup. If the basis states of this invariant subspace are relabeled in the computational basis, this subspace becomes the logical encoded qubits. The errors that this encoding can correct are operators that anti-commute with the elements (or more compactly the generators) ofS in a manner that we shall shortly describe. Any quantum operation that commutes with the stabilizer generators will leave the error correcting properties of the code unchanged. In particular the normalizer subgroupM ofS if used as Hamiltonians generates a universal set of encoded operations on the qubits. To summarize let{E i } denote the set of the errors that can be corrected by the stabilizer code and let{S i } denote the generators of the stabilizer groupS. Then the error correction criteria is satisfied if {E i E j ,S k } = 0 orE i E j =S k for somek. The normalizer ofS is defined as all Pauli operators that commute with every element ofS. These operations leave the encoded subspace unchanged but induce logical oper- ations on the codewords. Stabilizers generators also provide a compact representation of the codewords. For example the three qubit bit flip correcting code is a stabilizer 24 code with generators given by{Z 1 Z 2 I 3 ,Z 1 I 2 Z 3 } where the indices represent the phys- ical qubit labels. The generators of the normalizer in this case are simply the encoded ¯ X = X 1 X 2 X 3 and ¯ Z = Z 1 I 2 I 3 operations. While the normalizer elements constitute an encoded version of the Pauli group on the logical qubits, they are far from universal. This is the content of the Gottesman-Knill theorem that states that any computation by means of the normalizer operators (E.g., X i Z j Z k or CNOT’s) can be efficiently sim- ulated on a classical computer. The proof is indeed based on the compactness of the stabilizer notation for describing the evolution of the state. Thus we require another operator or another method for universal computation on the qubits. For a class of stabi- lizer codes called the CSS codes (after Calderbank, Stean, and Shor) where the stabilizer generators can be written as products of physicalX or physicalZ qubit operators, it is possible to include certain prepared states along with measurements to achieve fault- tolerance [Rei04]. In other cases one may use unitary elements whose Hamiltonians are from the normalizer subgroup for universality. This in practice can be difficult as these elements can in general involve many body terms that are not directly available physi- cally and have to be simulated via the available two body interactions that are typically not fault-tolerant in the process. Thus the actual design of a fault-tolerant circuits should go alongside the physical requirements of the available operations and prepared states. Even theoretically, the geometric structure of the circuit, availability of flying qubits (for interaction at will but at the speed of light between faraway parts of the system), paral- lelism, and almost instantaneous classical computational need to be taken into account for a fault tolerant quantum computer that has a working error threshold (See [AGP06] for a example). 25 2.5.2 Decoherence free subspaces Certain symmetries in the structure of error operators allow us to construct subspaces of a larger Hilbert space that are naturally decoherence free [LCW98, LW03a]. Deco- herence free subspaces (DFS) are in a way dual to quantum error correcting codes where an error correcting subspace is constructed for the given error operators. Let H DFS = Span{|ψ α i} be a subspace in the Hilbert space of a (possibly larger) quantum system. The subspaceH DFS is a decoherence free subspace iff for the error generators E i (See Eq.(2.2) for the evolution of the system satisfy) and the DFS basis states E i |ψ α i =c i |ψ α i for everyi andα, where c i ∈ C. The DFS condition has a different but similar form for a Hamiltonian description of operators in the system-bath dogma. While the relative simplicity of the DFS condition makes it rather special in a natural setting, there are important physical quantum settings where it can be used as we shall see in the coming chapters. In general when the error operators affect the system in a manner that makes them oblivious to the labeling of the qubits (a case of permutational symmetry), we can find decoherence-free subspaces that are asymptotically large and efficient for encoding purposes. Furthermore, it is possible to construct quantum operations for control that leave the decoherence free subspace invariant in principle [KBLW01a]. In practice, the question of fault tolerance should be answered similar to quantum error correcting codes by considering the details of the physical implementation. 26 2.5.3 Dynamical Decoupling Operators in quantum mechanics do not always commute. This allows for sequences of operations to behave differently based on the order of application. Considering uni- tary operators such as the free evolution periods or quantum gates, it is possible to have sequences that act collectively and effectively like a free evolution period with a par- ticularly different Hamiltonian. In fact, it is possible to start with a given Hamiltonian for a free evolution periods and interrupt it at points with unitary operators, to engi- neer a new Hamiltonian. This fact has been used extensively in NMR spectroscopy [Fre98] to allow for recoupling and decoupling of various inter-nuclear couplings. In the setting of quantum error correction, this idea forms the basis of dynamical decou- pling [VL98, VL, VKL99]. In dynamical decoupling fast quantum unitary operations are applied to a system interacting with the environment in sequence. The resulting effective Hamiltonian for the interaction with the environment can then be switched off to zero and will hence not affect the quantum system, resulting in arbitrary quantum state preservation. As dynamical decoupling will form the main basis of research in this thesis, we have allocated a separate chapter for discussing it in more detail. 2.6 Physical Examples In this section we briefly review two physical proposals for quantum computation that are important as they are referred to in this thesis and are of particular realization poten- tial. 2.6.1 Ion Traps as Qubits The ion trap proposal originally due to Cirac and Zoller [CZ95] is a direct attempt at finding a quantum system with well defined, accessible qubits. Each qubit is identified 27 with the two (electronic) states of an ion in an ion trap. The levels can be separated due to the hyperfine splitting or the splitting can be induced by an external magnetic field producing the Zeeman effect. Single qubit operations on each qubit are realized by Stark shifts of a dedicated laser beam on the corresponding ion. The qubits are coupled via their motional degree of freedom (longitudinal phonons) [Ste97]. The ion qubits and the phonons can be encoded in a state that allows us to manipulate all of the qubits by applying properly constructed laser pulses on the ions [CC00]. As it turns out the ion trap qubits can be localized and cooled and currently the major noise factor is the random fluctuations of the laser field; as passive correction methods such as composite pulse techniques allow us to overcome systematic errors in the laser field [GRL + 03]. 2.6.2 Nuclear Spins as Qubits The nuclear spins of atoms and molecules can be directly manipulated by external mag- netic fields using nuclear magnetic resonance (NMR) techniques [Cal91]. More con- cretely, in a given molecule, various atoms can be identified by their natural magnetic resonance frequencies. A magnetic field that oscillates at a the resonance frequency of a particular nucleus ideally will only affect this nucleus and can be used for single qubit operations [NC00]. The interactions between the qubits are introduced by manip- ulation of direct dipolar couplings. As it turns out the single qubit operations can be used to stop, start and manipulate this interaction as well. The most important prob- lem of this approach is the lack of scalability to higher numbers of nuclei, due to the fact that molecules typically employed in NMR have a small number of distinguishable atoms and because the initial preparation technique of pseudo-pure states is not scalable in a straightforward manner 3 . Another important problem is the difficulty in the con- trol of decoherence due to undesired spin interactions. Despite these problems, NMR 3 One can achieve scalable preparations using algorithmic coooling. See [BMR + 02] 28 quantum information processing is still an important quantum information proposal and there are a number of ideas for improvements [VC04]. Furthermore NMR methods and ideas are directly used in other spin based proposals for quantum information processing [BEL00]. 2.7 Conclusion No system has yet been found in nature or has been engineered to have the ideal or close to the ideal setting for quantum computation. Typically quantum control suf- fers from serious problems and may simply not exist in some circumstances. The the- ory of quantum error correction is a response to these problems. The starting point of the theory is the characterization of errors that can affect a set of quantum con- trol operations. Another aspect of the theory that we skipped in this chapter is the actual identification of these errors in a given experimental setting. Once the error regimes and channels are identified, one may apply a combination of different error correction techniques to correct the errors. The threshold theorem is a positive result that encourages experimental and technological advances for reducing the errors in quantum devices so that they can be integrated into concatenated circuits to form a working robust quantum computer. While the threshold theorem holds for abstract error models, realistic considerations are expected to modify the value of the thresh- old [AGP06, SCCA05, SDT06, AHH + 02, AAK06, Kni04, Kni05] for the better or the worse. the study of fault tolerance is expected to dominate the research landscape in the- oretical quantum information science for some time and is still a source of excitement. 29 Chapter 3: Dynamical Decoupling 3.1 Effective Hamiltonian Theory The difference between time scales in the course of a quantum evolution allows us to define effective Hamiltonians that describe the evolution of the system under a time- varying Hamiltonian. For example consider the propagator of a qubit generated by the following time-dependent Hamiltonian for timest∈ [0,2τ +δ]: H(t) = gZ for 0<t<τ π 2δ X +gZ forτ <t<τ +δ gZ forτ +δ <t< 2τ +δ π 2δ X +gZ for 2τ +δ <t< 2τ +2δ (3.1) Consider the limiting case where δ → 0. In this case the Hamiltonian approaches a delta-function at time τ and 2τ and we can ignore the term gZ in the 2nd and 4th intervals in Eq.(3.1). The propagator can then be easily calculated (modulo a phase): U =U 2τ+2δ 2τ+δ U 2τ+δ τ+δ U τ+δ τ U τ 0 =X exp(−igτZ)Xexp(−igτZ) = exp(+igτZ)exp(−igτZ) =I Where we have used the following identity that holds for any invertible operatorA and any operatorM: Ae M A −1 =e AMA −1 (3.2) 30 We can interpret the identity propagator as an effective zero Hamiltonian. We can visu- alize Eq.(3.1) as the evolution of a qubit under the HamiltoniangZ interrupted atτ and 2τ by a “π pulse” that generates a bit flipX. The pulses have effectively averaged out the intrinsic Hamiltonian evolution. The simplicity of the cancelation is of course due to the limitδ→ 0, namely, the limit of infinitely strong, infinitely narrow pulses. More generally, we can formally define an effective HamiltonianH eff for the propa- gatorU of any (time-varying) HamiltonianH(t) over the interval[t,t+T]: e −iTH eff :=U. (3.3) From the above example, we already observe that the free evolution of a quantum sys- tem with a constant Hamiltonian, interrupted by unitary pulses can be thought of as an evolution under a piece-wise constant Hamiltonian. We will discuss this in more details in the next section. 3.2 Dynamical Decoupling: First Order Canonical Set- ting Quantum error correction deals primarily with the decoherence caused by undesired interactions with the environment. If these interactions can be effectively made zero, decoherence will vanish. Dynamical decoupling is a paradigm for quantum error cor- rection in which the control fields acting on a system with a (relatively) unknown inter- action with the environment can be effectively canceled or more realistically reduced. This is an example of low level error correction. In low level error correction, error correction proceeds with a close knowledge of the nature of errors in the basis blocks as opposed to high level error correction where correction procedures use semi-abstract 31 quantum elements. The previous section already contained the simplest dynamical decoupling sequence also known as CPMG [CP54, MW83] or bang-bang decoupling. In high resolution nuclear magnetic resonance spectroscopy [Hae76], such methods are in widespread use and are successful for transformation of inherent couplings under the guise of refocusing. Furthermore, transformations of available physical Hamiltonians form the basis of many proposals for quantum information processing and quantum sim- ulation algorithms [STH + 99, DNBT02b, WJB]. While dynamical decoupling methods are conceptually simple to understand and design, the question of actual effectiveness of dynamical methods in a given physical setting is a difficult one and is ultimately inter- twined with the time scales of the coupling between the system and the environment and as we shall see later, the time scales of the environment itself. Consider a quantum system (S) interacting with the environment (B) with a total Hamiltonian given by H =H S +H SB +H B (3.4) whereH S ,H B ,H SB denote the system, the environment, and the interaction Hamilto- nians respectively. As we have described earlier, it is important to be economic in the “choice” ofH B as one should exclude Hamiltonians that do not interact with the system fromH B . This can be problematic since depending on the time-scale, one is required to include those degrees of freedom of the environment that eventually “wake up to the sys- tem”. For simplicity let us assumeH S = 0, except for infinitesimally small times during 32 which pulses generating quantum operations, are applied 1 . Consider now a sequence of unitary quantum operations on the system given by P 1 f [τ] P † 1 P 2 f [τ] P † 2 ...P n f [τ] P † n whereP i andP † i denote unitary system operators and their corresponding inverses and f [τ] denotes a free evolution or “NO OPeration” period of length τ and the sequence is read from right to left as if it represents the actual multiplication of propagators cor- responding to the different segments. We call the above sequence canonical since the intervals are identical. It is not hard to see that any sequence of operations can be brought to the above form. The propagator corresponding to the whole sequence, can indeed be written as (using Eq.(3.2)) U =P 1 exp[−iτ(H B +H SB )]P † 1 ...P n exp[−iτ(H B +H SB )]P † n = exp[−iτ(H B +P 1 H SB P † 1 )]...exp[−iτ(H B +P n H SB P † n )] (3.5) 1 This is not necessary if the intervals between the operations are all of the same length, as one may use the interaction picture with respect toH S . In this pictureH B does not change butH SB and actual operations are transformed. Technically, one must then apply the operations in this reference frame. If all intervals are of the same length one cannot distinguish between the transformedH SB in each interval and thus they still can be effectively averaged out without any extra knowledge ofH S . 33 Now let us consider the limit where (in addition to infinitely narrow pulse times) the interval τ is sufficiently small: IfknτHk≪ 1, we can ignore the non-commutative properties of the operators appearing in Eq.(3.5) and write (see Eq.(eq:magnus2)) U = exp h −inτH B +τ n X i=1 P i H SB P † i − 1 2 τ 2 X i<j [H B +P i H SB P † i ,H B +P j H SB P † j ]+··· i | {z } =:W , (3.6) wherekWk =O(n 2 τ 2 kH 2 k) Notice that approximately and effectively the interaction part of the Hamiltonian is transformed in the following way H SB 7→ 1 nτ n X i=1 P i H SB P † i (3.7) The simple transformation in Eq.(3.7) forms the basis of 1st order dynamical decou- pling. For example, takeH SB = gB z ⊗Z whereB z is an operator of the environment andg is a coupling strength. For the pulses takeP i = X(I) for odd(even)i . We thus have the following transformation of the interaction Hamiltonian: H SB 7→ 0 Apparently, the interaction Hamiltonian is approximately canceled. Let us review the assumptions in this derivation once more: (i) We considered our pulses to be ideal, i.e. each pulse resulted in a propagator that indeed acted on the system as the unitary operation that it was intended to produce. (ii) We assumed the intervals of free evolution or the pulse rate inverse to be small. These assumptions cannot be satisfied in reality: 34 Ideal pulses do not exist and infinite pulse rates are not available either. In a realistic setting the cancelation will not be exact and we will be left with “un-decoupled” terms. A direct approach to the effective Hamiltonian is the “average Hamiltonian theory” and the Magnus expansion [Mag54]. For a time-dependent HamiltonianA(t) the prop- agatorU T 0 for0<t<T is given bye −iA eff , whereA eff can be written as A eff =A (1) eff +A (2) eff +... and A (i) eff are Hermitian operators approximatingA eff in increasingly higher powers of A(t). The first terms are often quoted in the literature and are of great importance in the study of accurate numerical solutions of ordinary differential equations on manifolds. See [Ise02] for a recent and introductory review. We quote the expansion here up to order3 for future reference: A (1) eff = Z T 0 A(t 1 )dt 1 (3.8) A (2) eff =− 1 2 Z T 0 Z t 1 0 [A(t 2 ),A(t 1 )]dt 2 dt 1 (3.9) A (3) eff = 1 4 Z T 0 Z t 1 0 Z t 2 0 [[A(t 3 ),A(t 2 )],A(t 1 )]dt 3 dt 2 dt 1 + 1 12 Z T 0 Z t 1 0 Z t 1 0 [A(t 3 ),[A(t 2 ),A(t 1 )]]dt 3 dt 2 dt 1 (3.10) The Magnus expansion converges ifA(t) is bounded and Z T 0 kA(t)dtk<c, wherec≈ 1.08687 [Ise02]. 35 Let us replace the pulsed evolution of the Hamiltonian, with a piece-wise constant time-varying HamiltonianH(t) obtained from Eq.(3.5) and use the Magnus expansion to obtain the effective Hamiltonian. Provided thatknτHk<c, we can write H (1) eff = 1 nτ n X i=1 P i H SB P † i +H B H (2) eff =− 1 2 n X i=1 i−1 X j=1 [P j H SB P † j ,P i H SB P † i ] . . . where the first order effective HamiltonianH (1) eff corresponds to our approximate result in Eq.(3.6). As we shall see later, the Magnus expansion provides an almost systematic way of obtaining undecoupled error terms in dynamical decoupling. 3.3 Universal Dynamical Decoupling The examples of the previous section worked for a specific type of the system-bath cou- pling Hamiltonian for a qubit, namelygB Z ⊗Z. It turns out that one can do better if there is a sufficient availability of system unitary operators even without any explicit assumptions on the form of the environment operators. Thus we may have a universal dynamical decoupling pulse sequences as pointed out in [Zan99, VKL99]. The possi- bility of approximately reducing the effects of decoherence due to unknown couplings to the environment might seem counter-intuitive at first but can be understood as the interference effect of adding mutually destructive pathways of the quantum system up to the leading order. In this section we follow the notation of [VKL99]. 36 Consider the following general form of the Hamiltonian for the system plus the environment: H = X B α ⊗S α (3.11) whereS α (B α ) are operators on the system (environment). Without loss of generality, let us assume thatS 0 = I S is the identity operator on the system, resulting in identifying B 0 with the pure-environment HamiltonianH B . Consider now the so called decoupling groupG represented on the system Hilbert space by{g i } i=1,...,n and consider the effect of the centralizer mapping of this group on a system operatorS: S7→ ¯ S = 1 |G| X g i ∈G g † i Sg i The resulting operator ¯ S commutes with every g i . Now consider the following pulse sequence DD G [τ] =g † 1 f [τ] g 1 g † 2 f [τ] g 2 ...g † n f [τ] g n (3.12) which approximately corresponds to the following propagator (the small operatorh is bounded in norm bykHk and is used to simplify the notation): U = exp[−iτ X α X i B α ⊗g † i S α g i ]+O(τ 2 h 2 ) = exp[−iτ X α B α ⊗ ¯ S α ]+O(τ 2 h 2 ) In particular, if the groupG coincides with the so called unitary error basisC [Zan99], the centralizer mapping of the Hamiltonian becomes the identity operator on the sys- tem. Thus the sequence DD C [τ] approximately and effectively cancels all interaction terms in the propagator and generates a dynamics that acts purely on the environment. Decoupling using the unitary error basis of the system is therefore a method of arbitrary 37 quantum state preservation: the state of the system will almost be the same after the pulse sequence despite the presence of the interactions with the environment. The unitary error basis can be constructed for all finite quantum systems. In the case of a qubit, this basis is given byC ={I,X,Y,Z} and the corresponding pulse sequence is given by (note the order of the group elements is arbitrary) DD C [τ] =f [τ] Xf [τ] XZf [τ] ZYf [τ] Y =f [τ] Xf [τ] Yf [τ] Xf [τ] Y, (3.13) where we have usedXZ =Y after algebraic simplification. The apparent length of the sequence is4τ and it consists of4 pulses. Another example is a system ofN non-interacting qubits indexed byk. More pre- cisely, by non-interacting we mean: given the decomposition in Eq.(3.11), eachS α is of the formI 1 ⊗...⊗S k ⊗...I N , whereS k is an operator on thek-th qubit. The decou- pling group in this case is given simply by n N N k=1 I k , N N k=1 X k , N N k=1 Y k , N N k=1 Z k o . In contrast, if we consider the possibility that all system qubits may interact with each other, the decoupling group has to be drastically enlarged to{I,X,Y,Z} ⊗N which is of order(size)4 N , thus requiring4 N single qubits pulses. 3.4 Decoupling of the Spin-Boson Model The spin-boson model that we refer to is given by the following Hamiltonian H = X i ω i a † i a i | {z } B 0 ⊗I + 1 2 X i g i (a i +a † i )⊗Z | {z } Bz⊗Z (3.14) 38 wherea i are bosonic operators of different modes (indexed withi) of a field environment interacting with a qubit system through the second term in Eq. (3.14). The operatorsa i satisfy [a i ,a † j ] =δ ij [a i ,a † j ] = 0 The algebra of the bath operatorsa i is thus nil-potent. In fact the above Hamiltonian can be exactly diagonalized. Notice also the way in which operatorsB 0 andB z correspond to the operators on the environment as defined in Eq.(3.11). It is not hard to verify that the decoupling group for the HamiltonianH is given by G ={I,X} =Z 2 . The simple decoupling sequence in this case is given by s 1 =D Z 2 [τ] =Xf [τ] Xf [τ] . The propagatorU for the evolution under this simple sequence is: U s 1 =X exp(−iτH)X exp(−iτH) = exp[−iτ(B 0 −B z ⊗Z)]exp[−iτ(B 0 +B z ⊗Z)], (3.15) which was discussed in Section 3.2. In the remainder of this section we use the notation and reproduce the results origi- nally appearing in [VL98]. To understand the effect of decoherence we shall first obtain the dynamics of the off-diagonal density matrix elements of the qubit. The density matrix is obtained by tracing out of the environment: ρ s (t) = X i,j ρ ij (t)|iihj| = Tr B [ρ(t)]. 39 Let us also introduce the following function: ξ i (Δt) = 2g i ω k 1−e iω k Δt We can use the decay of the off-diagonal elements of the density matrix as a measure of decoherence. For the free evolution, the off-diagonal elements decay as ρ 01 (t) =e −Γ(t,t 0 ) ρ 01 (t 0 ) wheret 0 is some initial time and the decay function is given by Γ(t,t 0 ) = Γ(t−t 0 ) = X i |ξ i (t−t 0 )| 2 2 coth ω i 2T where T is the temperature of the environment as it enters the calculations through tracing out. Note that in the decoupling limit where theg i approach zero, Γ(t,t 0 ) also approaches zero. Likewise we can calculate the decay rate when 2N pulses, each of durationτ are applied so that the whole evolution is generated by applyingU s 1 ,N times as given in Eq.(3.15). In this case the decay factor is given by Γ(N,τ) = X i |ξ i (t−t 0 )| 2 2 coth ω i 2T |1−f i (N,τ)| 2 where f i (N,τ) = 2 ξ i (τ) ξ i (2Nτ) N X n=1 e 2i(n−1)ω i τ . Now taking the simultaneous limits ofτ→ 0 andNτ =T we have: lim τ→0 f i (N,τ) = 1 40 and therefore lim N→∞ Γ(N,T) = 0. Remember that Γ is the measure of decoherence. Therefore, in the limit of infinitely tightly spaced pulses, decoherence simply vanishes regardless of the coupling strengths and the temperature. Realistically the limit is justified for reducing decoherence as long as τω c ≪ 1 whereω c is the highest frequency for the modes of the environment or the ultra-violet cut-off of the model. If these conditions are not met we will obtain a partial decoupling. 3.5 Dynamical Decoupling as Means of Quantum Error Correction By itself, dynamical decoupling can be used to manipulate effective Hamiltonians over time. As we described earlier, it can also help to remove undesired interactions with the environment to ensure that a quantum memory stays intact over time. This is an example of quantum error correction. Nonetheless, as with any other quantum error correction scheme, dynamical decoupling has problems associated with both the theo- retical assumptions and practical implementation issues. In this section we look at some of these problems in more detail. The fundamental shortcoming of dynamical decoupling is that it works in the limit of extremely small switching times. As described in the previous section, except in excep- tional cases it is rather unlikely to come up with a sequence for removing the undesired Hamiltonian terms in an efficient way such that the error scales down exponentially with the number of pulses used. If that were the case, serious restrictions on the efficiency of 41 operations and environment time scales could be removed. Consider a general method of decoupling the following Hamiltonian for the system and the environment H =B 0 ⊗I +B z ⊗Z Consider a single segment of evolution with a lengthτ i enclosed by two pulsesP i and P † i 2 . The propagator for this segment is given by P i exp[−iτ i (B 0 +E)P † i where B 0 = B 0 ⊗I and E = B z ⊗Z. The operators E can be redefined for any other starting Hamiltonian such as the one given in Eq.(3.11). Notice that when many segments are combined, the effective Hamiltonian will generally be a member of the Lie algebra generated byB 0 andE i . Also note that it is possible in principle to get terms that are pure-bath other thanB 0 itself. For example consider the following commutator expression: [B z ⊗Z,[B 0 ⊗I,B z ⊗Z]] = [B z ,[B 0 ,B z ]]⊗I This is slightly unexpected but becomes clear when we are reminded that the environ- ment modes are simply being activated by the application of pulses on the system. Such terms might show up even ifB 0 = 0 but will disappear for the case whereE =B z ⊗Z. Let us now consider a series ofN segments of lengthτ i each. The propagator for the whole process can be obtained by replacing the overall Hamiltonian by a piece-wise constant time-varying HamiltonianH(t) such that H(t) =B 0 +E i forδ i−1 <t<δ i 2 This can be assumed without loss of generality: Even if the interval is enclosed by two pulsesP and Q, we can writeQ =P † R, and useR for the immediate segment coming after. 42 whereδ 0 (= 0) andδ i (= P i j=1 τ j ) fori> 1, denote the times when each segment starts and we setδ n =T . Let us also defineE(t) :=H(t)−B 0 . The propagator for the series is now given by U =T + [e −i R δn 0 H(t)dt ] whereT + [·] denotes the time-ordering and just gives a symbolic solution. To obtainU we can use the interaction picture with respect to the bath generated by the Hamiltonian B 0 . We first factor out a pure-bath term: U = exp(−iδ n B 0 ) ˜ U. Here ˜ U is generated by the interaction picture Hamiltonian ˜ E(t) given by ˜ E(t) =e itB 0 E(t)e −itB 0 =e itB 0 E i e −itB 0 = ˜ E i (t) ifδ i−1 <t<δ i . Before we proceed, we also note that ˜ U is indeed the part of the propagator that we are interested in for decoupling. In fact exact decoupling would occur when ˜ U =I. Let us now obtain a Dyson series expansion for ˜ U: ˜ U =I +(−i) Z δn 0 ˜ E(t 1 )dt 1 +(−i) 2 Z δn 0 Z t 2 0 ˜ E(t 1 ) ˜ E(t 2 )dt 1 dt 2 +··· =I +(−1) n X i=1 Z δ i δ i−1 ˜ E i (t)dt+··· where in the last line the higher multiple integrals can also be written in terms of multiple ordered sums of multiple integrals. If B 0 is not present or is sufficiently smaller than 43 E i , we can approximate ˜ E i (t)≈ E i and the first sum will correspond to the first order Magnus expansion. As we showed in Sec. 3.2, this sum can be canceled, if X i τ i E i = X i τ i P i EP † i ≡ 0. (3.16) Notice the appearance ofτ i in this case as the periods are assumed to be different. Given, the algebraic condition for first order decoupling we have ˜ U =I +(−1) n X i=1 Z δ i δ i−1 ∞ X k=0 [iB 0 t,···[iB 0 t,E i ]] | {z } k times dt+O(T 2 h 2 ) whereh is bounded bykHk and we have replaced terms that are second order or higher inE i with an asymptotic bound. In order to obtain a more accurate bound, let us also introduce positive small quantitiesJ andβ to be bounded bykE i k andkBk respectively. We can thus write U =I +(−i) X i τ i E i +(−i) n X i=1 ∞ X k=1 (i) k δ k+1 i −δ k+1 i−1 k +1 [B 0 ,···[B 0 ,E i ]] | {z } k times +O(T 2 J 2 ) (3.17) Estimating (by upper bounds) the double sum in Eq.(3.17), leaves us with [using Eq.(3.16)] U =I−i X i τ i E i +O(T 2 Jβ +T 2 J 2 ) =I +O(T 2 J(J +β)) We thus have reduced the departure of the propagator from identity from ( P i τ i )E(= O(TJ)) to O[T 2 J(J +β)]. Since the departure of the propagator from identity is a measure of undesired evolution of the qubit state, dynamical decoupling results in higher fidelities for quantum state preservation as long asTJ < 1. This estimate is somewhat heavy handedly pessimistic. It sets a finite limit on the total pulse sequence duration 44 which is given by the inverse of the system-bath coupling strength. In chapter 7 we re-derive the strength of the so called “undecoupled” terms in more detail for periodic pulse sequences and revise the above estimates accordingly. In practice all pulses used in decoupling have an intrinsic width. It can be shown that this will lead to a decoupling error proportional to the pulse widths. Certain pulse sequences are robust with respect to pulse errors such as those due to the widths but in general imperfect pulses represents a real problem for dynamical decoupling if extremely high fidelities are expected. Another similar observation is the issue with the timing errors. We note that such timing errors directly affect the efficiency of dynamical decoupling as Eq.(3.16) is no longer true. This is an important issue and generally becomes more severe when the number of pulses used for dynamical decoupling grows. For exactly this reason, shorter sequences are practically more attractive. While dynamical decoupling works well for preserving the state of a qubit, it cannot be used in conjunction with quantum operations. For example consider the following naive scenario: We need to perform a simpleX rotation on the qubit while pulses are used for decoupling it from the environment. If the operation is performed as an almost instantaneous pulse and we have good control on the timing of the operations, we might place it right after a decoupling cycle where it would perform seamlessly. However this is too optimistic. If we enjoyed the freedom of applying infinitesimally narrow pulses at the right moment, we certainly would not need to wait for decoherence to kick in and would perform all operations in a quantum computer algorithm without delay in an infinitesimal time. This is realistically not the case. The pulses have width and it would be certainly practically convenient to be able to to initiate them at any point in time. We will discuss this problem in greater detail in the chapter 7. 45 3.6 Conclusion In this chapter we reviewed simple foundations of dynamical decoupling. We reviewed the algebraic conditions for it to work for decoupling a quantum system from the envi- ronment. We also gave explicit constructions of dynamical decoupling basic pulse sequences for universal decoupling of a qubit and examined in more detail the error correcting properties of dynamical decoupling. We also commented briefly on the prob- lems associated with dynamical decoupling and hinted at some solutions in the upcom- ing chapters. 46 Chapter 4: Fault-Tolerant QC in the Presence of S.E. 4.1 Introduction The most severe obstacle in the path towards the dramatic speedup offered by future quantum information processing (QIP) devices is decoherence: the process whereby a quantum system becomes irreversibly entangled with an uncontrollable environment (“bath”). This causes information loss and may degrade the operation of a quantum computer to the point where it can be efficiently simulated classically [ABO96]. One can formally model decoherence processes in QIP as being due to operators{S i } acting on the system qubits{i}, that are coupled to bath operatorsB i in a system-bath interaction HamiltonianH SB = P i S i ⊗B i . Two of the main proposals to combat decoherence in QIP are quantum error correction codes (QECCs) and decoherence free subspaces (DFSs). In QECCs multi-qubit states define quantum “code words”, with the special property that they are distinguishable (orthogonal) after the occurrence of errors, i.e., decoherence. Appropriate non-destructive measurements yield an “error syndrome”, which can be used for recovery from the errors [KCL98, Got98]. The DFS approach similarly invokes code words, but it does not require active measurement and recovery, since the encoded states are chosen so as to be immune from decoherence: a state|ψ n i is decoherence-free if S i |ψ n i = c i |ψ n i, where c i is a scalar that does not depend on |ψ n i [ZR97a, LBW99]. This condition, which we refer to as the DFS condition below, assumes that there is a symmetry in the system-bath interaction, such as “collective decoherence”, whereinH SB is qubit-permutation-invariant [ZR97b, DG98]. A number of studies have pointed out the advantages of combining the QECC and DFS approaches 47 [LBW99, LBKW01, ABC + 01]. Of particular relevance is the recent work by Alber et al. [ABC + 01], who introduced a new class of hybrid DFS-QECC codes, known as “detected-jump correcting (DJC) quantum codes”. These codes, which we review below, are particularly useful in the case of spontaneous emission errors: S i =|0i i h1|, where|1i i (|0i i ) is the excited (ground) state of, e.g., an atomi. The DJC codes improve upon earlier work on QECC in the presence of spontaneous emission [PVK97] in that they take advantage of knowing where the emission event occurred (which qubit). This assumes that the mean distance between qubits exceeds the wavelength of the emission. The work by Alber et al. [ABC + 01] left open the question of computation with these codes 1 We show here how to perform universal, fault-tolerant quantum computation (QC) on a class of the DJC hybrid codes, in the presence of spontaneous emission and collective dephasing errors. The latter are errors that arise when the system-bath interaction can be written as H SB = S z ⊗ B z , where S z = P i σ z i , (σ z i is the Pauli σ z matrix acting on the i th qubit), and have been extensively discussed before, both theoretically [DG98, LBW99, BKLW00, KBLW01b, WL02a] and experimentally [KBAW00, KMR + 01]. We show below that in order to accomplish this we need only control the coupling constantsJ z ij and/orJ ij appearing in an anisotropic, exchange-type system Hamiltonian:H S = P J ij (σ x i σ x j +σ y i σ y j )+J z ij σ z i σ z j . The caseJ z ij 6= 0 (J z ij = 0) is known as the XXZ (XY) model. These types of Hamiltonians naturally appear in a number of promising proposals for implementing quantum computers, in which sponta- neous emission, as well as collective dephasing errors, are important sources of decoher- ence. E.g., the quantum Hall [MPG01], quantum dots [IAB + 99], dimer atoms in a solid host [PK02], and atoms in cavities [ZG00] proposals are all of the XY type and suffer 1 Except a brief discussion in [ABC + 01](b), where it was shown that the Heisenberg exchange interac- tion can be used to perform single qutrit operations for one instance of the DJC code. See also [BBBK00]. 48 from photon and phonon emission, while the electrons on helium proposal [PD99] is of the XXZ type and suffers in addition from ripplon emission. The phonon-mediated ion-ion interaction in the Sørensen-Mølmer (SM) scheme for trapped-ion QC [SM99] is equivalent to an XY model, and this proposal too suffers from spontaneous emission of photons and phonons, as well as from collective dephasing [KMR + 01]. Other sources of decoherence can also appear in all proposals, but as shown in [WL02a], using appro- priate pulse sequences generated by the XY Hamiltonian, they can be reduced to the collective dephasing type. The idea of universal QC using the XY or XXZ interaction has been considered before, starting with [IAB + 99], where the XY interaction had to be supplemented with arbitrary single-qubit operations. In [LW02] it was shown how to perform universal QC using the XY interaction supplemented with static single-qubit energy terms (e.g., a Zeeman splitting) and an encoding into 2 qubits; in [KBDW01] universal gate sequences were given for the XY interaction alone, using an encoding into 3 qubits; and in [LW02, WL02c] universal gate sequences using the XXZ interac- tion were found for encodings into 2 or more qubits. Here we use an encoding into 4 or more qubits, that has the additional, significant advantage of offering protection (using a QECC) against spontaneous emission errors. 4.2 Detected Jump-Corrected Codes In the DJC codes method, the Markovian quantum trajectories approach [PK98] is used to describe decoherence. This approach is equivalent to the Lindblad semigroup master equation [Lin76]. The evolution is decomposed into two parts: a conditional non-Hermitian HamiltonianH C , interrupted at random times by application of random errors. For errors such as spontaneous emission, where the jump can be detected by observation of the emission, the quantum trajectories approach also provides a way to 49 combine QECCs and DFSs [ABC + 01]. The DFS takes care of the conditional evo- lution, whereas the QECC deals with the random jumps that couple DFS states with states outside of the DFS. Formally, the conditional Hamiltonian is given by [PK98]: H C =H S − i 2 P i κ i S † i S i , whereκ i are (in our case) the spontaneous emission rates. The DFS in the quantum jump approach is given by the eigenspace of the collective operator C≡ P i κ i S † i S i . The symmetry that leads to the DFS condition being satisfied isκ i ≡κ. Forn qubits and spontaneous emission errors we then haveC =κ P n i=1 |1i i h1|, and the DFS with maximal dimension is comprised of (“computational”) basis states with n 2 1’s and n 2 0’s. It has dimension n n/2 and eigenvaluen/2 underC. From here on we work exclusively with this DFS. Consider such a DFS encoding into n = 4 qubits (n = 2 qubits already yields a logical qubit, but n = 4 is the smallest such generalizable example, in the sense of the multi-encoded-qubit scheme discussed below). It protects against the conditional evolution, so what remains is to protect against the jumps. As shown in [ABC + 01], if we assume knowledge of the position of errors by observing the emission, then one can use states in this DFS in order to construct a QECC that encodes one logical qubit: |0i L = |1010i+|0101i √ 2 |1i L = ±(|0110i+|1001i) √ 2 , (4.1) where the choice of sign is + (−) ifJ 12 < 0 (> 0), as will be clarified below, and for simplicity we assume from here on thatJ z ij ≥ 0. The general QECC condition [KCL98] that keeps the errors from scrambling the code words|ψ n i takes the following form, provided we know which of the errors indexed byi has occurred: hψ m |S † i S i |ψ n i = Λ i δ mn , (4.2) 50 where Λ i is a number independent of the code words [ABC + 01]. This is easily veri- fied for the code in Eq. (4.1). Therefore this code offers complete protection against the detected-jump spontaneous emission process. Note that in addition to the states in Eq. (4.1) the state|2 L i = (|0011i+|1100i)/ √ 2 also satisfies the QECC condition (4.2), and is inside the DFS. Alber et al. [ABC + 01] gave a combinatorial design-theory method for generalizing the code of Eq. (4.1). We now describe a class of these codes that come with natural encoded qubit operations, that allow for universal, scalable, and fault tolerant QC. Our protocol is as follows: Computation is performed during the conditional evolution peri- ods, while the system is in a DFS. If a jump is detected, it must first be corrected (as in QECC), before computation can resume. We note that the performance of DJC codes in the presence of imperfections such as detection inefficiencies, unequal decay rates κ i , and time delay between error detection and recovery operations, has been analyzed in [ABC + 01](c), with favorable conclusions. 4.2.1 Example: Encoded Logic for 4-Qubit DJC Code In order to perform universal QC we first identify a set of generators of all encoded single qubit transformations. As is well known, arbitrary single qubit transformations can be generated from Hamiltonians via time evolution, using a standard Euler angle construction: e −iωn·σ = e −iβσ z e −iθσ x e −iασ z . This is a rotation by angle ω about the axis n, given in terms of three successive rotations about the z and x axes. Let us now suppose that we have at our disposal a controllable XXZ Hamiltonian, as defined above. This gives us the ability to switch on/off, separately, the Hamiltonian termsT ij ≡ 1 2 (X i X j +Y i Y j ) andZ i Z j , whereX i ≡σ x i , etc. These operators preserve the number of 0’s and 1’s [WL02b, WL02c]. Since this implies that they cannot take states outside of the DFS, it follows that they are naturally fault tolerant [BKLW00, KBLW01b]. Now 51 suppose that we turn|J 12 | (J z 13 ) on for a timet such that|J 12 |t/~ = θ (J z 13 t/~ = θ). Then: e −iθT 12 |ǫi L = cosθ|ǫi L −isinθ|¯ ǫi L , e −iθZ 1 Z 3 |0i L = e −iθ |0i L , e iθZ 1 Z 3 |1i L =e iθ |1i L (4.3) whereǫ = 0 or 1, and ¯ ǫ = (ǫ + 1) mod 2 =NOT(ǫ). These equations show thatT 12 and Z 1 Z 3 have precisely the action of single qubit σ x and σ z transformations, on the code states in Eq. (4.1), and that this code space is perfectly preserved under T 12 and Z 1 Z 3 . We denote logicalX (Z) operations on thei th encoded qubit by ¯ X i ( ¯ Z i ). Thus ¯ X 1 = T 12 and ¯ Z 1 = Z 1 Z 3 and we have the ability to generate arbitrary encoded single qubit transformations in the XXZ model. This is particularly relevant for the electrons on helium proposal [PD99]. However, in many QC proposals of interest it is either inconvenient to separately control J z ij , or such exchange interactions vanish [MPG01, IAB + 99, PK02, ZG00, SM99]. We must then resort to controlling only the XY term. Now, as shown in [LW02], using the “encoded recoupling” method, it is possible to generateZ 2i−1 Z 2j−1 operations with arbitrary i,j as long as one can control an XY Hamiltonian. Define C φ A ◦B≡ exp(−iφA)Bexp(iφA), then 2 [WL02c, LW02]: 2C π/2 1 2 T 2i,2j−1 ◦(C π/2 T 2i−1,2i ◦T 2i−1,2j−1 ) =Z 2i−1 Z 2j−1 . (4.4) The procedure given in Eq. (4.4) is a 5-step implementation of the Ising interaction Z 2i−1 Z 2j−1 . Fori = 1,j = 2 this yields ¯ Z 1 , and we have all we need for encoded single qubit transformations in the XY model. 2 Here we have neglected a term−Z 2i−1 Z 2i /2 since it is constant on the code space, i.e., it has equal action on|0 2i−1 1 2i i,|1 2i−1 0 2i i. 52 The one apparent disadvantage of the procedure in Eq. (4.4) is that in 1D it requires next-nearest neighbor interactions (this is inevitable with an XY interaction in 1D [WL02c]), but note that these interactions are still nearest neighbor on a 2D triangu- lar qubit lattice. Let us also note that application of T 2i−1,2j−1 [as arises in Eq. (4.4); e.g.,T 13 is needed for the implementation of ¯ Z 1 ], maps the code state|1i L to a superpo- sition of|1i L and|2i L . While|2i L is not part of our encoded qubit it is part of the DJC code [it is in the DFS and satisfies the QECC condition (4.2)], so that the fault tolerance of our procedure is not violated. 4.2.2 Generalization: DJC Code Encoding Several Qubits We now introduce an encoding that generalizes the code in Eq. (4.1) to arbitrary numbers of encoded qubits. Let | ˜ 0i i ≡|0 2i−1 1 2i i, | ˜ 1i i ≡−sign(J 2i−1,2i )|1 2i−1 0 2i i. We then define a code as follows: |ǫ L i 1 ⊗···⊗|ǫ L i n−1 = |˜ ǫi 1 ···|˜ ǫi n−1 | ˜ 0i n +conj. √ 2 , (4.5) whereǫ = 0 or1, and “conj.” denotes the bitwise NOT of the first ket. The rate (number of encoded per physical qubits) of this class of codes isr = n−1 2n . As in the case of a single encoded qubit, Eq. (4.3), the generators of encodedσ x andσ z transformations are ¯ X i = 1 2 (X 2i−1 X 2i +Y 2i−1 Y 2i ), ¯ Z i =Z 2i−1 Z 2n−1 , (4.6) 53 as is easily verified by checking their action on| ˜ 0i i ,| ˜ 1i i . Using the Euler angle formula we may construct arbitrary encoded single-qubit operations from ¯ X i and ¯ Z i , using oper- ations from within the XY or XXZ models only. The fact that we can apply such single encoded qubit operations on the code in Eq. (4.5) shows that this code is equipped with a (formal) tensor product structure, and allows for scalable QC. At this point we are ready to show how to implement a controlled-phase (CP) gate, CP|x,yi = (−1) xy |x,yi (wherex,y are 0 or 1), which together with arbitrary single- qubit operations is universal for QC [NC00]. As is well known, the CP gate is generated by an Ising interactionZ⊗Z [NC00]. Thus to generate a CP gate between encoded qubitsi,j we must consider ¯ Z i ¯ Z j = (Z 2i−1 Z 2n−1 )(Z 2j−1 Z 2n−1 ) = Z 2i−1 Z 2j−1 . In the XXZ model such a two-body Ising interaction is directly controllable. In the XY model, we can generate it using the 5-step procedure of Eq. (4.4). Furthermore, since a CP gate can be used to construct a SWAP gate [NC00], we need only use at most next nearest- neighbor interactions (in 1D; nearest neighbor in 2D) in order to couple arbitrary pairs of encoded qubits. Finally, we stress that the combination of Eqs. (4.3),(4.4),(4.6), and the result above for ¯ Z i ¯ Z j , is an explicit prescription for constructing arbitrary quantum circuits in terms of the XY and/or XXZ interactions. 4.3 Fault Tolerant Measurement and Recovery An inherent assumption in the DJC codes method is that it is possible to observe which of the physical qubits underwent spontaneous emission [ABC + 01]. This is a manifestly fault-tolerant measurement [Got98], in the sense that observing an error on a specific qubit cannot cause errors to multiply. Now consider the recovery from spontaneous emission errors. When a spontaneous emission error occurs on qubiti, each code word given by Eq. (5) collapses to the component in which at the error locationi there used 54 to be a|1i before the error, with|1i changed to|0i after the error. Let X i and W i respectively denote the bit flip and Hadamard gates on qubit i, and let CX i else denote a collective controlled-NOT (CNOT) gate from the control qubit i to all other qubits. The recovery operation after spontaneous emission from qubiti is observed is given by U rec = X i (CX i else )W i . Now, in order to perform these recovery operations we must assume that in addition to an XY or XXZ Hamiltonian we have the ability to control single-qubit energies (i.e., control terms of the form ω i Z i ) and perform a Hadamard [W = 1 √ 2 1 1 1 −1 ] gate, which is certainly reasonable in optics-based QC pro- posals [IAB + 99, PK02, ZG00, SM99] (where such single-qubit operations are executed through the application of laser pulses). This requirement is harder to satisfy in solid- state QC proposals that use gate voltages for single qubit operations [MPG01, PD99], but is not unreasonable. Note that the assumption that we can perform single qubit operations is made only to enable recovery from spontaneous emission errors. It is needed since the XY and XXZ Hamiltonians preserve the number of0’s and1’s in each codeword, while spontaneous emission lowers the number of 1’s. Now, with the extra assumption we are able to constructCX 1 and CX 2 . This is easily done by by combining CNOT gates:CX i else = Q j6=i CX i j , whereCX i j is the CNOT gate with qubiti as control and qubitj as target. This formula forCX i else is not necessarily optimal – we expect sim- pler implementations to exist depending on the range of theXY orXXZ interactions, possibly combined with single qubit terms. This procedure can be made fault tolerant by concatenating blocks of qubits encoded according to Eq. (4.5), and using the standard arguments of concatenated fault tolerant quantum error correction (e.g., [Got00]). 55 4.3.1 State Preparation and Read-Out Finally, we must also show that our encoded states can be reliably prepared and read out. A general preparation technique is cooling to the ground state of a Hamiltonian. For this procedure to work there should be an energy gapΔ between the code subspace and other states. Diagonalization of the XY HamiltonianJ ij T ij = J ij 2 (X i X j +Y i Y j ) in the subspace of qubitsi,j yields, depending on whetherJ ij > 0 or< 0, either the singlet state|si ij = 1 √ 2 (|0 i 1 j i−|1 i 0 j i) or the triplet state|ti ij = 1 √ 2 (|0 i 1 j i +|1 i 0 j i), as the ground state, with energy−|J ij |. Consider the case of a single encoded qubit and assumeJ ij > 0: the ground state of the XY HamiltonianJ 12 T 12 +J 34 T 34 is|s 12 i⊗|s 34 i, which is exactly 1 √ 2 (|0i L +|1i L ), in terms of the code states of Eq. (4.1) with the choice of “−” for|1i L . I.e., cooling prepares a state that is in the code subspace, and application of the encoded logical operations derived above can rotate this initial state to any other desired encoded state. To prepare a state in the code subspace of 2n physical qubits we turn on the pairwise XY Hamiltonian P n i=1 J 2i−1,2i T 2i−1,2i , keep the temperature below Δ, and wait. The resulting ground state is⊗ n i=1 |si 2i−1,2i , and a simple calculation shows that this state is in the code space: ⊗ n i=1 |si 2i−1,2i =⊗ n−1 j=1 (|0 L i j +|1 L i j )/ √ 2. Identical conclusions hold when assumingJ ij < 0, with|ti 2i−1,2i replacing|si 2i−1,2i . Thus cooling always prepares a state in the code subspace and can serve as an ini- tialization procedure for our protocol. Measurement can be be done analogously, i.e., by using the energy difference to distinguish a singlet from a triplet state on pairs of qubits encoding a logical qubit [WL02b]. Thus, to distinguish|0 L i j from|1 L i j we first apply an encoded Hadamard gate to physical qubits 2j− 1,2j, mapping|0 L i j → 56 (|0 L i j +|1 L i j )/ √ 2 and|1 L i j → (|0 L i j −|1 L i j )/ √ 2, which by the preparation argu- ments above correspond to singlet and triplet states, depending on sign(J 2j−1,2j ). 4.4 Conclusions We have studied a class of “detected-jump” codes that is capable of avoiding collective dephasing errors and correcting spontaneous emission errors on a single qubit. These codes are a hybrid of decoherence-free subspaces and active quantum error correction, and use2n qubits to encoden−1. We have shown how to quantum compute universally and fault tolerantly on this class of codes, using Hamiltonians (XY- and XXZ-type) that are directly relevant to a number of promising solid-state and quantum-optical proposals for quantum computer implementations [MPG01, IAB + 99, PK02, ZG00, PD99, SM99]. 57 Chapter 5: Efficient Error Correction and QC in the presence Spontaneous Emission 5.1 Introduction Decoherence [GJK + 96] remains the most daunting obstacle to the realization of quan- tum information processing, coherent control, and other applications requiring a high degree of quantum coherence. As quantum computation (QC) moves into the experi- mental realm it becomes increasingly important to design methods for overcoming this main obstacle to realization, that are tailored to particular systems and the resulting errors that afflict them. Here we show how to perform universal, fault-tolerant QC in the presence of decoherence due to spontaneous emission (SE). Since SE is a conse- quence of the inevitable coupling to the vacuum field [SZ97], it cannot be “engineered away” and must eventually be dealt with, in all QC proposals. Several methods have been designed to this end, that may roughly be classified as “hardware” and “software”: In the former category are proposals to construct quantum computers in materials where SE is strongly suppressed, e.g., placing atomic qubits in a photonic band-gap structure [VRJ01]. In the latter category are various error correction, avoidance, and suppres- sion methods [MZ96, CPZ96, PVK97, ABC + 01, KL02, BBBK00, ASW01]. With the exception of the2π pulsing method of [ASW01], a unifying theme of these methods is to place the system under continuous observation. It is then well known that the Markovian quantum master equation can be unravelled into a set of quantum trajectories, consisting of a conditional evolution (governed by a non-Hermitian conditional HamiltonianH c , 58 defined below), randomly interrupted by quantum jumps (wavefunction collapse) into different observed decay channels [DCM92, GZ00, Car93, PK98]. The time evolution conditional to a given set of time-ordered observations is called “a posteriori dynamics” [BB91], and is not Markovian. The continuous observation can lead to a Zeno-effect type suppression of decoherence, a fact that was exploited in [BBBK00], in conjunction with an encoding into a decoherence-free subspace (DFS) [ZR97b, BKLW00], in order to resist SE. Quantum error correcting codes (QECCs) can correct both the conditional evolution and the jumps [CPZ96], but more efficient constructions are possible when one considers subspaces of the full system’s Hilbert spaces that are invariant under the conditional evolution. It is then necessary to correct only the errors arising due to the quantum jumps [MZ96, CPZ96, PVK97, ABC + 01, KL02]. The first proposal along these lines, [MZ96], did not consider QC. A simple, but non fault-tolerant QC scheme, encoding a logical qubits into two physical qubits (four atomic levels), tailored to SE of phonons in trapped-ion QC, was subsequently presented in [CPZ96]. A QECC correct- ing one arbitrary single-qubit error and invariant underH c was given in [PVK97], using an encoding of one logical qubits into 8 physical qubits. When one makes the assump- tion that the qubit undergoing the quantum jump can be identified (“detected-jump”), a more efficient encoding is possible. A family of such “detected-jump codes” (DJC) was first developed in [ABC + 01], using a DFS to construct a subspace invariant underH c . In [KL02] we showed how to perform fault-tolerant universal QC on a subclass of such codes encodingn−1 logical qubits into2n physical qubits. Here we present a new method for reducing and correcting SE errors. Rather than constructing a code subspace invariant underH c , we dynamically eliminateH c by apply- ing dynamical decoupling (or “bang-bang”, BB) pulses [VL98, SL04]. We then con- struct a QECC that deals with the remaining jump errors, under the detected-jump assumption. The advantage of this method compared to the previous methods using 59 encoding is that it is significantly more economical in qubit and pulse timing resources: It uses a QECC in whichn logical qubits are encoded into onlyn+1 physical qubits; and, while in [ASW01] the pulse interval has to satisfy the standard BB condition of being shorter than the inverse of the bath high-frequency cutoff [VL98, SL04], in our case the requirement is that the pulses are faster than the average time between photon emission events, which can be orders of magnitude longer. Furthermore, our method is fully compatible with universal QC using Hamiltonians that are naturally available in a large variety of QC proposals [LW03b], so unlike [VRJ01] does not rely on one specific architecture. The idea of using a hybrid BB-encoding approach to suppress decoherence was first proposed in [BL02], where it was pointed out that BB is fully compatible with encod- ing into a QECC or DFS. In particular it was observed there that one could use BB to suppress phase-flip errors, thus leaving the QECC with the need only to correct bit-flip errors. However, no method specifically tailored for SE errors was given. An experi- mental NMR implementation of a hybrid BB-QECC was presented in [BPF + 02], where decoupling was used to remove coherent scalar coupling between protons (environment) and carbon qubits, together with QECC used to further correct for fast relaxation due to dipolar interactions modulated by random molecular motion. Clearly, correcting for SE errors is only a part of a general procedure for offsetting decoherence, as additional decoherence sources will inevitably be present in any QC implementation. The methods we present here therefore will have to become part of this more general procedure, either as a first level of defence (in the case that SE is dominant), or at higher levels in a concatenated QECC scheme [Got98], after other, more dominant errors have been accounted for. The importance of the results presented here lies in the fact that SE is always present and therefore can never be ignored. A code 60 that is optimized with respect to this type of error can potentially offer flexibility and significant savings in resources and overhead. The structure of this chapter is the following. In Section 5.2 we show how the conditional evolution during SE can be eliminated using a sequence of simple, global BB decoupling pulses. In Section 5.3 we construct a simple and economical QECC that corrects for the remaining quantum jump errors. We address fault tolerance and various imperfections in Section 5.4. We then show how to quantum compute in a universal and fault tolerant manner over our QECC, using a variety of model Hamiltonians pertinent to a wide class of promising quantum computing proposals. We conclude in Section 5.6. 5.2 Eliminating the Conditional Evolution using BB pulses ConsiderN qubits that can each undergo SE, under the detected-jump assumption. This localizability of the SE events implies that the mean distance between qubits exceeds the wavelength of the emission. Note that this optical distinguishability between qubits does not limit our ability to couple the qubits via non-optical interactions, of the type we consider in Section 5.5 below. The ground and excited states of each qubit are denoted by|0i and|1i respec- tively. Let σ − i = |0i i h1| denote the SE error generator acting on the ith qubit and let κ i denote the corresponding error rate. We use the quantum trajectories approach [DCM92, GZ00, Car93, PK98] to describe the dynamics of the decohering system. The evolution is decomposed into two parts: a conditional non-Hermitian HamiltonianH c , interrupted at random times by occurrence of random jumps, each corresponding to an observation of decay channels in a quantum optical setting. For errors such as SE, 61 where the jump can be detected by observation of the emission, the quantum trajecto- ries approach also provides us with a way to combine QECCs and BB, in analogy to the way this was done for QECC and DFS in [ABC + 01, KL02]. The BB pulses take care of the conditional evolution, whereas the QECC deals with the random jumps. The conditional Hamiltonian is given in the SE case by [DCM92, GZ00, Car93, PK98]: H c =− i 2 P N i=1 κ i σ + i σ − i , whereσ + i = σ − i † . In [KL02] we assumed that the environ- ment effectively does not distinguish among the qubits that undergo SEs (κ i = κ) and the conditional Hamiltonian would then become:− i 2 κ P i |1i i h1|. This assumption is not necessary in the current work. From here on operators X i ,Y i ,Z i refer to the cor- responding Pauli matrices acting on the ith qubit, and I denotes the identity matrix. Now suppose that we apply a collectiveX≡⊗ N j=1 X j pulse to the system, at intervals T c /2≪ 1/γ, whereγ is the SE rate. 1 Under this condition, and usingX i σ − i X i = σ + i we can write the evolution after a fullT c period as: U = exp(−i T c 2 H c )X exp(−i T c 2 H c )X = exp(− T c 4 X i κ i |1i i h1|)exp(− T c 4 X i κ i |0i i h0|) = exp(− T c 4 X i κ i )I, where I is the identity operator. Therefore the decohering effect of the conditional Hamiltonian (that distinguishes states with different numbers of 1’s) is removed and replaced by an overall shrinking norm. When the jumps are included in the dynamics, the state must be renormalized [DCM92, GZ00, Car93, PK98], so this shrinking dis- appears. Note that we have not eliminated Markovian decoherence using BB pulses, 1 We re-emphasize that this time can be much longer than the bath correlation time typically assumed to set the time-scale for BB operations (for an exception see [SL04]). 62 since we have considered only a single trajectory. In fact, a comparison of the coher- enceC = Tr(ρ 2 ) (where ρ is the qubit density matrix) shows that if the results are ensemble-averaged over the a posteriori dynamics (recovering the Markovian master equation), and the jump errors are not corrected, then there is no advantage in using a BB sequence. More specifically, when comparingC for the (1) free and (2) every T c /2X-pulsed evolution of a single qubit undergoing SE with rateγ, we find: C 1 = 1−γT c (β 2 )+O(γ 2 ) C 2 = 1−γT c (α 4 +β 4 )+O(γ 2 ) where the initial qubit state|ψi =α|0i+β|1i is normalized:α 2 +β 2 = 1. Averaging over a random sample of initial states chosen from a uniform distribution (withα andβ subject to normalization), we havehC 1 i =hC 2 i , so as expected for purely Markovian dynamics, there is no improvement after using just BB pulses. 5.3 Correcting Spontaneous Emission Jumps with a QECC We now introduce a very simple QECC that corrects the remaining part of the decoher- ence process, the random jumps. Since the error correction process by necessity takes place during the conditional evolution (the jump is instantaneous and the QECC takes time), we must ensure that the QECC keeps its error correcting properties under the con- ditional Hamiltonian and BB pulses. A minimal example of such a “decoupled-detected 63 jump corrected” code is given by a subspaces of theN =n+1 qubit Hilbert spaceC n , spanned by the codewords |xi L ≡|x 1 ,...,x n i L = |x 1 ,...,x n ,0i+|x 1 ,...,x n ,1i √ 2 . (5.1) where{x i } is the binary representation of the n-qubit state|xi and x is an inverted x in which 1 and 0 are interchanged. For example, for n = 2, the codeC 2 is (up to normalization by factors of √ 2): |00i L = |000i+|111i |01i L =|010i+|101i |10i L = |100i+|011i |11i L =|110i+|001i. ThatC n is a QECC against the jump errors follows from the fact that a spontaneous emission error at a given qubit position i eliminates the component of the codeword with 0 in that position, which by construction results in a surjective mapping between the original codewords and the resulting states that are orthogonal to each other. More specifically, the sufficient condition that a QECC must satisfy is that orthogonal code- words must be mapped to orthogonal states after the occurrence of errors, so that the errors can be resolved and undone [KL97a]. Recall that here we are assuming that we know the location of the error, after recording the position of the SE. Hence we need only compare orthogonal codewords after the action of an error in a known locationi: L hy|σ + i σ − i |xi L = δ xy /2 if y i =x i 0 if y i 6=x j 64 The second line is explained in the following way: If x i 6= y i , then either x i or y i is 0. Suppose y i is zero, then the component of the|yi L codeword that remains after the SE is σ − i |y,1i and the component of|xi L that remains is σ − i |x,0i which are always orthogonal to each other. Thus the QECC condition is satisfied. To see that recovery from the errors is indeed possible, we describe a simple (non fault- tolerant) scheme. The recovery operation after the detection of an error in position i, is given by U = Q n6=i X n Q n6=i CNOT in H i , where H i is a Hadamard operation on qubit i and CNOT ij is a CNOT gate with qubit i (j) as the control (target), i.e., CNOT ij |x i ,x j i =|x i ,x j ⊕x i i. That this unitary operation fixes the SE error can be seen as follows by considering the transformation of the codewords after the error and recovery for the two cases: x i = 1 andx i = 0. Ifx i = 1 (x i = 0) then the codeword after the error becomes|x 1 ...0...x n+1 i (|x 1 ...0...x n+1 i). It is easy to verify that applyingU to this state indeed returns the original logical codeword|xi L . To illustrate this we discuss in detail the case ofC 2 . The conditional evolution, under the collective BB pulse X = X 1 X 2 X 3 , has the sole effect of shrinking the norm of all codewords in Eq. (5.2) equally. Thus the BB-modified conditional evolution does not change the orthogonality of the codewords. Now suppose SE from the first qubit has been observed. Then an arbitrary encoded state|ψi L = a|00i L +b|01i L +c|10i L +d|11i L changes into|ψ err i =a|011i+b|001i+c|000i+d|010i. To reverse the error we use the uni- tary operator U = X 2 X 3 CNOT 12 CNOT 13 H 1 . The erred state is then transformed to U|ψ err i = (1/ √ 2)(a|00i L +b|01i L +c|10i L +d|11i L ) =|ψi L . 65 5.4 Fault Tolerant Preparation, Measurement, and Recovery So far we have assumed perfect error detection, recovery, and gates. Of course, in reality these assumptions must be relaxed. Here we discuss the implications of imperfections. In general, a procedure is said to be fault tolerant if the occurrence of an error in one location does not lead (via the applied procedure) to the catastrophic multiplication of errors in other locations [Got98], an event that the code cannot correct. Let us first discuss preparation of the encoded qubits. The code word|0i L is pre- pared by cooling all physical qubits in their ground state (|0i), which can be done, e.g., via cooling, a strong polarizing field, or repeated strong measurements of all qubits, fol- lowed by a Hadamard on then+1-th qubit, and a collective CNOT from then+ 1-th qubit to all the other physical qubits 1 to n. Once|0i L has been prepared computa- tion proceeds using the fault-tolerant logical operations given in Section 5.5 below, so any other state can be reached fault-tolerantly. Readout is also simple: First apply the same collective CNOT and then measure the firstn physical qubits. The measurement procedure must be tailored to the specific implementation, but our only assumption is that single-qubit measurements are possible, and that these measurements do not cou- ple qubits. The measurement procedure is then fault-tolerant. If means of applying fault-tolerant CNOT are available then both preparation and readout are fault tolerant. Next consider recovery. The codeC n is an especially simple example of CSS sta- bilizer codes [Got96], with stabilizer generated by the single element⊗ n+1 j=1 X j . It is well known how in general to perform fault-tolerant recovery from this class of codes [Got98] (see also [LBKW01]), so we will not repeat the general construction here, which involves preparing and measuring encoded ancilla qubits (note that this typically dou- bles the number of physical qubits required, even before concatenation). 66 Finally consider detection of SE events. Above we assumed that it is possible to perfectly identify the position of a qubit that underwent SE. Note that this measurement is in itself fault-tolerant, in the sense that observing an SE event on a specific qubit cannot cause errors to multiply. Clearly, detecting which qubit emitted a photon is very demanding experimentally, and can in practice only be done to some finite precision (though there is no fundamental limit, provided the distance between the qubits is larger than the wavelengths of emitted photons), and at the cost of introducing potentially cumbersome detection apparati. The same difficulty is shared by previous “detected- jump” schemes [ABC + 01, KL02, BBBK00]. More specifically, in reality there is a finite probability that the emitted photon will (i) Go undetected; (ii) Be attributed to the wrong atom (misidentification). The latter possibility applies also to other qubit measurements; (iii) In case (ii), there is the addi- tional possibility of an error by applying the correction step to the wrong qubit. In general, fault-tolerance results again come to the rescue: provided that the probability of an undetected photon and/or misidentification can be kept sufficiently low, concate- nated QECC guarantees that the procedure will remain robust [Got98, KLZ98a, Ste03]. However, several additional comments are in order. First, we note that the perfor- mance of DJC codes in the presence of imperfections such as detection inefficiencies and time delay between error detection and recovery operations, has been analyzed in [ABC + 03], with favorable conclusions regarding fidelity degradation. We expect similar conclusions for our current method. Second, unlike the case of DJC codes [ABC + 01, KL02, ABC + 03], we do not require equal error ratesκ i . Hence our qubits need not be identical: qubits can be tuned to different cavity modes and therefore emit distinguishable photons. This should enable a significant reduction in the misidentifica- tion error rate. Third, we can take advantage of the fact that after any SE event, each codewords is transformed to a state which is orthogonal to all original codewords. Thus, 67 we can perform an extra measurement (of the stabilizer X 1 ...X n+1 ) that determines whether an error has occurred at all. This is done by adding one more ancilla qubita (initalized into the|0i state) that functions as a syndrome-measurement bit. Now repeat- edly apply Q n+1 i=1 H i CNOT i a H i and periodically check the qubita to see if it has changed to|1i. In such a case if the position of the SE is undetected the computation has to be restarted and the ancilla qubit has to be reset; otherwise the recovery procedure may still be applied. Fault tolerant procedures are known for syndrome measurement as well [Got98, KLZ98a, Ste03]. There is also the possibility of SE on the ancilla qubit, but this can only be caused by two successive spontaneous emissions (one on the code qubits and one on the parity qubit), which has a lower probabilityp 2 , wherep is the probability of two SE errors occurring during the same observation period, before the first one is detected. Note that the parity bit also helps preventing the error of applying a correction step without an SE event having taken place. 5.5 Fault Tolerant Computation So far we have described a fault tolerant implementation of quantum memory in the presence of SE. Now we describe how to perform universal quantum computation fault tolerantly on our code. Formally, one can use the formalism of normalizer group oper- ations, together with a non-normalizer element such as theπ/8 or Toffoli gate [Got98]. However, here we are interested in how to carry this out from the perspective of the nat- urally available interactions for a given physical system. Similar questions have been raised recently under the heading of “encoded universality”: the ability to quantum com- pute universally directly in terms of a given and limited set of Hamiltonians, possibly by use of encoded qubits (see, e.g., [LW02, KBDW01] and references therein). The problem then translates into finding sets of Hamiltonians that generate a universal set of 68 logic gates on the code. There are many options, depending on the set of naturally avail- able interactions. Nevertheless, all encoded universality constructions rely on showing that the well-known universal set of all single-qubit operations and a single entangling gate can be generated, on the encoded qubits. Underlying this are a few elementary identities. Let us define conjugation as: A B,ϕ →≡e −iϕB Ae iϕB . Then for any threesu(2) generators{J x ,J y ,J z }: J x Jz,ϕ −→J x cosϕ+J y sinϕ. (5.2) This can be lifted to unitary evolutions using Ue A U † =e UAU † , (5.3) valid for any unitaryU. Hence where convenient we present our arguments in terms of transformed Hamiltonians. Eqs. (5.2),(5.3) show that given two su(2) generators one can generate a unitary evolution about any axis. This is also the basis for the well- known Euler angle construction, used to argue that all single qubit operations can be generated fromσ x andσ z Hamiltonians: an arbitrary rotation by an angleω around the unit vectorn is given by three successive rotations around thez andx axes: e −iωn·σ = e −iβσ z e −iθσ x e −iασ z [NC00]. Eqs. (5.2),(5.3) show that this is true also for “encoded Hamiltonians”, which we define as Hamiltonians that have the same effect on encoded 69 states as do regular Hamiltonians on “bare” (unencoded) qubits. We denote encoded Hamiltonians by a bar. For the code states (5.1) these are given by: ¯ Z i =Z i Z n+1 , ¯ X i =X i , (5.4) and generate su(2). Therefore controllable Z i Z n+1 and X i Hamiltonians suffice to generate arbitrary single encoded-qubit transformations. To complete the set of uni- versal logic gates we require some non-trivial (entangling) gate [DNBT02b], such as controlled-phase: CP = diag(1,1,1,−1), in the computational basis. CP can be generated from the Ising interaction Z i Z j as follows: CP ij = e −i π 4 (Z i +Z j ) e −i 3π 4 Z i Z j . An entangling gate can also be generated from the Hamiltonian X i X j [one way to see this is to note that it can be rotated to Z i Z j using Y i and Y j in Eqs. (5.2),(5.3)]. Encoded CP can thus be generated from the encoded Hamiltonians ¯ Z i ¯ Z j = Z i Z j or ¯ X i ¯ X j = X i X j . Note that in both cases the physical interaction is also the correspond- ing encoded Hamiltonian. Thus the sets of controllable Hamiltonians{X i ,Z i Z j } or {X i ,Z i Z n+1 ,X i X j } suffice for encoded universal QC on our code. Importantly, these sets moreover exhibit “natural fault-tolerance” [BKLW00]: they preserve the code sub- space and hence will not expose the code to uncorrectable errors. An accurracy error in the time over which the Hamiltonians are turned on can be dealt with using the tech- nique of concatenated QECCs [Got98]. The question now is how to generate these sets, or an equivalent fault tolerant universal set, from the given, naturally available interac- tions. We will consider here the most important cases, extending methods developed in [LW02, WL02b, WL02c]. Note that the decoupling procedure requires us to assume in any case the ability to apply a global (non-selective)X pulse, and the recovery proce- dure requires the ability to apply a CNOT gate. We comment on these requirements in each of the cases we next analyze. 70 5.5.1 Case 1: Natural{Z i ,X i ,X i X j } The Hamiltonians Z i ,X i X j are naturally available, e.g., in the Sørensen-Mølmer scheme for trapped-ion QC [SM00], and in proposals using Josephson charge qubits [YTN02]. However these do not form a universal set for our code and hence we must assume the ability to turn on spin-selectiveX i Hamiltonians. This will also be sufficient for producing the encodedZ i Z j coupling. 5.5.2 Case 2:{Z i ,X i ,XY Model} Members of a relatively large class of promising QC proposals (quantum dots [IAB + 99, QJ99], atoms in a cavity [ZG00], quantum Hall qubits [MPG01], subradiant dimers in a solid host [PK02], capacitively coupled superconducting qubits [V AC + 02]) have a con- trollable Hamiltonian of the XY form:H XY ij =J ij (X i X j +Y i Y j ). LetT ij ≡ 1 2 (X i X j + Y i Y j ). Then|01i T 12 ←→|10i, and annihilates|00i,|11i. I.e., the XY Hamiltonian cannot change the total number of 1’s in a computational basis state [WL02b, WL02c]. There- fore by itself, or even if supplemented withZ i Hamiltonians, it cannot generatesu(2) on our code. This conclusion is unchanged even if one considers conjugatingH XY ij with H XY ik : then{T 12 ,T 13 ,−Z 1 Z 2 T 23 } close assu(2) , and still preserve the total number of 1’s. Therefore in this case we must assume the ability to tuneX i Hamiltonians as well, to obtain universality. Now, X i X j (T jk )X i X j = 1 2 (X j X k −Y j Y k ), which commutes with T jk . Therefore, using Eq. (5.3), we have X i X j e −iθT jk X i X j e −iθT jk = e −iθX j X k , showing that the HamiltonianX j X k can be generated in four steps. At this point we have the same set of Hamiltonians as in Case 1, so that universal encoded computation is possible, as are the globalX pulse and recovery. 71 5.5.3 Case 3:{Z i ,X i , Heisenberg interaction} Next we consider the case of single-qubit X-Z control together with the Heisenberg interaction H Heis ij = J ij (X i X j + Y i Y j + Z i Z j ). Heisenberg interactions prevail in QC proposals using spin-coupled quantum dots [LD98, Lev01, FRS + 03] and donor atoms in Si [Kan98, VYW + 00] . This case is similar to that of the XY model, since H Heis ij also preserves the total number of 1’s in a computational basis state. There- fore, as in the XY case, we must assume the ability to generate anX i X j pulse. Then, X i X j (H Heis jk )X i X j =J jk (X j X k −Y j Y k −Z j Z k ), which commutes withH Heis jk , so that X i X j e −itH Heis jk X i X j e −itH Heis jk = e −2itJ jk X j X k , and we are back to Case 1. There is now another option for generating an entangling gate: we can generate a pureZZ interac- tion usingZI(H Heis )ZI =−XX−YY +ZZ, which commutes withH Heis , so that e −itH Heis e −i π 2 ZI e −itH Heis e −i π 2 ZI = e −2itJZZ . This is a four-step, naturally fault tolerant procedure. The decoupling pulse and recovery are now the same as in Case 1. Finally, there remains the issue of compatibility between the encoded logic opera- tions and the decoupling pulses that are being constantly applied to the system. All three interaction Hamiltonians we have considered commute with the globalX BB-pulse, so are fully compatible with the BB operations. Furthermore, the logical single-qubit terms also commute with theX pulse. Thus whenever use of a single bodyZ i Hamiltonian is required, it must be synchronized to be applied only after an even number of col- lective BB pulses, to ensure the compatibility of quantum manipulation and dynamical decoupling. 5.6 Conclusions We have proposed a new method for performing universal, fault tolerant quantum com- putation in the presence of spontaneous emission. The method combines dynamic 72 decoupling pulses with a particularly simple and efficient quantum error correcting code, encoding n logical qubits into n + 1 physical qubits. Computation is performed by controlling single-qubitσ x andσ z terms together with any of three major examples of qubit-qubit interaction Hamiltonians, applicable to a wide range of quantum computing proposals. The proposed method offers an improvement over previous schemes for pro- tecting quantum information against spontaneous emission in that the code is at least twice as efficient in terms of qubit resources, and the method is fully compatible with computation using physically reasonable resources and interactions. 73 Chapter 6: Fault-Tolerant Quantum Dynamical Decoupling 6.1 Introduction In spite of considerable recent progress, coherent control and quantum information pro- cessing (QIP) is still plagued by the problems associated with controllability of quan- tum systems under realistic conditions. The two main obstacles in any experimental realization of QIP are (i) faulty controls, i.e., control parameters which are limited in range and precision, and (ii) decoherence-errors due to inevitable system-bath interac- tions. Nuclear magnetic resonance (NMR) has been a particularly fertile arena for the development of many methods to overcome such problems, starting with the discovery of the spin-echo effect, and followed by methods such as refocusing, and composite pulse sequences [Fre98]. Closely related to the spin-echo effect and refocusing is the method of dynamical decoupling (DD) pulses introduced into QIP in order to overcome decoherence-errors [VKL99, Zan99]. In standard DD one uses a periodic sequence of fast and strong symmetrizing pulses to reduce the undesired parts of the system-bath interaction Hamiltonian H SB , causing decoherence. Since DD requires no encoding overhead, no measurements, and no feedback, it is an economical alternative to the method of quantum error correcting codes (QECC) [e.g., [KLV00, Ste03, BT05], and references therein], in the non-Markovian regime [FTP + 05]. Here we introduce concatenated DD (CDD) pulse sequences, which have a recur- sive temporal structure. We show both numerically and analytically that CDD pulse sequences have two important advantages over standard, periodic DD (PDD): (i) Signif- icant fault-tolerance to both random and systematic pulse-control errors (see [VK03] 74 for a related study), (ii) CDD is significantly more efficient at decoupling than PDD, when compared at equal switching times and pulse numbers. These advantages sim- plify the requirements of DD (fast-paced strong pulses) in general, and bring it closer to utility in QIP as a feedback-free error correction scheme. 6.2 The noisy quantum control problem The problem of faulty controls and decoherence errors in the context of QIP, as well as other quantum control scenarios [HRK00], can be formulated as follows. The total HamiltonianH for the control-target system (S) coupled to a bath (B) may be decom- posed as:H =H S ⊗I B +I S ⊗H B +H SB , whereI is the identity operator. The com- ponentH SB is responsible for decoherence inS. We focus here on the single qubit case, but the generalization to many qubits, withH SB containing only single qubit couplings, is straightforward. We shall interchangeably use X,Y,Z to denote the corresponding Pauli matricesσ α , andσ 0 to denoteI. The system Hamiltonian isH S = H int S +H P , where H int S is the intrinsic part (self Hamiltonian), and H P is an externally applied, time-dependent control Hamiltonian. We denote all the uncontrollable time-independent parts of the total Hamiltonian byH e , the “else” Hamiltonian:H e :=H int S +H B +H SB . We assume that all operators, exceptI, are traceless (traceful operators can always be absorbed into H S ⊗I B , I S ⊗H B ). We further assume thatkH e k <∞ 1 . Note that this is a physically reasonable assumption, even in situations involving theoretically infinite-dimensional environments (such as the modes of an electromagnetic field), since in practice there is always an upper energy cutoff 2 . 1 We use the spectral radiuskAk := max hψ|ψi=1 |hψ|Aψi|; a convenient unitarily-invariant operator norm, which, for normal operators, coincides with the largest absolute eigenvalue. 2 This has been observed also in QECC work dealing with general error models [KLV00, BT05] but as has been pointed out there, ifkH e k=∞ it must be appropriately redefined. 75 We consider “rectangular” pulses [piece-wise constantH P (t)] for simplicity; pulse shaping can further improve our results [Fre98]. An ideal pulse is the unitary system- only operatorP(δ) =T exp[−i R δ 0 H P (t)dt], whereT denotes time-ordering and~ = 1 units are used throughout. A non-ideal pulse,U P (δ) =T exp[−i R δ 0 {H P (t)+W P (t)+ H e (t)}dt], includes two sources of errors: (i) Deviations W P from the intended H P . Such deviations can be random and/or systematic, generally operator-valued; (ii) The presence ofH e during the pulse. 6.2.1 Periodic DD In standard DD one periodically applies a pulse sequence comprised of ideal, zero- width π-pulses representing a “symmetrizing group”G = {P i } |G|−1 i=0 (P 0 = I), and their inverses. Let f τ 0 = T exp[−i R τ 0 0 H e (t)dt] denote the inter-pulse interval, i.e., free evolution period, of duration τ 0 . The effective Hamiltonian H (1) e for the “sym- metrized evolution” Q |G|−1 i=0 P † i f τ 0 P i =: e i|G|τ 0 H (1) e is given for a single cycle by the first-order Magnus expansion: H (1) e ≈ H eff = 1 |G| P |G|−1 i=0 P † i HP i [VKL99]. This result is the basis of an elegant group-theoretic approach to DD, which aims to elim- inate a given H SB by appropriately choosingG [VKL99, Zan99]. The “universal decoupling” pulse sequence, constructed fromG UD := {σ 0 ,σ 1 ,σ 2 ,σ 3 }, proposed in [VKL99], plays a central role: it eliminates arbitrary single-qubit errors. For this sequence we have, after using Pauli-group identities (XY = Z and cyclic permuta- tions), p 1 := e iτ 1 H (1) e = Q 3 i=0 P † i f τ 0 P i = f τ 0 Xf τ 0 Zf τ 0 Xf τ 0 Z, where τ 1 = 4τ 0 . The idea of dynamical symmetrization has been thoroughly analyzed and applied (see, e.g., [Vio04, FTP + 05] and references therein). However, higher-order Magnus terms can in fact not be ignored, as they produce cumulative decoupling errors. Moreover, standard PDD is unsuited for dealing with non-ideal pulses [VK03]. 76 6.2.2 Concatenated DD Intuitively, one expects that a pulse sequence which corrects errors at different levels of resolution can prevent the buildup of errors that plagues PDD; this intuition is based on the analogy with spatially-concatenated QECC (e.g., [Ste03]). With this in mind we introduce CDD, which due to its temporal recursive structure is designed to overcome the problems associated with PDD. Definition 1 A concatenated universal decoupling pulse sequence: p n+1 := p n Xp n Zp n Xp n Z, wherep 0 ≡ f τ 0 andn≥ 0. Several comments are in order: (i)p 1 is the “universal decoupling” mentioned above, but one may of course also concatenate other pulse sequences; (ii) One can interpretp 1 itself as a one-step concatenation: p 1 :=p X Yp X Y , wherep X := fXfX (f:=f τ 0 ) and Pauli-group identities have been used. (iii) Any pair, in any order, of unequal Pauli π-pulses can be used instead ofX andZ, and furthermore a cyclic permutation in the definition of p 1 is permissible; (iv) The duration of each sequence is given by T . τ n := 4 n τ 0 (after applying Pauli-group identities); (v) The existence of a minimum pulse intervalτ 0 and finite total experiment timeT are practical constraints. This sets a physical upper limit on the number of possible concatenation levelsn max in a given experiment duration; (iv) Pulse sequences with a recursive structure have also appeared in the NMR literature (e.g., [HW68, CTPG86]), though not for the purpose of reducing decoherence on arbitrary input states. We next present numerical simulations which compare CDD with PDD. 77 6.3 Numerical Results for Spin-Bath Models For comparing the performance of CDD vs PDD, we have chosen an important example of solid-state decoherence: a spin-bath environment [PS00]. This applies, e.g., to spec- tral diffusion of an electron-spin qubit due to exchange coupling with nuclear-spin impu- rities [SDS03], e.g., in semiconductor quantum dots [GBD99], or donor atom nuclear spins in Si [Kan98]. Specifically, we have performed numerically exact simulations for a model of a single qubit coupled to a linear spin-chain via a Heisenberg Hamiltonian: H e = ω S σ z 1 +ω B ( P K a=2 σ z a ) + P K a>b≥1 j ab ~ σ a ·~ σ b . The system spin-qubit is labeled 1; the second sum represents the Heisenberg coupling of all spins to one another, with j ab = jexp(−λd ab ), whereλ is a constant andd ab is the distance between spins. Such exponentially decaying exchange interactions are typical of spin-coupled quantum dots [GBD99]. The initial state is a random product state for the system qubit and the envi- ronment. The goal of DD in our setting is to minimize (the log of) the “lack of purity” of the system qubit, l ≡ log 10 (1− Tr[ρ 2 S ]), where ρ S is the system density matrix obtained by tracing over the environment basis. At given CDD concatenation level n we also implement PDD by using the same minimum pulse intervalτ 0 as in CDD and the same total number of pulses N . 4 n ; this ensures a fair comparison. In all our simulations we have set the total pulse sequence duration T = 1, in units such that (ω S T,ω B T,λd j,j+1 ) = (2,1,0.7). Longer pulse sequences correspond to shorter pulse intervalsτ 0 . Note that we have chosen our system and bath spins to be similar species. Qualitatively, the number of bath spinsK had no effect in the tested range 2≤K≤ 7, while quantitatively, and as expected, decoherence rises withK. DD pulses were imple- mented by switchingH P =hσ α 1 ,α∈{x,z}, on and off for a finite durationδ > 0; note thatn≤n max (T,δ). We define the pulse jitterW P as an additive noise contribution to H P . It is represented asW α P = ~ r α ·~ σ 1 , with~ r α being a vector of random (uniformly distributed) coefficients. We distinguish between systematic (W α P fixed throughout the 78 0.05 0.1 0.15 0.2 8 6 4 2 0 l |w | P 1 3 5 n 0.025 0.075 8 6 4 l 1 3 5 n |w | P Figure 6.1: Left: Performance of CDD (solid line,n = 4) vs PDD (dot-dashed line) as a function of random jitter fraction|w P | :=kW P k/kH P k, with pulse-widthδ = 10 −5 T , coupling strength j = .2/T (T is the total evolution time), and number of bath spins K = 2, averaged over 90 jitter realizations. For comparison, the horizontal dashed line corresponds to free evolution. Right: CDD as a function of random jitter fraction|w P | and concatenation leveln. The vertical axis denotesl = log 10 (1− purity) here and in Figs. 2 and 3. 1 3 5 n 0.01 15 30 jT 10 5 0 l 1 3 5 1 3 5 n 0.01 15 30 jT 10 7.5 5 2.5 0 l 1 3 5 n Figure 6.2: CDD (left) and PDD (right), as a function of system-bath couplingj; pulse widthδ = 10 −4 T , number of bath spinsK = 5, and without jitter (W P = 0). Note the l-axis scale difference between CDD and PDD. pulse sequence, but different for eachα) and random (W α P changing from pulse to pulse) errors. Our simulation results, shown in Figs. 6.1-6.3, compare CDD and PDD as a function of coupling strength, relative jitter magnitude, and number of pulses. Fig. 6.1, left, com- pares CDD and PDD at fixed number of pulses. CDD outperforms PDD in the random jitter case with noise levels of up to almost10%. Fig. 6.1, right, shows the performance of CDD as a function of jitter magnitude and concatenation level: the improvement is systematic as a function of the number of pulses used. Figure 6.2 contrasts CDD and 79 0 2 4 6 n 0 2.5 5 7.5 10 |w | P 8 6 4 2 0 l 0 2 4 6 0.2 0.4 0.6 0.8 1 10 8 6 4 2 0 |w | P l Figure 6.3: Left: CDD performance as a function of concatenation level and systematic jitter. The pulse widthδ = 10 −4 T , number of bath spinsK = 5,jT = 15.0, averaged over 7− 80 realizations (more realizations for higher n). Right: CDD (solid line) vs PDD (dot-dashed line) as a function of systematic jitter forn = 5,δ = 10 −5 T ,K = 5, jτ 0 = 3.0, averaged over 14−80 realizations. The dashed line is pulse-free evolution. CDD performance is unaffected up to∼ 20% jitter level. PDD in the jitter-free case, as a function of system-bath couplingj. As predicted in the analytical treatment below, CDD offers improvement compared to PDD in decoherence reduction over a wide range ofj values. Figure 6.3 compares CDD and PDD as a func- tion of systematic jitter. Superior performance of CDD is particularly apparent. These results establish the advantage of CDD over PDD in a model of significant practical interest, subject to a wide range of experimentally relevant errors. We now proceed to an analytical treatment. 6.4 Imperfect decoupling Consider DD pulse sequences composed of ideal pulses. Let us partition H e as H e = H ⊥ X +H k X , where H ⊥ X = Y⊗B y +Z⊗B z and H k X = X⊗B x +H B . The super/subscripts⊥,X andk,X correspond to terms that anti-commute and commute withX⊗I B , respectively. Thus the effect of p X =fXfX in PDD can be viewed as a projection ofH SB onto the component “parallel” toX, i.e.,H k X . For theY pulses in p 1 =p X Yp X Y we can similarly write H k X = H ⊥ Y +H k Y , where⊥,Y (k,Y ) denotes anti-commutation (commutation) with Y , whence H k Y = H B . Then the role of the 80 X H X H X X f f Y Y X X p p (1) SB H (2) SB H (3) SB H ) b ( X H X H X X f f Y Y X X p p ) a ( Figure 6.4: Projections involved in DD. (a) Perfect cancelation in first-order Magnus case. (b) Extra rotation induced by higher-order Magnus terms, and effect of concate- nation. Y pulses is to projectH k X ontoH k Y , which eliminatesH SB altogether, i.e., transforms H e =H SB +H B into a “pure-bath” operatorH B . This geometrical picture of two suc- cessive projections is illustrated in Fig. 6.4(a). However, these projections are imperfect in practice due to second-order Magnus errors. Indeed, instead of a sequence such as fXfX, one has, after pulseP i=X,Y ,I E,i := exp[−iτ(H k i −H ⊥ i )]exp[−iτ(H k i +H ⊥ i )], where H e = H k i + H ⊥ i , and we have accounted for the sign-flipping due to P i . Using the BCH formula (e.g., [SK89]), we approximate the total unitary evolution as I E,i = exp[−i(2τ)H eff,i +O(λ 3 i )], where H eff,i :=D P i (τ)[H e ] = e −iτH ⊥ i /2 H k i e iτH ⊥ i /2 , whereλ 3 i := τ 3 kH ⊥ i k 2 kH k i k, and it is assumed that, sincekH e k <∞, one can pickτ such thatλ i ≪ 1. The mappingD P i (τ)[H e ] clearly has a geometric interpretation as a projection that eliminatesH ⊥ i , followed by a rotation generated byH ⊥ i . This rotation produces extra system-bath terms besidesH k i , hence imperfect DD. This is illustrated in Fig. 6.4(b): in the first-order Magnus approximation the transition fromH (2) SB toH (3) SB suffices to eliminateH SB , i.e.,H (3) SB = 0. But in the presence of second-order Magnus errors H (3) SB 6= 0. The difference between CDD and PDD is precisely in the manner 81 in which this error is handled: in PDD theH (3) SB error accumulates over time since the same procedure is simply repeated periodically. However, in CDD the process of pro- jection+rotation is continued at every level of concatenation, as suggested in Fig. 6.4(b) (red arrow aboveH (3) SB ). In CDD, H (m) SB is shrunk with increasing m, in a manner we next quantify. 6.4.1 Convergence of CDD in the limit of zero-width pulses Decoupling induces a mapping on the components of H e . For a single qubit, writing H e = P α=x,y,z σ α ⊗B α , we haveH e p 1 7→H (1) e = P α σ α ⊗B (1) α , where a second-order Magnus expansion yields: B (1) 0 = B 0 , B (1) x = iτ 0 [B 0 ,B x ], B (1) y = iτ 0 1 2 ([B 0 ,B y ]− i{B x ,B z }),B (1) z = 0. Let us defineβ :=kB 0 k andJ := max(kB X k,kB Y k,kB Z k), where we assumeJ <β <∞ 3 . Comparing with the model we have used numerically, J = O(λ) andβ = O(ω B ). It is possible to show that a concatenated pulse sequence p n can still be consistently described by a second-order Magnus expansion at all levels of concatenation, provided the (sufficient) conditionτ n β≪ 1 is satisfied. We can then derive the recursive mapping relations forH (n−1) e pn 7→H (n) e = P α σ α ⊗B (n) α :B (n≥0) 0 = B 0 , B (n≥1) x = (iτ n−1 )[B 0 ,B (n−1) x ], B (n≥2) y = 1 2 (iτ n−1 )[B 0 ,B (n−1) y ], B (n≥1) z = 0. The propagator corresponding to the whole sequence is exp(−iτ n H (n) e ), which in the limit of ideal performance reduces to the identity operator. These results forB (n) α allow us to study the convergence of CDD, and bound the success of the DD procedure, as measured in terms of the fidelity (state overlap between the ideal and the decoupled evolution). This fidelity is given by [BT05] f n ≈ 1−||τ n e H (n) e || 2 ≈ 1−(τ n h (n) ) 2 =: 1−(Φ CDD ) 2 , (6.1) 3 This assumption is made to simplify some of our convergence arguments; while it is not essential, it is reasonable, since we expect only a small number of bath degrees of freedom to be coupled to a given qubit, whereas no restriction exists on the bath self-Hamiltonian 82 where e H is the system-traceless part ofH, andh (n) := max{||B (n) x ||,||B (n) y ||}. We find that Φ CDD ≤ (βT/N 1/2 ) n (JT), (6.2) where T = Nτ 0 . τ n = 4 n τ 0 is the total sequence duration, comprised of N pulse intervals. In contrast, Φ PDD =Th (1) yields Φ PDD = 2(βτ 0 )(JT) = 2(βT/N)(JT). (6.3) Note that forN = 4, Φ CDD = Φ PDD as expected. There is a physical upper limit to the number of concatenation levels, imposed by the conditionβτ n ≪ 1. Using this condi- tion in the formβ = c/T , wherec is some small constant (such as 0.1), and fixing the value ofβ, we can back out an upper concatenation leveln max =−log 4 βτ 0 c ; inserting this into Eq. (6.2) we have Φ CDD ≤ (cβτ 0 ) − 1 2 log 4 βτ 0 c (JT). We can now compare the CDD and PDD bounds in term of the final fidelity: 1−f CDD 1−f PDD ≤ (cβτ 0 ) −log 4 βτ 0 c 4(βτ 0 ) 2 βτ 0 →0 −→ 0. (6.4) This key result shows that CDD converges super-polynomially faster to zero in terms of the (physically relevant) parameterβτ 0 , at fixed pulse sequence duration. However, it is important to emphasize that our bound on Φ CDD is unlikely to be very tight, since we have been very conservative in our estimates (e.g., in applying norm inequalities and estimating convergence domains). Indeed, in our simulations (above)βτ n ≈ 2, which is beyond our conservatively obtained convergence domain. 83 6.4.2 Finite width pulses We now briefly consider the more realistic scenario of rectangular pulses with a unitary propagator given byT exp[−i R δ 0 {H P (t)+H e (t)}dt], where the pulse duration or width δ is much smaller thanτ 0 . In this case we can derive a modified form of the condition βτ n ≪ 1, required for consistency (of using a second-order Magnus expansion at all levels of concatenation): cτ n β +d δ τ n ≪ 1, (6.5) wherec,d∼ 1 are pulse sequence-specific numerical factors. The consistency require- ment (6.5) validates the analysis of convergence of CDD forδ6= 0, and we can repro- duce the advantage of CDD over PDD forδ = 0 [manifest in Eq. (6.4)]. As expected Eq. (6.5) imposes a more demanding condition on the total duration τ n , at fixed bath strength β. While Eq. (6.5) cannot be called a threshold condition (in analogy to the threshold in QECC), since it depends on the total sequence duration, it does provide a useful sufficient condition for convergence of a finite pulse-width CDD sequence, and introduces the concept of error per gate which is fundamental in QEC. 6.5 Conclusions and outlook We have shown that concatenated DD pulses offer superior performance to standard, periodic DD, over a range of experimentally relevant parameters, such as system-bath coupling strength, and random as well as systematic control errors. Here we have addressed the preservation of arbitrary quantum states. Quantum computation can in principle be performed, using CDD, over encoded qubits by choosing the DD pulses as the generators of a stabilizer QECC, and the quantum logic operations as the corre- sponding normalizer [BL02, BL03]. Another intriguing possibility is to combine CDD and high-order composite pulse methods [KB04]. 84 Chapter 7: Efficiency Analysis of Dynamical Decoupling 7.1 Introduction Arbitrary quantum state preservation is a fundamental imperative for quantum informa- tion processing, but undesired interactions of a candidate quantum system with uncon- trollable external systems (the environment/bath) results in poor control and fidelity loss. Even if the structure of these interactions is approximately known, the statistical uncertainty in the state of the environment invariably results in decoherence [Zur91]. While undesired couplings to the environment are inevitable (even at zero temperature), strong control fields applied to the system can be used to effectively manipulate the cou- plings. Nuclear magnetic resonance is an excellent example where techniques such as refocusing and composite pulses are readily used to generate reliable and high precision quantum dynamics [Fre98, VC04]. Similar in execution but applicable in a generic set- ting, dynamical decoupling (DD) is a method for the effective renormalization of the system-bath interaction Hamiltonian via the application of strong system-control fields. Usually the goal is the cancellation of all coupling terms. In the context of quantum information processing, DD can be used for feedback-free quantum error suppression without encoding overhead [VKL99]. Dynamical decoupling is most efficient against bounded environments, when the pulse switching times are short on a scale set by the bath spectral density high-frequency cutoff [VL98, P. 05], or when the spectral density is rapidly decaying [SL04]. Within these assumptions different flavors of DD can be designed. Of course, technology limits how strongly, rapidly and accurately we can modulate the system Hamiltonian, and cool 85 the system. Dynamical decoupling can be implemented with the pulse sequence chosen deterministically, e.g., periodically [VL98, BL01, PS06], or randomly [FLP04, VK05]. Randomized decoupling is expected to perform better in the case of varying/fluctuating (effective) Hamiltonians while deterministic methods perform better in cases where the undesired terms in the system-bath Hamiltonian are sufficiently weak [SV06, VS06]. Hybrid schemes with optimized performance have also been considered [VK05, KA05]. The analysis of DD schemes is often performed within an interaction picture. Here we consider explicitly the internal dynamics of the bath in terms of its effect on the performance of DD. Dynamical decoupling strategies, some of which can be derived from group theo- retical considerations [Zan99], are typically based on a universal DD pulse sequence [VKL99]: a short sequence of unitary operators designed to completely cancel errors up to the first order in the Magnus expansion [Mag54]. Here we consider two deterministic decoupling schemes: (i) In periodic DD (PDD), the universal decoupling sequence is repeated periodically for the duration of the quantum state preservation. (ii) In concate- nated dynamical decoupling (CDD) [KL05b], the universal pulse sequence is recursively embedded within itself. We provide an analytic leading-order study of the performance of the above strategies. Our first basic finding is a verification in the DD-setting of a result familiar from NMR, that even when ideal (zero-width) pulses are used for decou- pling, the corrections from second and higher order Magnus terms impose an upper performance bound on PDD. A central result of our approach is that the coupling terms responsible for errors undergo an effective renormalization transformation by the externally applied pulse sequences. This process is conveniently described via the Magnus expansions for deriva- tion of effective coupling Hamiltonians [see Eqs. (7.41)-(7.44) below]. The renormaliza- tion approach leads to the view of DD as a dynamical map, whose convergence to a fixed 86 point (ideally, the cancellation of the system-bath interaction) depends on whether the norm of the coupling terms decreases under repeated applications of the pulse sequence. In support of our earlier study [KL05b], within the technological constraints of finite pulse numbers and the bounds imposed by the applicability of DD in general, we ana- lytically prove the asymptotic superiority of CDD over PDD. In addition, we present here new pulse sequences, inspired by the Trotter-Suzuki expansion [Suz76], with even better convergence properties than CDD. In a more abstract setting, we show that in fact any application of unitary operators on the system cannot cause an increase in the strength of the undesired couplings. Our conclusions are valid within the convergence domain of our expansions. We find that the Magnus expansion itself sets the most stringent limit on convergence domains, in the sense that it includes or coincides with the regime of applicability of DD. In the worst case, this corresponds to the limit of slow internal bath dynamics. We also analyze robustness with respect to systematic pulse errors. The decoupling error of any deterministic scheme is thus a result of the environmental coupling errors, and errors in the decoupling operations. In the case of realistic, imperfect pulses, we show that the performance of DD, when pulses of finite width (or uniform error rate) are used, is determined by a condition based on both the pulse switching times and pulse widths. In this case, unless pulse profiles and timings are adjusted in case of systematic pulse errors, even the first order (and thus dominating) terms in the Magnus expansion will be non-zero. While undesired coupling terms are responsible for fidelity loss, the relationship between the two is complicated and depends on the details of the environment and pos- sible physical energy cutoffs. This motivates us to perform a generic analysis based on operator-norm estimates. Our conservative estimates provide a worst-case analy- sis for decoupling performance. We expect these estimates to be useful as guidelines 87 for choosing and combining dynamical decoupling strategies when constraints such as pulse switching times and pulse errors are considered. This chapter is organized as follows: in Section 7.2 we review the basic universal DD cycle which suffices to decouple a qubit from an arbitrary non-Markovian envi- ronment to first order in the Magnus expansion. In Section 7.3 we provide a detailed analysis of this sequence in terms of the Magnus expansion, for both ideal (zero-width) and non-ideal (finite-width) pulses. In Section 7.4 we compare two deterministic decou- pling strategies founded on the basic universal DD sequence: periodic and concatenated sequences. We calculate a fidelity measure associated with the two strategies and show that the concatenated one strictly outperforms the periodic one. In Section 7.5 we intro- duce a new decoupling strategy, based on the Trotter-Suzuki expansion. Even though this decoupling sequence has implementations problems and is not as robust as CDD, we find it interesting in light of its superior convergence properties. This Section also includes a table which compares the three deterministic decoupling strategies (PDD, CDD, and Trotter-Suzuki), and clearly illustrates and summarizes their relative perfor- mance. Finally, in Section 7.6 we present a general result concerning the behavior of error norms under pulse sequences: we show that for sufficiently narrow pulses, pulse sequences cannot increase error norms. This result has impact also on fault tolerance theory using quantum error correcting codes. A summary and discussion is presented in Section 7.7. Supporting calculations can be found in the Appendices. 7.2 Universal Dynamical Decoupling for a Qubit In the absence of driving terms, the terms in the system-bath interaction Hamiltonian are responsible for decoherence and loss of quantum information. Removal of these terms is sufficient (but not necessary [LW03a]) for preservation of arbitrary quantum 88 states. Dynamical decoupling schemes use strong and fast control Hamiltonians acting on the quantum system only, to effectively remove/modify various terms in the system- bath interaction Hamiltonian 1 . In particular, a pulse sequence designed to remove every term in the interaction Hamiltonian is referred to as universal dynamical decoupling. In this work, we focus on the universal DD of a single qubit. Extensions to multiple qubits [VKL99] and higher dimensional quantum systems exist [LAW02, BLWZ05], but we will not consider these here. We use X, Y , andZ to denote the standard 2× 2 Pauli matrices σ x = 0 1 1 0 , σ y = 0 −i i 0 , σ z = 1 0 0 −1 acting on a single qubit, and work in units of ~ = 1. System and environment are assumed to inhabit different Hilbert spaces i.e., we do not consider leakage, which can also be treated using DD [LAW02, BLWZ05]), and all Hamiltonian operators are taken to be traceless without loss of generality. Consider a qubit with a Hamiltonian H(t) =H ctrl (t)+H e (t), (7.1) whereH ctrl refers to a time-dependent controllable system-only part andH e includes all other terms, i.e., the internal bath, internal system, and interaction Hamiltonians: H e =H B ⊗I S +I B ⊗H S +H SB . (7.2) HereI denotes the identity operator. We have implicitly excluded a pure-bath Hamil- tonian termH 0 B ⊗I S satisfying [H 0 B ⊗I S ,F] = 0, whereF is any element of the Lie 1 This is not the only possibility; one may consider controlling the environment as well. See, e.g., [PR06]. 89 algebra generated byH SB ,H B ⊗I S , andI B ⊗H S . The reason is that such a term on the one hand does not impact the system dynamics, but on the other hand will increase the operator norm that arises below in our calculations of decoupling errors. With this in mind, ideally one would like to haveH e = 0. The “error Hamiltonian”H e can always be expanded as: H e =B 0 ⊗I S +B X ⊗X +B Y ⊗Y +B Z ⊗Z (7.3) whereB α (α = 0,X,Y,Z) are operators acting on the environment. We are allowing theB α to include the identity operator, i.e., from now on we are incorporatingH S into H SB . I.e., assuming H S = X α=x,y,z ω α σ α , (7.4) withω α all non-zero frequencies, and writing H SB = X α=x,y,z b α ⊗σ α , (7.5) yields B α = ω α I B +b α α∈{x,y,z}, B 0 = I B +H B . (7.6) Note that the first term inH e ,B 0 ⊗I S , is a pure-environment term and simply generates the environment’s internal dynamics. It also includes the global phase generating term I B ⊗I S . Obviously if b X = b Y = b Z = 0, thenH e = B 0 ⊗I S has no effect on the system dynamics. 90 Universal DD of a qubit with ideal pulses removes every term inH e exceptB 0 ⊗I S , by applying the following pulse sequence: fXfZfXfZ, where f denotes a “pulse- free” period of fixed duration. The pulses are generated by H ctrl . This universal DD sequence is a simplification of (I S fI S )(XfX)(YfY)(ZfZ), where X,Y,Z and the identity I S represent the decoupling groupG on the qubit. The universal decoupling group has the property that for every Hamiltonian H acting on the system, the sum P {P i ∈G} P i HP † i acts trivially on the system [VKL99]. Since this sum is the lead- ing order generator in the Magnus expansion, the universal DD sequence completely removes any system-bath interaction to first order (we revisit this in detail in subsection 7.3.2). A complete analysis of the performance of DD needs to take into account details of the environment (participating modes, energy cutoffs, temperature, (dis)equilibrium, etc.) but we shall minimize these considerations and focus on the model-independent features of DD. Our analysis is mathematically constrained by convergence domains that are explicit in our approximations. We expect certain unbounded systems such as bosonic environments to be within the realm of our theoretical framework after the introduction of spectral cutoffs. 7.3 Analysis of the Universal Decoupling Pulse Sequence In this section we derive the transformation of various Hamiltonian terms under the basic universal DD pulse sequence. We first consider ideal pulses, then amend our discussion to allow for non-ideal (finite width) pulses. We will see that the system-bath interaction Hamiltonian is effectively renormalized under the DD pulse sequence. As long as this renormalization transformation is norm-reducing, the DD procedure is effective. 91 7.3.1 Ideal Pulses Let us first assume that the pulses used are ideal, i.e., infinitely strong and narrow. For example,H ctrl (t) = π 2 δ(t−t 0 )X generates an idealX pulse at timet 0 (δ is the Dirac δ-function). The propagator corresponding to free evolutionf of periodτ 0 is given by: U f = exp(−iτ 0 H (0) ), H (0) :=H e . (7.7) To obtain the cycle propagator we use the identityMe A M −1 = e MAM −1 , valid for any operatorA and invertibleM. Define H 1 ≡B 0 ⊗I +B X ⊗X +B Y ⊗Y +B Z ⊗Z =IH e I, H 2 ≡B 0 ⊗I +B X ⊗X−B Y ⊗Y−B Z ⊗Z =XH e X, H 3 ≡B 0 ⊗I−B X ⊗X +B Y ⊗Y−B Z ⊗Z =YH e Y, H 4 ≡B 0 ⊗I−B X ⊗X−B Y ⊗Y +B Z ⊗Z =ZH e Z, (7.8) The free evolution propagator can then also be written as f= exp(−iτ 0 H 1 ). Using Eqs. (7.8) we can write the total propagator corresponding to the universal DD cycle, (IfI)(XfX)(YfY)(ZfZ), in terms of four effective Hamiltonians describing the dif- ferent evolution segments, as: U 1 =e −iτ 0 H 1 e −iτ 0 H 2 e −iτ 0 H 3 e −iτ 0 H 4 . (7.9) 92 A time-varying piecewise constant Hamiltonian,H(t) varying over four intervals each of length τ 0 , can generate this propagator. The total propagator can then be used to define the effective HamiltonianH (1) : U 1 = 4 Y i=1 exp(−iτ 0 H (0) i ) =: exp(−i4τ 0 H (1) ), (7.10) where we have added superscripts(0) to the HamiltoniansH i of Eq. (7.8), in anticipation of the concatenation procedure that we consider in subsection 7.4.3. 7.3.2 Magnus Expansion The cycle propagatorU 1 [Eq. (7.10)] can be approximated using a Magnus expansion (for an alternative method of analysis that is particularly useful for the design of periodic sequences of soft pulses see [SP05]). Consider a time-dependent HamiltonianH(t) generating the propagator U(t) from time 0 to t. In the Magnus expansion (a type of cumulant expansion) we have U(t) = exp ∞ X i=1 A i (t), (7.11) withA 1 andA 2 given by: A 1 =−i Z t 0 dt 1 H(t 1 ), (7.12) A 2 =− 1 2 Z t 0 dt 1 Z t 1 0 dt 2 [H(t 1 ),H(t 2 )]. (7.13) Higher order terms are given by higher order commutator expressions [Mag54]. A recent bound for the convergence radius of the Magnus expansion [MO01] translates 93 in our case into maxkH(t)kt < 2. 2 In many situationskB 0 k (norm of the environ- ment’s internal Hamiltonian) is expected to dominate the Hamiltonian and we may as well usekB 0 kt. 1 as a conservative convergence domain. This bound can, however, be superficial since not all degrees of freedom of the environment might actually be involved in the dynamics. For example, in a spin bath, bath spins far away from the sys- tem will not immediately contribute to the dynamics but including them will nonetheless increasekB 0 k without changing the real convergence radius. A precise analysis of the actual convergence radius requires us to estimate the next-to-leading-order terms – see Sec. 7.8. For the piecewise constant evolution of the Hamiltonian in Eq. (7.9), we can calcu- lateA 1 andA 2 in terms ofH(t) ={H j for(j−1)τ 0 ≤t≤jτ 0 } 4 j=1 , i.e., the sign-flipped Hamiltonians appearing in Eq. (7.9): A (1) 1 =−iτ 0 (H (0) 1 +H (0) 2 +H (0) 3 +H (0) 4 ) (7.14) A (1) 2 =− 1 2 τ 2 0 X 1=i<j=4 [H (0) i ,H (0) j ]. (7.15) Again, the superscripts are included in anticipation of the CDD analysis below. Note that the pure-environment parts ofA (1) i , i.e., terms of the formB⊗I, have no effect on the dynamics of the qubit to first order inτ 0 , but do have an effect to second order inτ 0 , through the commutator terms. Clearly, pure-environment terms are not renormalized under the DD procedure. 2 Throughout this work we usekAk to denote a unitary invariant operator norm, e.g., the maximum eigenvalue for traceless operators, or the absolute difference between the smallest and largest eigenvalues [Bha97]. 94 Using Eqs. (7.8),(7.14),(7.15) we find: A (1) 1 = −i(4τ 0 )B 0 ⊗I A (1) 2 = 4τ 2 0 [B 0 ,B X ]⊗X +2τ 2 0 ([B 0 ,B x ]−i{B X ,B Z })⊗Y (7.16) This shows that up to first order in the Magnus expansion (theA (1) 1 term) the universal decoupling cycle completely removes the coupling to the environment. However, there is a leading second order correction due toA (1) 2 in which the coupling to the environment has not been removed. Note that a pure-environment term appears only in A (1) 1 and, due to our particular choice of DD sequence, fXfZfXfZ, there is no term involving⊗Z inA (1) 2 . We now define two norms which will play a central role in our analysis: β :=kB 0 k<∞ (7.17) J := max(kB X k,kB Y k,kB Z k)<∞. (7.18) Recall thatB α = ω α I B +b α forα∈{x,y,z}, andB 0 = I B +H B ; unless otherwise specified, we assume thatJ < β in order to simplify our convergence arguments. This conservative assumption is reasonable for systems where only a small number of envi- ronment particles (or degrees of freedom) are coupled to a given qubit (such as electron spins coupled to a nuclear spin bath [A VK02]) – which translates into a smallkb α k – whereas no restriction exists on the environment self-Hamiltonian (e.g., on the number of particles). The distinction betweenJ andβ is a reflection of the different rolesH SB and H B play in the dynamics of the system. In simple terms, J quantifies the direct 95 coupling strength whileβ quantifies typical bath frequencies. Consider, e.g., the sim- ple case of a spin qubit coupled to another spin-1/2 particle via a Heisenberg coupling: H e =ωZ B ⊗I +c(X B ⊗X +Y B ⊗Y +Z B ⊗Z), wherec is the coupling coefficient. In this case we have:J =O(c) andβ =O(ω). The Magnus terms can be bounded using these quantities:kA (1) 1 k = O(τ 0 β) and kA (1) 2 k = O(τ 2 0 βJ). Higher order Magnus terms, A (1) i>2 , will contain all orders of τ i 0 J k β i−k where 1≤ k≤ i− 1, and the leading term is always given byO(τ i 0 Jβ i−1 ) sinceJ <β. As long asτ 0 β≪ 1 we can safely neglectA (1) i>2 : ||A (1) i>2 ||≪||A (1) 2 ||<||A (1) 1 ||. (7.19) Note that our derivations are based on the separation of the coupling termsB 0 ⊗I and B X ⊗X +B Y ⊗Y +B Z ⊗Z in the error HamiltonianH e . A similar separation can be done for decoupling schemes on systems other than a single qubit. The approximate effective HamiltonianH (1) corresponding to the basic dynamical decoupling cycle, Eq. (7.10), is now: H (1) ≈ 1 −i4τ 0 (A (1) 1 +A (1) 2 ) := X α B (1) α ⊗σ α . (7.20) The renormalized environment operators B (1) α , which can easily be read off from Eqs. (7.16), are the main result of the DD procedure. The important message emerg- ing from the analysis in this subsection is that even when ideal (infinitely strong and narrow) pulses are used, the universal decoupling cycle only removes (the system-bath component of) the lowest order Magnus term, and renormalizes the higher order Mag- nus terms. The success of dynamical decoupling ultimately depends on whether the mapping to the renormalized environment operators is significantly norm-decreasing, an issue we address in detail below. 96 7.3.3 Pulses of Finite Width Ideal pulses that act as system-only unitary operators are simplified mathematical abstractions. In this subsection we model and analyze the effect of the finite width of pulses in decoupling. For simplicity we consider rectangular pulsesP with a width δ. The ideal finite-width pulse is, P = exp(−iδH P ), (7.21) whereH P is a fixed control Hamiltonian. For realistic pulses we must includeH e in the pulse propagator: U P = exp(−iδ(H P +H e )). (7.22) We call such a pulse “non-ideal”. Extremely narrow pulses with||H P ||≫||H e || are thus desirable to minimize the effect of the unwanted terms in the pulse Hamiltonian. Here we build upon the approximation of instantaneous pulses in subsection 7.3.2, by decomposing the non-ideal pulses into products of the ideal unitary operator of the pulse P and some effective pulse error unitary operator E P . We explicitly approximate the operatorsE P for rectangular pulses on a qubit, but the decomposition of the actual pulse into an ideal pulse and a “pulse error” unitary can be reproduced for other pulse shapes as well. The periods of the universal dynamical decoupling cycle need to be adjusted in order to incorporate the time delays associated with finite pulse widths. Therefore, assume 97 that all free evolution periods, with propagatorU f , are adjusted to lengthτ 0 −δ. The propagator for the cycle can then be written as: U (1) =U f U X U f U Z U f U X U f U Z =U f E X XU f XE ′ Z YU f YE ′ X ZU f ZE Z =:U f 1 E X U f 2 E ′ Z U f 3 E ′ X U f 4 E Z (7.23) where U α =e −iδ(ησα+He) α =X,Z, (7.24) δη = π/2, and, in order to fit the formulation of subsection 7.3.2, we have defined the pulse-error operators as follows: E X X :=U X , YE ′ X Z :=U X , (7.25) ZE Z :=U Z , XE ′ Z Y :=U Z . (7.26) Note that since the errorsE α are unitary and are produced during an intervalδ, we may formally associate them with effective Hamiltonians defined through E α =: exp(−iδH E,α ). (7.27) Using these definitions, the propagator defined in Eq. (7.23) is equivalent to the evo- lution due to that of a piecewise constant Hamiltonian H(t), given by the sequence {H 1 ,H E,X ,H 2 ,H ′ E,Z ,H 3 ,H ′ E,X ,H 4 ,H E,Z } withH i given in Eqs. (7.8), at appropriate times. We ignore terms of orderδ 2 ||B α || 2 andδτ 0 kB α k 2 , which allows us to treat the components ofH E,α as c-numbers instead of operators, since no commutators will be involved. Using Eqs. (7.12),(7.13), we can repeat the calculation ofA (1) 1 andA (1) 2 , this 98 time including the pulse segments withH E,α as their effective Hamiltonians, and con- sider the limit of narrow pulses,δ≪τ 0 . In this limit, we can safely truncate the Magnus expansion afterA (1) 2 , provided we assume: cτ 0 β +d δ τ 0 ≪ 1, (7.28) wherec andd are numerical factors ofO(1) (recall thatβ :=kB 0 k). This inequality is derived in Sec. 7.9. Note that it implies an optimal pulse intervalτ 0 = p dδ/cβ, which minimizes the left-hand side of (7.28) for givenβ and a fixed minimal pulse widthδ. Note further that Ineq. (7.28) is not as strict as the condition for the convergence of the Magnus expansion, which reads (when β ≫ J): βT < 1, where T is the total experiment duration. The components of the effective Hamiltonian can be calculated explicitly: B (1) 0 =B 0 , (7.29) B (1) X =i(τ 0 −δ)[B 0 ,B (0) X ]+ δ τ 0 ( 1 2 B (0) X − 1 π B (0) Y ), (7.30) B (1) Y = i 2 τ 0 ([B 0 ,B (0) Y ]−i{B (0) X ,B (0) Z }) + i 2 δ([B 0 ,B (0) Y ]−2i{B (0) X ,B (0) Z }−2iB (0) Z B (0) X ) + 1 π δ τ 0 B (0) Z , (7.31) B (1) Z =iδ 2 π B (0) X (B (0) Z +B (0) X )+B (0) Y B (0) X + δ τ 0 B (0) Z (7.32) The only modifications associated with the pulse widthδ are of orderO(J(δJ +δ/τ 0 )), associated with the new small parametersδJ andδ/τ 0 . In this case the decoupling is not exact and even the first order Magnus terms contribute decoupling errors of orderδJ, the “per-pulse-error”. This is an important effect which will adversely affect decoupling schemes not designed to compensate for such finite pulse-width errors. 99 We note that it is possible to design a piece-wise constant profile for the control HamiltonianH ctrl such that the first order Magnus corrections due to systematic errors in the control Hamiltonian are zero [VK03]. The separation of non-ideal pulses into ideal and error pulses, as above, still applies to this “Eulerian decoupling” scheme, as do most bounds we obtain here. Finally, we note that treatments of decoupling and refocusing errors similar to the above have been pursued in an NMR-specific setting [Hae76]. 7.4 Decoupling Strategies The universal DD cycle results in segments of evolution with HamiltoniansH i such that P H i acts trivially on the system. The derivations of the previous section show how the actual overall propagator contains higher order corrections that do not act trivially on the qubit. Nonetheless, the basic universal DD cycle provides us with the building blocks of general decoupling schemes that optimize decoupling performance with respect to constraints in switching times and pulse precision. Numerical simulations comparing and discussing some of the schemes (deterministic, randomized, or hybrid) are avail- able [SV05, VS06, KA05] (ideal pulses) and [SV06] (ideal and non-ideal pulses), and in this section we focus on system-independent analytic arguments. We deviate from our abstract treatment of bath operators by considering two limiting cases for the cou- pling strengths of system-bath and pure-bath, namely: (i)J <β and (ii)β≪J [recall Eq. (7.18)]. In case (i), the coupling to the environment induces slow dynamics while the environment itself has fast dynamics. In case (ii), the coupling to the environment is dominant but relatively stable due to the environment’s slow internal dynamics. This regularity makes case (ii) more attractive for dynamical decoupling, or methods such 100 as 3 [YLS07], while case (i) is a worst case scenario. As will be noted however the presence of higher-order commutators blurs out the distinction between the two cases in higher orders of concatenated decoupling (Subsection 7.4.3). Nonetheless, both cases are still within the convergence domain of Magnus expansion (kH e kT < 1, see subsec- tion 7.3.2). Outside the convergence domain (e.g. corresponding to a longer duration of the experiment), deterministic decoupling might be replaced by randomized decoupling methods, for which there is some evidence of better performance [SV06]. In practice the co-existence of various bath regimes makes hybrid decoupling methods a practical choice for long-time decoupling [SV05]. 7.4.1 Error Phase We require a measure of fidelity to quantify the performance of DD. To this end we define the error phase corresponding to a propagatorU =e −iTHe , describing an evolu- tion of total durationT generated by an effective HamiltonianH e , as Φ :=Th, (7.33) where h =kH e −B 0 ⊗Ik (7.34) is the norm of the non-pure-environment part of H e , which is effectively (like J) a measure of the coupling strength. The pure-environment part is explicitly excluded from the error phase since it does not affect the fidelity f (state overlap between ideal and decoupled evolution) up to the leading order in our expansion. In this manner the error phase now simply connects the coupling terms in the Hamiltonian to the infidelity and 3 The sequence of operations used in [YLS07] corresponds to the concatenated CPMG sequence explicitly given in Sec. 7.10. 101 decoherence. Indeed, for small error phases the infidelity,1−f, depends monotonically on Φ – see Eq. (7.50) below. In any physical implementation of dynamical decoupling we are limited by tech- nological constraints. Let N denote the number of pulses used during a decoupling experiment of durationT . Normally this number is bounded above due to a minimum pulse switching timeτ min : N < T/τ min . A basic pulse widthδ is used when required, to characterize the systematic error due to non-zero pulse widths. These technological constraints are incorporated below by evaluating the fidelity gain due to decoupling in terms ofN,T , andτ 0 . 7.4.2 Periodic Decoupling In periodic DD for a qubit, a basic universal sequence, such as fXfZ fXfZ, is repeated periodically over the whole interval T . If N pulses are used, there are then N intervals of lengthτ 0 =T/N that correspond to the free evolution periods, andN/4 repetitions of the basic sequence. The effective Hamiltonian for the total interval, is obtained from the results of the previous section, as long as we are within the con- vergence limit of the Magnus expansion, given byTkH e k < 1. Limiting the Magnus expansion to the first two terms, A 1 and A 2 , the propagator for the total evolution is given by U≈ exp[A (1) 1 +A (1) 2 ] = exp(−iTH (1) ) = exp " −iT B 0 ⊗I +iτ 0 X α,β,γ∈0,X,Y,Z F γ αβ B α B β ⊗σ γ !# (7.35) where the effective coupling coefficientsF γ αβ are given in Eqs. (7.16) and can be calcu- lated for any decoupling scheme (not just the universal sequence). 102 For brevity we introduce the parameterG := max(J,β). We read off the error phase from Eq. (7.35) as Φ (1) PDD(i) =Th =O(TJτ 0 G). (7.36) The above estimate for ideal pulses will be modified to the following if rectangular pulses of widthδ are used: Φ (1) PDD(i) (δ) =O[TJ(τ 0 G+δ/τ 0 )]. (7.37) Our estimates show that in if ideal pulses are used, using a higher number of pulses, at fixedT , leads to monotonic improvement in the error phase. I.e., the error phase is proportional to the pulse interval τ 0 . Technology sets a lower limit τ min on τ 0 , which implies that the infidelity is bounded from below by a monotonic function ofτ min TJG in the ideal pulse limit, and the fidelity gain scales with the number of pulsesN. From Eqs. (7.37) for non-ideal pulses, we expect the optimal pulse interval to be given approx- imately byτ 0 = max{τ min ,(δ/G) 1/2 }. In these expressions we have assumed that the pulse widthδ is already at the technological lower limit. 103 7.4.3 Concatenated Decoupling Definition Significant improvement over periodic DD can be obtained by constructing a concate- nated sequence, i.e., by recursively embedding the basic universal DD cycle within itself [KL05b]. This is done in the following manner: p 0 = f p 1 = p 0 Xp 0 Zp 0 Xp 0 Z . . . p n = p n−1 Xp n−1 Zp n−1 Xp n−1 Z. (7.38) Here p 0 (no pulses) is of durationτ 0 , p 1 is of durationτ 1 = 4τ 0 (in the limit of ideal, zero-width pulses), andp n is of durationτ n = 4τ n−1 = 4 n τ 0 (n levels of concatenation). As we are about to show, PDD dramatically outperforms CDD over a wide parameter range. At an intuitive level, this is attributable to the fact that CDD, with its self-similar structure, has error correcting capabilities at multiple resolution levels, whereas PDD allows errors to accumulate essentially as a random walk. First, however, let us remark that the aperiodic sequencep n may be simplified using Pauli matrix product identities when Pauli operators appear in succession (XY =Z and cyclic permutations), in the same manner that the universal decoupling sequence is sim- plified from (I S fI S )(XfX)(YfY)(ZfZ) to p 1 . The reduction in the number of pulses gained by such algebraic cancellations might not be strictly advantageous in a practi- cal setting, since experimentally it is not always possible to generate rotations around all three axes. Moreover, the simplification does not change the asymptotic behavior of the number of pulses as a function of the concatenation leveln. The simplification 104 does affect the formal recursive structure of the sequence. This has no physical effect when decoupling pulses are ideal. But with realistic, finite width pulses, this loss of self-similarity might adversely affect the robustness of the pulse sequence against sys- tematic errors. On the other hand one could argue that the product of two slightly wrong pulses is worse than one, which would be an argument in favor of simplification. A clear decision one way or the other must be made in a context-specific setting. Effective Hamiltonian Due to its recursive definition, the propagator corresponding top n−1 is generated by an effective Hamiltonian H (n−1) e , which is then decoupled with p n and in turn generates H (n) e . The interval length is multiplied by 4 in each such recursive step in which the effective Hamiltonian is renormalized. The truncation of the Magnus expansion beyond the first two terms must be justified, so that the higher order terms do not accumulate as the concatenation level goes up – we do this in Sec. 7.8. We can then construct the higher order effective Hamiltonians by truncating the Magnus expansion and recursively obtainingA (n) i fromH (n−1) e . In the following,H (n) i are constructed as in Eqs. (7.8) and are reproduced each time from the Magnus expansion: H (n) e = i τ n (A (n) 1 +A (n) 2 ) = 1 4 (H (n−1) 1 +H (n−1) 2 +H (n−1) 3 +H (n−1) 4 ) − i 32 τ n X 1=i<j=4 [H (n−1) i ,H (n−1) j ], (7.39) The sum of the operators H (n−1) i is independent of τ 0 and contributes to the pure- environment part. Nonetheless, H (n) e contains the commutator terms that do include 4 n τ 0 (and contribute to the error-terms acting on the system). The commutator [H (n−1) i ,H (n−1) j ] must compensate for this exponential growth. 105 For the qubit case we can derive the explicit form of the sequence of effective Hamil- toniansH (n) e , by finding theB (n) α . We already have the first step of the concatenation in Eq. (7.16). This also serves to initialize the recursion. Proceeding recursively we obtain, for the next iteration: B (1) 0 7→ B (2) 0 =B 0 , B (1) X 7→ B (2) X =iτ 1 ([B (1) 0 ,B (1) X ] =iτ 1 [B 0 ,B (1) X ], B (1) Y 7→ B (2) Y =iτ 1 1 2 ([B (1) 0 ,B (1) Y ]−i{B (1) X ,B (1) Z }) =iτ 1 1 2 [B 0 ,B (1) Y ], B (1) Z 7→ B (2) Z = 0. (7.40) Therefore, for generaln: B (n) 0 =B 0 , forn≥ 0 (7.41) B (n) X = (iτ n−1 )[B 0 ,B (n−1) X ], forn≥ 1 (7.42) B (n) Y = 1 2 (iτ n−1 )[B 0 ,B (n−1) Y ], forn≥ 2 (7.43) B (n) Z = 0. forn≥ 1 (7.44) These equations capture the essence of the renormalization transformation the error terms experience under the pulse sequence. This renormalization transformation could loosely be considered as the cancelation of higher order Magnus expansion terms in the effective Hamiltonian. We note however, that the efficient cancelation of arbitrary higher order Magnus expansion is an open problem in a general setting for generic bath operators (B i ’s in our notation). For example a palindromic pulse sequence (which is the same if applied backwards, in reverse time, e.g., XfYfXfYffYfXfYfX for a qubit) designed to cancel first order Magnus terms will also cancel the second order Magnus 106 terms, since these terms are invariant under time-reversal and therefore must vanish. Sometimes the algebraic structure of the bath operators can be exploited to obtain such efficient sequences more easily. For example, in [Uhr07], a pulse sequence was found that cancels the higher order Magnus terms exactly for a simple spin-boson model, using the fact that the bosonic bath operators form a nilpotent Lie algebra. Convergence and Performance Next we study the convergence conditions of the latter recursive relations. Let us define h (n) := max{kB (n) X k,kB (n) Y k}. (7.45) SinceH (n) e = P α B (n) α ⊗S α ,h (n) is closely related to a bound onH (n) e . We consider the analog of case (i) in PDD, i.e., letJ < β. Then, after recursively applying Eq. (7.42) and the inequalityk[A,B]k≤ 2kAkkBk (valid for bounded oper- atorsA andB [Bha97]), we have B (n) X ≤ n−1 Y i=0 (2τ i β) B (0) X =J n−1 Y i=0 (2×4 i τ 0 β) = 2 n 2 (βτ 0 ) n J. (7.46) Because of the factor of1/2 inB (n) Y [compare Eqs. (7.42),(7.43)] we also have: B (n) Y ≤ B (n) X for sufficiently largen. (7.47) Therefore, at the final concatenation leveln =n f h (n f ) ≤ 2 n 2 f (βτ 0 ) n f J. (7.48) 107 The total durationT is given in terms ofτ 0 andn f as: T =τ n f = 4 n f τ 0 =Nτ 0 . (7.49) Let us now give the connection between the fidelity and the error phase. The propa- gator corresponding to the whole sequence is given by exp h −iτ n f H (n f ) e i , and the ideal evolution of the system is given by the identity operator. If fidelity is measured as the “state overlap between the ideal and the decoupled evolution”, we can write [BT05]: f≈ 1− TH (n f ) 2 = 1− τ n f h (n f ) 2 = 1−Φ 2 CDD , (7.50) whereH refers to the traceless part ofH that acts trivially on the system. Generally, a different universal DD pulse sequence as the basic cycle of concate- nation will modify Eqs. (7.42)-(7.43) but the (asymptotic) form of Eq. (7.48) remains the same. The overall error phase Φ CDD = Th (n f ) can be bounded from above using Eqs. (7.48),(7.49): Φ CDD ≤ (βT/N 1/2 ) log 4 N (JT). (7.51) We expect the above bound to be satisfied provided the convergence condition βT/N 1/2 < 1, i.e.,β < (Tτ 0 ) −1/2 , is satisfied. However, this condition is less strict than the condition appearing in Sec. 7.8, for the truncation of the Magnus expansion:β≪ 1/T [Eq. (7.74)]. For comparison, we also estimate Φ PDD = Th (1) , i.e., simply take n f = 1 in Eq. (7.48): Φ PDD = 2(βτ 0 )(JT) = 2(βT/N)(JT). (7.52) Indeed, this agrees with Eq. (7.36). Note that forN = 4, and taking the equality signs in Eq. (7.51), we have as expected Φ CDD = Φ PDD . Now, we may conclude, by comparing 108 Eqs. (7.51) and (7.52), that when β < (Tτ 0 ) −1/2 , Φ CDD converges quickly to zero as the number of pulses increases, while no such convergence is observed for PDD over the same total sequence duration. In particular, while the fidelity gain in PDD scales with the number of pulsesN, it scales with (N 1/2 /c) log 4 N (≫N forN≫ 1) for CDD, where c = βT is a small constant regulated by the actual convergence domain. On the other hand, this convergence domain puts a physical upper limit on the number of concatenation levels, imposed byβ < (Tτ 0 ) −1/2 or the stricterβ≪ 1/T (c≪ 1). Another way to compare CDD and PDD is as follows. By fixing the value ofc and β, we can back out an upper concatenation level: n max f =−log 4 (βτ 0 /c). Inserting this into Eq. (7.51) we have: Φ CDD ≤ (cβτ 0 ) − 1 2 log 4 βτ 0 c (JT). (7.53) We can now compare the CDD and PDD bounds: Φ CDD Φ PDD ≤ (cβτ 0 ) − 1 2 log 4 βτ 0 c 2(βτ 0 ) βτ 0 →0 −→ 0, (7.54) which serves to show that CDD is indeed superior to PDD in the (relevant) limit of smallβτ 0 . This key result was first reported in our previous study [KL05b], without a full proof. When the dynamics is dominated by direct system-environment coupling, namely β≪J [case (ii)], we find that the third order Magnus term dominates the effective pure bath term. We find that the effective coupling is then bounded by: h (n f ) ≤c n 2 f (τ 0 max(β ′ ,β)) n f J 4 forn f > 2 (7.55) 109 wherec =O(1) and β ′ =O(τ 2 0 J 3 ) (7.56) is an effective pure-bath term that arises in the 3rd order Magnus expansion and kicks in at the second level of concatenation. The asymptotic behavior in the effectiveness of dynamical decoupling is thus the same as case (i) [compare to Eq. (7.48)], however, due to the dependence on a higher power ofJ, we see that this is a more favorable scenario for CDD. Any universal decoupling sequence (e.g., higher order sequences) can be concate- nated and our analysis still applies. When this is done, Eq. (7.51) becomes Φ CDD ≤ Φ 0 (αN −a ) logN N b (7.57) whereΦ 0 is the error phase for a free evolution of the system for timeT . The parameter α is generally bounded by a power ofkH e kT and is required to be small [it is analogous toβT in Eq. (7.51)]. The parametersa andb areO(1). The parametersα,a,b all depend on the basic decoupling cycle used and are all positive. 7.4.4 Finite Width We can analyze the finite-width pulse CDD procedure by using the pulse error unitary operators introduced in subsection 7.3.3. The systematic error associated with these pulses at each level leads naturally to a decoupling error, and is corrected at the next level of concatenation. For convergence, we require the coupling strength to shrink as a function of concatenation level. We obtain the following condition for convergence of the concatenation procedure for rectangular pulses, derived in Sec. 7.9: c ′ τ 0 β +d ′ δ τ 0 < 1. (7.58) 110 where c ′ and d ′ are numeric factors of O(1). This inequality is a special case of Ineq. (7.28), so the latter is already sufficient to guarantee convergence. As men- tioned there, the finite-width convergence condition implies an optimal pulse interval τ 0 = p d ′ δ/c ′ β, for givenβ and a fixed minimal pulse width δ. We reiterate that we have requiredδ≪ τ 0 , but we expect this requirement to be inessential as evident from the exact numerical simulations reported in [KL05b]. For example when a technolog- ical lower limit on the smallest switching times was used, we obtained a “closed-pack sequence” withδ≈τ min that still provides significant decoupling. Generally, errors associated with non-zero pulse widths are a combination of sys- tematic and random pulse errors. By construction, the recursive nature of CDD tolerates a significant level of systematic pulse errors, since the decoupling error at each level is “cleaned up” at the next level of decoupling. Random pulse errors are tolerated to some extent as well [KL05b]; this is reminiscent of randomized decoupling techniques where pulses implementing random unitary operators on the system are utilized for decoupling. In fact randomized decoupling techniques have been shown to be efficient in the limit of fast time-varying/fluctuating system-bath Hamiltonians for which the deterministic methods (PDD/CDD) are relatively ineffective [SV06]. 7.4.5 Example: Decoupling in Spin Quantum Dots In this subsection we apply our analysis to the case of an electron spin in a quantum dot coupled via the hyperfine interaction to a bath of nuclei. The Hamiltonian for the 111 interaction of an electron with spinS (the qubit), confined in a semiconductor quantum dot, with a collection of nuclear spinsI n , is given by: H = H S +H SB +H B = ΩZ + X n A n I z n ⊗Z + X n<m B nm [I n I m ] (7.59) whereA n andB nm are coupling constants andB nm [I n I m ] is a shorthand for the diago- nal and off-diagonal dipolar coupling terms among the nuclei. The intra-nuclear dipolar coupling is short-ranged (B nm ∝ r −3 nm ). The hyperfine interaction between the elec- tron spin and the nuclei is a Fermi contact interaction, whose magnitude is given by A n , which is proportional to the electronic wave function magnitude at the position of nucleus n. The parameters relevant to our bounds are given by J = O(I P n A n ) and β = O(I 2 P n<m B nm ), where I(= O(1)) denotes the nuclear spin. The num- ber of nuclei within the radius of the electron wavefunction is of the order of 10 5 and for GaAs we have β ∼ 10kHz, and J ∼ 1MHz [IAM02, CGEL06]. Let us define an error rate as e≡ Φ/T for an evolution of length T with an error phase Φ. The uncorrected (pulse-free) error ratee 0 = Φ/T isO(J)∼ 1MHz. Let us take our inter- pulse interval τ 0 = T/N, to be smaller than 1μs. At any pulse rate faster than this, PDD provides improvement: e PDD = J 2 τ 0 . 1MHz [Eq. (7.36)]. Note that this error rate is fixed and applying more pulses at this rate only retains it, resulting in an abso- lute error that grows linearly with time. For a detailed example of this improvement see [SSW05] where a Carr-Purcell-Meiboom-Gill (CPMG) [CP54, MG58] sequence is used to decouple the electron spin from the nuclei (we give a concatenated version of the CPMG sequence in Sec. 7.10). 4 In a system-specific example such as that of 4 The CPMG sequence is a simple dynamical decoupling sequence consisting of fixed periodic spin flips corresponding toX pulses in our notation. See Sec. 7.10 for a detailed discussion. 112 Refs. [SSW05, YLS07, WS07], many of our conservative bounds on the commuta- tors appearing in the effective Hamiltonian can be improved.These works incorporate stochastic assumptions on the mesoscopic structure of the bath or use explicit construc- tions of higher order corrections to the dynamics of the bath. In contrast, we have used a simplified spin chain model. The sequence used in [YLS07] is in fact a concatenated CPMG sequence (See Sec. 7.10) and the analysis in Refs. [WS07, SSW05] corresponds to a periodic application of the CPMG sequence. Our example in this case serves as a theoretical demonstration using the underlying quantum Hamiltonian rather than a more accurate treatment utilizing the phenomenology of the mesoscopic bath. The error rate e CDD for CDD is given by dividing Eq. (7.51) by T (or more gen- erally by Eq. (7.57)), which decreases super-polynomially with the number of pulses used. Note that, while initially we start in a regime where β ≪ J for low levels of concatenation, the effective (renormalized) value ofJ quickly decreases with increas- ing concatenation level, and we are in a limit whereβ (or the effectiveβ ′ [Eq. (7.56)]) dominates through the undecoupled error term. Finally, we note that as long as pulse widths that are negligible with respect to our pulse intervals are used, the condition for convergence of concatenation [Eq. (7.58)] is satisfied in our example. We performed a numerically exact simulation for comparing different pulse sequences for a qubit coupled to a small environment. We consider a spin chain of length N, where the central spin is from a different species (the electron in the dot versus the surrounding nuclei) and acts as the system qubit. The (undesired) couplings between the spins is given by Heisenberg interaction terms, so that the coupling Hamiltonians between two spinsi andj at a distanced (measured in units of the lattice constant) is given by c 2 d (X i X j +Y i Y j +Z i Z j ), wherec depends on the species of the spinsi and j. We fix the coupling strengths and the number of spins so that the coupling strengths roughly correspond to GaAs our example: J = 1MHz andβ = 10KHz. The effective 113 12 34 56 7 8 -150 -125 -100 -75 -50 -25 0 PDD (ideal pulses) CDD (ideal pulses) log (N) log (1-purity) 10 4 log ( /s ) ¿ n 10 -5 . 6 -6 . 2 -6 . 8 -7 . 4 -8 . 0 -8 . 6 -9 . 2 -9 . 8 Figure 7.1: (Color online) Simulation of PDD (squares) and CDD (circles) for concate- nation levels 1 to 8. Results shown are for one minus purity of the system qubit as a function of the number of pulses. The pulses used are ideal. Note the fantastically high purity achieved with CDD. In the case of PDD the horizontal axis giveslog 4 of the total number of pulses, while for CDD the numbers on the horizontal axis denote the con- catenation level at the (bottom). The shortest pulse intervalτ n =T/4 n is also indicated (top). The total length of the sequence is fixed atT . errors at the end of the concatenated cycles are so low (purity loss of around 10 −100 – see Fig. 7.1) that we are forced to use extremely high precision linear algebra that sig- nificantly burdens simulation performance. To get practical results, we argue based on Lieb-Robinson bounds (see [Osb06]) that within the duration of the simulation, enlarg- ing the spin chain will not modify the results. This was verified independently for a test case where there was no significant qualitative difference between N = 3,5,7. Therefore the results presented here are forN = 3 and they capture the essence of our estimates. A more extensive numerical study of decoupling of an electron spin in quan- tum dots was performed in [ZDS + 07], where a more realistic bath is considered and various DD methods are compared and contrasted. 114 01 23 4 56 7 8 log (N) -12 -10 -8 -6 -4 -2 0 log (1-purity) PDD, pulse width=100ps PDD, pulse width=10ps PDD, pulse width=1ps CDD, pulse width=100ps CDD, pulse width=10ps CDD, pulse width=1ps 10 4 log ( /s ) ¿ n 10 -5 . 6 -6 . 2 -6 . 8 -7 . 4 -8 . 0 -8 . 6 -9 . 2 -9 . 8 Figure 7.2: (Color online)Simulation of PDD (squares) and CDD (circles) for concate- nation levels 1 to 8 with non-ideal pulses. The pulse widths are set at 1ps, 10ps, and 100ps. Axes as in Fig. 7.1. We fix the overall duration of the pulse sequence at 10 −5 s so that βT < 1 [see Eq. (7.28)]. This allows us to concentrate on the inter-pulse period and pulse-widths that are the main technological challenge for decoupling. Higher concatenation levels thus correspond to log(number of pulses) [or to log(1/inter-pulse period)]. In Fig. 7.1, we compare CDD and PDD purities for ideal pulses at various levels of concatenation. The vertical axis displays the loss of purity of the system qubit, i.e.,1−Tr[(ρ S ) 2 ], where ρ S = Tr B (ρ), andρ is the joint system-bath density matrix at the final timeT . The higher order data for PDD are obtained by simply repeating the basic sequence and shrinking the pulse-interval. This is necessary to get the long-term behavior of decoupling. The graphs show a progressive improvement in purity of the initial qubit state, |0i+|1i √ 2 , as a function of concatenation level. The environment part of the spin chain is initialized in a thermal state at a temperature of 1K. In Fig. 7.2 we have depicted the effect of 115 realistic pulse widths on decoupling. We note that in the case of PDD the performance of the pulse sequence deteriorates with increasing pulse width (as expected), but also (after an initial improvement, as in the ideal pulse case) as a function of the number of pulses used. This latter deterioration is somewhat surprising, and is due to the pulse width errors that simply accumulate over time. The point of deterioration shifts to the right as the pulse width is made smaller, as expected. The improvement seen for the 100ps case at log 4 (N) = 7 can be understood as being due to the essentially random- walk-on-a-circle-like nature of the accumulated errors, which will occassionally result in a recurrence. Concatenated sequences are naturally robust against these errors, as can be seen clearly in the figure, where CDD results in a saturation of the purity. The asymptotic purity level is roughly equal to the square of the per-pulse-errorδJ (recall subsection 7.3.3). For example, withδ = 1ps andJ = 1MHz we find(δJ) 2 = 10 −12 , in agreement with Fig. 7.2. While it is impossible to go to error rates below per-pulse-error, with concatenation we are able to maintain the error rate at this minimum. 7.5 Higher Order (Trotter-Suzuki) Universal Decou- pling Suppose a universal dynamical decoupling sequence is known, i.e., essentially a series of unitarily transformed HamiltoniansH j = P j H e P † j such that P j P j H e P † j = B⊗I S for some environment operatorB. As we saw previously, the sequential application of propagators generated by these Hamiltonians acts trivially on the system only up to the first order inkB α kτ 0 whereτ 0 is a typical free-evolution period duration. Trotter-Suzuki expansion allows us to construct a sequence of these propagators that act trivially on the system up to any ordern. Suppose{ǫA j } k j=1 are dimensionless Hermitian operators 116 such that P A j = A andǫ is a small parameter. Trotter-Suzuki expansion allows us to find a sequence of indices{n i } N i=1 (n i ∈{1...k}) and real numbers{c i } N i=1 such that e ǫc 1 An 1 e ǫc 2 An 2 ...e ǫc N An N =e ǫA+O(ǫ n ) (7.60) The number N of propagators required, scales with e n 2 . It is worth emphasizing that while numerous, the sequence of coefficientsc i and indicesn i can be constructed recur- sively [Suz76, HS05]. For practical purposes it is advantageous to use a variation on this expansion in which the coefficients c i are rational numbers. Once the sequence of propagators is known we can construct the Trotter-Suzuki decoupling sequence (TSDS) accordingly: Set up A j = iH j and A = iB⊗I based on the universal DD cycle and employ the following DD sequence using the smallest pulse switching timeτ min available: P n 1 f[c 1 τ min ]P † n 1 P n 2 f[c 2 τ min ]P † n 2 ...P n N f[c N τ min ]P † n N , (7.61) where f[τ] denotes a free evolution period of duration τ. By dimensional analysis, the error in Eq. (7.60), ǫ n , translates into left-over terms asymptotically bounded by products of operators B α and is thus bounded by O[(τ min maxkB α k) n ]. We can thus asymptotically estimate the error phase as: Φ TSDS = O[(τ min max{kB α k}) n ] = O h (T max{kB α k}/N) √ logN i , (7.62) A serious problem is that negative c i ’s routinely appear in the TSDS (except for n = 3). Clearly, this presents a major problem since one cannot have negative times in the free evolution segments in Eq. (7.61). For this reason, as presented the TSDS is 117 DD Scheme Φ forJ <β Φ forβ≪J Periodic DD T 2 βJ/N T 2 J 2 /N Concatenated DD (βT/N 1/2 ) log 4 N (JT) N(JT/N) γlog 4 N Trotter-Suzuki (βT/N) √ log 4 N (JT/N) √ log 4 N Table 7.1: Asymptotic comparison of the error phase Φ for deterministic decoupling schemes. T is the total experiment duration,N the total number of pulses,J andβ are defined in Eq. (7.18). The constantγ =O(1) depends on the basic cycle used. not physically implementable. A solution for this problem is not yet available to us, but does not seem impossible. Table (7.1) summarizes the asymptotic performance of PDD, CDD and TSDS in the limit of large number of pulses and the two regimes ofJ <β andβ≪J. The TSDS performs remarkably well and is oblivious to the distinction between pure-bath and system-bath dynamics. Nonetheless, its performance is sensitive to small errors in pulse operation and the switching times that need to be precisely set (not required in PDD and CDD and randomized decoupling schemes). Thus it may not be a robust alternative to CDD, in spite of its superior convergence properties, but it may be used as a basic dynamical decoupling sequence at the base of a new concatenated pulse sequence. This is particularly useful with the TSDS atn = 3. Finally, we note that the Trotter-Suzuki expansion was also recently used by Brown et al. in a study of arbitrarily accurate composite pulse sequences [KB04]. There the goal was to overcome systematic errors in the system control Hamiltonian, without con- sidering decoherence. 118 7.6 Decoupling with Very Narrow Pulses Cannot Increase Error Norms In this section we return to DD with ideal, zero width pulses, and argue that such DD sequences can never effectively strengthen the undesired terms and cannot cause extra errors. We then argue that even with finite-width, but sufficiently narrow pulses, error norms cannot increase under DD. These results are of independent interest and apply to any pulse-based error suppression strategy, including closed-loop quantum error correc- tion. Consider a sequence of ideal unitary operations P i applied to a quantum system (measurements can also be included by enlarging the Hilbert space), and suppose a sequence of intervalsτ i separates these operations, such that the pulse sequence is given by: P 1 f τ 1 P 2 ...f τ n−1 P n . In the absence of environmental couplings, the overall effect of the pulse sequence would be given byQ n =P 1 ...P n – the “ideal operation”. But in the presence of the environmental couplings inH e , the overall propagatorU is modified by error terms. LettingQ i =P 1 ...P i , we can define an effective unitary error operator corresponding to the whole sequence in the following manner: U =P 1 e −iτ 1 He P 2 e −iτ 2 He P 3 ...P n−1 e −iτnHe P n =Q 1 e −iτ 1 He Q † 1 Q 2 e −iτ 2 He Q † 2 ...Q n−1 e −iτ n−1 Q † n−1 Q n =e −iτ 1 Q 1 HeQ † 1 e −iτ 2 Q 2 HeQ † 2 ...e −iτ 1 Q n−1 HeQ † n−1 Q n =: exp(−iTH ′ e )P 1 ...P n (7.63) Note that in the last line we have defined the overall durationT = P τ i , and isolated the effective error HamiltonianH ′ e from the desired, ideal unitary operationQ n =P 1 ...P n . 119 As before, we use a unitary operator normk.k to compare the strength of the error HamiltoniansH e andH ′ e . Unitary operator norms [Bha97], such as the absolute differ- ence between the largest and smallest eigenvalues, are invariant under unitary transfor- mations (for unitaryU,kUAU † k =kAk) and can be used as measures of fidelity errors in Hamiltonian error correction theory [BT05]. We now use the following existential theorem due to Thompson [Tho86, CHN03]: LetA 1 ,A 2 be Hermitian matrices. Then there exist unitariesU 1 andU 2 and a Hermitian matrixA such that e iA 1 e iA 2 =e iA ; A =U 1 A 1 U † 1 +U 2 A 2 U † 2 . This theorem can be extended (induction on the number of exponentials) to products involving more than two matrix exponentials:e iA 1 ···e iAn =e iA . Using Eq. (7.63) and Thompson’s theorem we have: TH ′ e = P i τ i U i Q i H e Q † i U † i , and we have the following inequality for the norm ofH ′ e : kH ′ e k = 1 T k X i τ i U i Q i H e Q † i U † i k ≤ 1 T X i τ i kU i Q i H e Q † i U † i k =kH e k (7.64) Thus the norm of the effective error Hamiltonian does not increase under the action of ideal unitary operators. In the case of non-ideal pulses carrying systematic errors per pulse (e.g., due to finite pulse-width), we can model the pulse error as a unitary error operator immediately preceding the ideal pulse. For single-qubit pulse errors due to finite pulse widths, one can again show thatkH ′ e k≤kH e k provided the pulse widths are small enough. This allows us to use Thompson’s theorem to show that our argument in this section also 120 applies to pulses of sufficiently narrow width on a single qubit. We expect this argument to apply to multi-qubit near-perfect operators in the presence of a bounded bath. As a special case, this argument applies to dynamical decoupling. While positively reassuring that with ideal pulses the undesired couplings do not increase in strength, the present argument does not quantify the efficiency of dynamical decoupling. For this purpose we employed, above, approximations based on the Magnus expansion. The same argument applies in a quantum error correction codes setting [KLZ98b], in particular in non-Markovian fault-tolerance theory [AAK06, AGP06, BT05], where a “time-resolved fault path” expansion has recently been used to decompose the action of errors in the course of a general evolution. Our argument can be used to further rationalize such expansions based on the fact that errors are well-behaved (in the sense ofkH ′ e k≤kH e k) in the non-Markovian regime. 7.7 Summary and Discussion Dynamical decoupling (DD) cannot be exactly analyzed without concrete reference to the details of the system-environment coupling, but an abstract picture of the interaction in terms of bounded environment operators – as pursued here – can yield useful per- formance estimates. Within this framework, we have provided an analytic estimate of the leading order decoupling error associated with the basic universal decoupling cycle for a qubit. We have analyzed and compared the performance of periodic DD (PDD) and concatenated DD (CDD) schemes. We have provided detailed calculations sup- porting the conclusion reported in [KL05b], that CDD significantly outperforms PDD within practical boundaries of pulse parameter space. We have distinguished between two different limiting cases of fast versus slow environment dynamics. Fast dynam- ics of the environment limits the performance of higher order deterministic dynamical 121 decoupling. This can be understood from the interaction picture, where the system-bath interaction Hamiltonian is fast fluctuating. Slow bath dynamics, on the other hand, can be exploited by CDD to result in super-exponential decoupling using an exponential number of pulses. Table 7.1 provides a convenient summary of the relative performance of PDD and CDD, as well as the new Trotter-Suzuki based pulse sequence we have introduced. Our discussion was based on a pulsed control mode, but it is known that higher fidelities are possible via fine-tuned navigation of the control Hamiltonian [VK03]. This direction can be especially useful when decoupling methods are to be used not for quantum state preservation but for performing an error-corrected quantum evolu- tion. Dynamically error corrected evolution can also be achieved via a hybrid decou- pling error correction method [BL02, BL03, KL03]: Consider a stabilizer quantum error correcting code characterized by a stabilizer group of Pauli operators [CRSS97, Got97]. Assume that this code corrects all single qubit errors, which means that each single qubit error anticommutes with at least one stabilizer generator. This anti-commutation condition translates into time reversal of the error, when exponentiated. Thus it is pos- sible to use the generator set of the stabilizer to form a universal decoupling sequence for decoupling single qubit error terms on the code space. In this way decoupling inte- grates seamlessly with the encoded quantum operations on the code space generated by Hamiltonians written as the sum of normalizer elements of the code. This also allows us to perform quantum error detection and recovery within a hybrid decoupling-error correction setting, thus allowing the correction of errors in both the Markovian and the non-Markovian regimes. The open-loop approach of DD appears at first sight to be conceptually and prac- tically very different from the method of closed-loop quantum error correcting codes [Ste99]. However, recent results on the theory of non-Markovian fault tolerant quantum 122 error correction (FTQEC) [BT05, AGP06, AAK06] suggest that the error bounds appli- cable in the fault tolerant, concatenated version of DD are in fact very similar to the bounds relevant to FTQEC. Specifically, in both CDD and FTQEC it is essential that the system-bath interaction Hamiltonian is norm-bounded (see Refs. [ALZ06, Ali07] for a critique of this assumption). In light of the much smaller degree of overhead involved in CDD, this suggests that in the non-Markovian regime one can profit significantly by incorporating CDD into a closed-loop QECC procedure. CDD can then remove the leading order bath-induced errors, while the QECC procedure can target primarily the random control errors for which CDD offers only limited protection. We note that it is possible to view the CDD procedure as a discrete time dynamical system, whose ideal fixed point is a vanishing system-bath interaction. In this manner it should be possible to characterize the region of correctable errors using tools from the analysis of fixed points, and to incorporate perturbations of the pulse sequence. An analysis of CDD from this perspective may well be a fruitful endeavor. Indeed, there exists a dynamical maps approach to concatenated quantum error correction, which has proven to be very convenient in the analysis of the fault tolerance threshold [RDM02, FKSS06]. As a final comment we should emphasize that the ability to perform arbitrarily pre- cise Hamiltonian control on the system Hamiltonian assumes a classical control mecha- nism, while every control system is really quantum in nature. Thus besides technological constraints, fundamental quantum fluctuations may limit the performance of DD as well, since perfect classical control simply does not exist. A systematic characterization of bounds on the fidelity of feedback-free error correction schemes such as DD, imposed by fundamental quantum fluctuations, is still an important open question. 123 7.8 Addendum: Validity of the Magnus Expansion We analyze the convergence domain of approximations and the dynamical renormaliza- tion process, specifically as it applies to CDD. This recursive renormalization can be written as: U (n+1) := exp(−iτ n+1 H (n+1) e ) = 4 Y j=1 exp(−iτ n H (n) j ) (7.65) whereτ n+1 = 4τ n , i.e., τ n = 4 n τ 0 , (7.66) and where H (n) 1 =B (n) 0 ⊗I +B (n) X ⊗X +B (n) Y ⊗Y +B (n) Z ⊗Z =IH (n) e I, H (n) 2 =B (n) 0 ⊗I +B (n) X ⊗X−B (n) Y ⊗Y−B (n) Z ⊗Z =XH (n) e X, H (n) 3 =B (n) 0 ⊗I−B (n) X ⊗X +B (n) Y ⊗Y−B (n) Z ⊗Z =YH (n) e Y, H (n) 4 =B (n) 0 ⊗I−B (n) X ⊗X−B (n) Y ⊗Y +B (n) Z ⊗Z =ZH (n) e Z, (7.67) are the recursive generalization of Eqs. (7.8). The Magnus expansion ofU (n) yields: U (n) = exp(−iτ n H (n) ) = exp( ∞ X i=1 A (n) i ), (7.68) whence τ n H (n) =i[A (n) 1 +A (n) 2 ]+τ n C (n) , (7.69) 124 where A (n) 1 =−iτ n−1 4 X i=1 H (n−1) i , (7.70) A (n) 2 =− 1 2 τ 2 n−1 X 1=i<j=4 [H (n−1) j ,H (n−1) i ], (7.71) −iτ n C (n) = ∞ X i=3 A (n) i = ∞ X i=3 m i τ i n−1 [H (n−1) j ,[H (n−1) k ,...]] | {z } i commutators , (7.72) wherem i is a numerical factor determined by explicit computation of theith order Mag- nus expansion, and C (n) is an operator-valued correction to the second order Magnus expansion. We would like to find an approximation for the B (n) α . To do so we will first show that it is consistent to use the second order Magnus expansion forH (n) e , in the sense that H (n) 1 =H (n) e ≈ i τ n (A (n) 1 +A (n) 2 ) =: X α=0,X,Y,Z ˜ B (n) α ⊗S α , (7.73) where we can safely neglectC (n) [i.e., allA (n) i>2 ] as long as τ n β≪ 1. (7.74) This is the recursive generalization of the result obtained above forn = 1. Then the ˜ B (n) α will be the desired approximation toB (n) α . The proof is by induction. We will require the following inequalities, satisfied for bounded operatorsA andB [Bha97] : k[A,B]k ≤ 2kAkkBk, (7.75) kABk ≤ kAkkBk, (7.76) kA+Bk ≤ kAk+kBk. (7.77) 125 Lemma 1 The following relations hold: ||C (n) ||≪ 1 τ n ||A (n) 2 || (7.78) and || ˜ B (n) α || = O(β), α =X,Y,Z (7.79) B (n) 0 = B 0 . (7.80) Proof We prove the lemma by induction. Let us call Eq. (7.78) “a(n)”, Eq. (7.79) “b α (n)”, and Eq. (7.80) “b 0 (n)”. We have already established the case a(1) [Eq. (7.19)], and b α (1) and b 0 (1) are based on our definitions and assumptions. We will show that (1) b α (n−1) & b 0 (n−1)⇒ a(n), (2) b 0 (n−1)⇒ b 0 (n), and then (3) a(n) & b α (n−1)⇒ b α (n). Recall all along that we have assumedJ <β. (1) Proof of a(n): We have, using Eqs. (7.71),(7.72) and inequality (7.75) τ n ||C (n) ||≤ ∞ X i=3 m i τ i n−1 ||[H (n−1) j ,[H (n−1) k ,...]] | {z } i commutators || =O ∞ X i=3 τ i−2 n−1 (||H (n−1) j ||||H (n−1) k ||···) | {z } i−2 terms ||A (n) 2 || bα (n−1)& b 0 (n−1) = O ∞ X i=3 τ i−2 n−1 β i−2 ||A (n) 2 || ! τ n−1 β≪1 = O(τ n−1 β||A (n) 2 ||)≪||A (n) 2 || 126 (2) Proof of b 0 (n): A (n) 1 =−iτ n−1 4 X i=1 H (n−1) i =−i4τ n−1 B (n−1) 0 ⊗I b 0 (n−1) = −iτ n B 0 ⊗I. (3) Proof of b α (n): From Eq. (7.73) we have H (n) e = X α=0,X,Y,Z B (n) α ⊗S α a(n) ≈ i τ n (A (n) 1 +A (n) 2 ) = X α=0,X,Y,Z ˜ B (n) α ⊗S α . Since A (n) 2 contains no pure-environment terms, it determines the part contributing to the sum overα =X,Y,Z: k X α=X,Y,Z ˜ B (n) α ⊗S α k = 1 τ n ||A (n) 2 || Eq. (7.71) = 1 τ n || 1 2 τ 2 n−1 X 1=i<j=4 [H (n−1) j ,H (n−1) i ]|| Ineq. (7.75)-(7.77) ≤ τ 2 n−1 4τ n−1 X 1=i<j=4 ||H (n−1) j ||||H (n−1) i || bα (n−1) = O(τ n−1 J 2 )<O(τ n−1 β 2 )<O(β). Since the system operators all have||S α || = O(1) and the environment operators ˜ B (n) α all have similar norm, we can conclude that, as required,|| ˜ B (n) α || =O(β). The upshot of this proof is the following: Corollary 1 The recursive second-order Magnus expansion, Eq. (7.73), is a valid approximation provided we assumeτ n β≪ 1. 127 The conditionτ n β≪ 1 of course puts a physical upper limit on the number of levels of concatenation. Provided this condition is satisfied, it follows that, schematically, we have H e ,τ 0 7→H (1) e ≈ i τ 1 (A (1) 1 +A (1) 2 ) =: X α ˜ B (1) α ⊗S α , H (1) e ,4τ 0 7→H (2) e ≈ i τ 2 (A (2) 1 +A (2) 2 ) =: X α ˜ B (2) α ⊗S α , . . . H (n−1) e ,4 n−1 τ 0 7→H (n) e ≈ i τ n (A (n) 1 +A (n) 2 ) =: X α ˜ B (n) α ⊗S α . (7.81) Note that in the body of the chapter, for notational simplicity we dropped the tilde, with the convention being that we are only considering the environment operators defined by the second-order Magnus expansion. 7.9 Addendum: Finite Pulse Width Analysis for CDD For brevity define β (n) α :=kB (n) α k, β α :=β (0) α , β :=β 0 , 128 and use Ineqs. (7.75)-(7.77) to reproduce the recursive inequalities corresponding to β (n) α from Eqs. (7.29)-(7.32): β (n) 0 =β, (7.82) β (n) X ≤ 2(τ n−1 −δ)ββ (n−1) X + δ τ n−1 ( 1 2 β X + 1 π β Y ) (7.83) β (n) Y ≤ (τ n−1 −δ)ββ (n−1) Y +(τ n−1 −2δ)β (n−1) X β (n−1) Z + 1 π δ τ n−1 β Z (7.84) β (n) Z ≤δ 2 π β (n−1) X (β Z +β X )+β (n−1) Y β X + δ τ n−1 β Z . (7.85) A necessary condition for convergence is given byβ (1) α < β (0) α ≡ β α forα = X,Y,Z. Let us define constantsa,b such that β Y =aβ X , β Z =bβ X . (7.86) Then we have for then = 1 case of Eq. (7.83) β (1) X ≤ 2(τ 0 −δ)ββ X + δ τ 0 ( 1 2 + a π )β X , (7.87) which must be smaller thanβ X . We thus find the necessary condition 2(τ 0 −δ)β + δ τ 0 ( 1 2 + a π )< 1. (7.88) Next consider then = 1 case of Eq. (7.85), and set it to be smaller thanβ Z : β (1) Z /β Z ≤ (δβ X ) 2 π (1+ 1 b )+ a b + δ τ 0 < 1. (7.89) 129 Both quantities δβ X ,δ/τ 0 are≪ 1 by our previous assumptions, so this inequality is automatically satisfied. Finally, consider then = 1 case of Eq. (7.84), and set it to be smaller thanβ Y : β (1) Y /β Y ≤ (τ 0 −δ)β +(τ 0 −2δ) b a β X + 1 π δ τ 0 b a < 1. (7.90) We can simplify these results somewhat, as follows. We can replace β X by J = max(kB X k,kB Y k,kB Z k)<β [Eq. (7.18)], and assume for simplicitya =b = 1. We can then replace Ineqs. (7.88),(7.90) by 2(τ 0 −δ)β +( 1 2 + 1 π ) δ τ 0 < 1 (7.91) (τ 0 −δ)(β +J)−δJ + 1 π δ τ 0 < 1. (7.92) We can safely replace J by β (since J < β) and drop δJ . This turns the second inequality into2(τ 0 −δ)β + 1 π δ τ 0 < 1, so it is subsumed by the first inequality. 7.10 Addendum: Concatenation of the CPMG pulse sequence Consider an error HamiltonianH e of the form H e =B z ⊗Z +B 0 ⊗I S . A simple dynamical decoupling sequence (also known as CPMG [CP54, MG58]) can decouple this error Hamiltonian: p 1 =XfXf 130 where f denotes a free evolution interval of durationτ. The propagator corresponding to this sequence is given by: U =e −iτ(−Bz⊗Z+B 0 ⊗I S ) e −iτ(Bz⊗Z+B 0 ⊗I S ) . A simple application of the Baker-Campbell-Hausdorf (BCH) [Rei00] expansion shows thatU can be written as U =e −i2τ(B 0 ⊗I S +F[B 0 ,B Z ]⊗Z) where F[B 0 ,B Z ] is a Hermitian operator in the Lie sub-algebra generated by B 0 and B Z . One can show thatkF[B 0 ,B Z ]k =O(kB 0 kkB Z kτ) in the limit ofτ→ 0. We can thus use the same sequence for decoupling the undecoupled term F[B 0 ,B Z ]⊗Z and construct 2nd and 3rd order CDD sequences: p 2 = fXffXf (7.93) p 3 =XfXffXfXfXffXf (7.94) Notice that concatenated CPMG requires far fewer pulses than concatenated universal DD. However, CPMG is not as robust with respect to systematic errors in the pulses as concatenated universal DD. 131 Chapter 8: Self-Correcting Single Qubit Unitary Operations 8.1 Introduction The ability to control and measure the evolution of large quantum systems is an immense resource and the efficiency in manipulating quantum evolutions is at the heart of quan- tum information processing. Technologies such as quantum computation and communi- cation are based upon such a premise. A quantum system may be controlled via control- lable parameters that characterize its Hamiltonian. In this setting, a quantum evolution can be realized by the unitary propagator of the system and is navigated by the control parameters of the Hamiltonian. For example, if the system is a spin qubit interacting with a controllable (in direction and magnitude) magnetic field, the parameters of this field can be controlled in a way that an arbitrary unitary evolution of the system is real- ized. In the first place, constraints on the controllable parameters of the system will limit our ability of efficiently controlling the system dynamics. For example, realisti- cally we cannot allow for infinitely strong magnetic fields switched over infinitesimally short periods. The description of what a quantum system can be made to do and how efficiently it may be done is the subject of quantum control [RR96]. In this chapter we look at another related physical constraint on the realization of quantum operations: the unavoidable and systematic interactions with the environment. In the context of quan- tum error correction, the interactions due to the environment are typically responsible for errors affecting the fidelity of operations and periods of “no operation” (NOOP). The dominant proposal for dealing with these errors in quantum error correction theory is based on the non-unitary open systems dynamics and the theory of error correcting 132 codes: an encoding of the quantum system into a higher dimensional Hilbert space, usu- ally combined with measurements is used to “correct” errors [KL97b]. This is a high level approach that utilizes the available quantum processing methods empowering the encoding and recovery. However, in certain error regimes a basic Hamiltonian descrip- tion of the errors can be used to design error correcting strategies. In contrast, this is an example of a low level approach, where the correction is performed at a more fundamen- tal physical level. Dynamical decoupling is the prominent example of such an approach where the free-evolution of the quantum system is corrected by simply applying ideal system unitary operators that result in an effective Hamiltonian that is approximately error-free [See chapter 3 for a review.]. Dynamical decoupling can preserve an arbitrary quantum state but in contrast to quantum error correcting codes is not capable of cor- recting quantum operations per se as it is designed to universally cancel all dynamics in its basic form. In this chapter, in a spirit similar to dynamical decoupling methods and the so called “composite pulse” techniques [LF81, CJ00, KB04], we show how to effectively cancel the errors of quantum operations and actually produce any desired dynamics as long as this desired dynamics is available in the absence of the errors. This is done without any assumptions on specific quantitative knowledge of the errors for a single qubit. The essence of our method is to treat all errors due to the undesired interactions with the environment as systematic and then try to algebraically formu- late their effects so that the action of the errors is characterized by a set of unknown, yet fixed operators acting on the environment/bath. Nonetheless, we are oblivious to the dynamics and the state of the bath. This results in the after-the-fact randomness in the occurrence of the errors. In a similar setting, methods based on optimization of operations exist for cases where the interaction Hamiltonian of a system plus (parts of) the environment is characterized quantitatively already [E.g., [KBG01, KG01]]. Our 133 method differs from such optimization technique fundamentally by making no assump- tion on the structure of the environmental interactions. On the other hand our approach is fundamentally based on composite pulse techniques that offer concrete advantages in correcting unknown couplings with simple structures. In this chapter, we simply do not take any error structure for granted and consider the most general case and give a recipe for extending it to higher orders. 8.2 Formulation and Characterization of Errors The starting point is the practical decomposition of the Hamiltonian of the quantum system and the environment: H =I B ⊗H S (t)+H B ⊗I S +H SB (8.1) where the subscriptS(B) refers to the system (environment/bath) Hilbert space and the corresponding operators and the subscriptSB refers to operators that act non-trivially on either the system or the environment. The system HamiltonianH S (t) contains the controllable partH c (t). We need to be economic in the choice ofH B with respect to the evolution time of the system. This ensures that we have the right measure for quantifying the strength of various parts of the Hamiltonian via unitarily invariant operator norms. One such norm is given bykAk := max hψ|A|ψi hψ|ψi , where|ψi is a vector in the combined Hilbert space of the system and the bath 1 . Henceforth we drop the identity operators on the system and the bath (I S andI B , resp.) and the tensor product symbol when there 1 This norm is best suitable for comparing operators that act on similar spaces. For example comparing the norms ofH B andH S can be meaningless. 134 is no ambiguity. We can assume that there exists a set of system basis operators S α includingS 0 =I S such that we can write H B ⊗I S +H SB = X α B α ⊗S α (8.2) for a system consisting of many qubits one such basis is given by the products of the Pauli spin operators and identity on each qubit. Consider a unitary quantum operation on the systemI B ⊗U({θ i }) where{θ i } = O(1), are the set of parameters associated with it such as Euler angles in the case of a qubit system. We will drop{θ i } when there is no ambiguity. WhenH SB is not present, and we have access to a universal set of Hamiltonians for control and depending on the constraints of this control, we can use a time-dependent system HamiltonianH S (t) such that U({θ i }) =T + [e −i R δ 0 H S (t)dt ] (8.3) where T + denotes the time-ordering and δ denotes the duration of the operation, the switching time, or the pulse width depending on the specifics of the implementation. For a finite dimensional Hilbert space, and for a given unitary quantum operation there can be many implementations in terms of the controlH S (t) but performing this control in the limit of δ → 0 incurs infinite costs. In a real world implementation, there is always a non-zero minimum switching time for Hamiltonians. Whenδ6= 0 and in the presence ofH SB the same control sequence given byH S (t) will not generateU({θ i }) exactly. We can model the effects of the environment on the unitary outcome of the control sequenceM ({θ i }), as M({θ i },δ) =T + [e −i R δ 0 (H S (t)+H SB +H B )dt ]. (8.4) 135 While sufficiently general for our purpose in this work, one could envisage scenarios where the above assumption is insufficient for modeling the errors due to the interac- tion with the environment such as randomly fluctuating Hamiltonians. Error correction in such scenarios will require active use of redundancies such as quantum error cor- recting codes as well. We use Eq.(8.4) to describe the non-ideal quantum operation in the presence ofH SB and non-zero pulse widthsδ. The “realistic” propagatorM is an approximation for the “ideal” propagatorU as quantified by an error unitaryE, defined as E({θ i },δ) := (U({θ i }) −1 M({θ i },δ) (8.5) where we have: lim δ→0 E({θ i ,δ) =I SB . (8.6) A Taylor expansion ofE({θ i },δ) as a function ofδ up to first order, will depart from identity by an operator with a norm bound byO(ǫ), where ǫ =δk X α B α ⊗S α k (8.7) In fact, sinceE is a unitary operator we can think of it as generated by an “error Hamil- tonian”: E({θ i },δ) = exp(−iH E ({θ i },δ)). (8.8) Using Eq.(8.6), we can Taylor expandE as a function ofǫ and write: E({θ i },δ) =E (1) ({θ i },δ)+O(ǫ 2 ), (8.9) where, E (1) ({θ i },δ) = exp(−iδ X α,β f αβ ({θ i })B α ⊗S β ). (8.10) 136 where we have defined the error structure constantsf αβ =O(1), as numerical constants defined by the ideal unitaryU. We interpret Eq.(8.10) by saying that the realistic prop- agator isǫ-perfect as the leading order term isO(ǫ). The error Hamiltonian defined in Eq. 8.8 will need to be reconsidered for a each mode of implementation but can always be obtained systematically to any order inǫ as long as the system control Hamiltonian is known. To see how we can manipulate and cancel generic error structures, we need to look at how the error structure is modified after a sequence of operations. Consider such a sequence of realistic operations: M 1 ,...,M k . The subscripts are carried to the corre- sponding ideal unitariesU i , first order error operatorsE (1) i (δ), etc. Notice that we are keeping everything only up the first order inǫ, therefore we do not need to include any term involving the commutators of B α ’s. The unitary propagator for the sequence is given by M =M 1 ...M k =U 1 E (1) 1 (δ)...U k E (1) k (δ) =U 1 ...U k F (1) 1 (δ)...F (1) k (δ) where F (1) i (δ) = (U i+1 ...U k ) −1 E (1) 1 (δ)(U i+1 ...U k ) are the transformed error opera- tors where we have separated the ideal and error parts of the propagator. The trans- formed error operators give rise to transformed (in the same way) error structure con- stants. These transformed structure constants can subsequently be added to form the effective error structure of the whole sequence. Notice however that f 00 component remains equal to 1 as it is the “pure-bath” part of the error Hamiltonian and acts trivially on the system up to the 1 st order and is still of no immediate consequence. 137 8.3 Single Qubit Operations 8.3.1 Error Structure We consider single qubit operations and for simplicity of exposition focus on the case of a rectangular pulse for the switching of the system Hamiltonian. Any other model where the pulse shape does not change with the intended quantum operation may in principle be incorporated by defining the proper error Hamiltonian defined in Eq.(8.8). Specifically, let us assume that the quantum operationU({θ i }) is generated by switching the HamiltonianH S =h x S x +h y S y +h z S z for a duration ofδ:U({θ i }) = exp(−iδH S ). The realistic operationM is given by: M = exp[−iδ(H S + X α B α ⊗S α )] (8.11) where{S α } α=0,1,2,3 ={I S ,X,Y,Z} denote the system identity operator, along with the Pauli spin operators. We note that any qubit Hamiltonian can be generated by lin- ear combinations ofS i andX,Y,Z generate the Lie-algebrasu(2). Consider the case of a rotation around the x-axis: U = e −iθX . The first order error term E (1) (θ,δ) is characterized by the following error structure constants: (f) αβ = 1 0 0 0 0 1 0 0 0 0 cosθsinθ θ sinθ 2 θ 0 0 − sinθ 2 θ cosθsinθ θ (8.12) Notice howf αβ reflect the (non-)commutative properties of the operators involved in the quantum operationU. 138 8.3.2 First-order perfect self-correcting qubit operations Is it possible to apply a series ofǫ-perfect operation to get any desiredǫ 2 -perfect oper- ation? This sequence must be composed of realistic operations whose switching-times are not considerably different from each other. For experimental purposes, they should preferably be integer multiples of a basic pulse-width. This ensures that the sequence will not challenge the already present implementation constraints by requiring stronger control fields. For the qubit case the answer is positive. Before we describe the steps involved in the construction of the sequence, we give the sequence in its final form. Let R α [θ,δ] denote the realistic operations corresponding to rotations around theα axis with an angleθ generated during a pulse of widthδ. After keeping only the first order terms inǫ we can show that e −iθX =R X [θ,0] =R Y [π/2,δ]R X [π,6δ]R Y [π/2,δ]R X [π/2,δ] ×R Y [π,2δ]R X [θ,δ]R X [−θ,δ]R X [π/2,δ] ×R X [θ,2δ]+ ˜ B⊗I S +O(ǫ 2 ) (8.13) Each operation as a pulse-width that is an integer multiple ofδ, the total length of the sequence is 16δ, and it is composed of 13 alternatingX andY rotations in total. The operator ˜ B⊗I denotes a pure-bath term that cannot be canceled but acts trivially on the system up to the first order. Note that this is simply a possible solution which might not be optimal. In principle the optimal solution can be found by numerical optimization for any given angle θ. Once found, the sequence is established once and for all and can be used in various implementations. As an added bonus, the above sequence can incorporate different sets of error generators{B i α } for the i ={X,Y} rotations. By swapping theX andY rotations in Eq.(8.13), we get a sequence with the same properties 139 forR Y [θ,δ]. The additivity of the first order errors allows us to combine theǫ 2 -perfect rotations in an Euler angle construction to generate any qubit unitary. The construction of Eq.(8.13) is based on finding a rotationC that cancels the first order errors inR X [θ] modulo the pure-bath terms. Since up to the first order the errors are additive, it suffices to cancel the components of f αβ individually. As we will see later, it is useful to be able to generate any error operator with f αβ =±cδ αβ where c is an arbitrary positive number for α,β 6= 0 . We call these generators given by E ± αβ [cδ] := exp[∓icδ(B α ⊗S β + ˜ B⊗I S )], the error correcting generators, where the pure-bath term ˜ B⊗I S has no dynamical effect in the leading order. We will need to reconsider this term when we consider the higher order constructions in Sec. 8.4.1. Next we give a list of these generators in terms ofǫ-perfect rotations. Let us parameterize an arbitrary unitary by the three angle-like parametersθ α forα∈ {X,Y,Z}, such that turning onH S = 1 δ (θ X X +θ Y Y +θ Z Z) for a duration ofδ in the absence of errors generatesU({θ α }) andM({θ α },δ) in the presence of errors. Define the imperfectX-flipX[a] (and similarlyY[a] andZ[a]) asX[a] = M({π/2,0,0},a). We define a “NOOP” for a no-operation operatorF[a] :=M((0,0,0),a). We also need a flipped NOOPSF P [a,b] :=P[a]F[b]P[a], wherea andb are durations (thus positive numbers) andP∈{X,Y,Z}. Up to first order inǫ we have: E + 1,1 [δ] =R X [2π,δ] E − 1,1 [δ] =SF X [δ,3δ]SF Y [δ,3δ]SF Z [δ,3δ] ×E + 2,2 [δ]E + 3,3 [δ] E + 3,2 [2δ] =R X [π/4, π 2 δ]H π/4 [2δ]E − 1,1 [ π 2 δ]E − 2,2 [δ]E − 3,3 [δ] E − 2,3 [2δ] =R X [π/4, π 2 δ]H −π/4 [2δ]E − 1,1 [ π 2 δ]E − 2,2 [δ]E − 3,3 [δ] 140 The remaining E ± αβ [δ] operators can be generated from the generic set above by per- mutations of the indices. Also notice that we do not need extremely narrow pulses for generatingE ± αβ [cδ] whenc is very small, since as long as we allow precise control over durations we can generate them byE ± αβ [δ(1+c)]E ∓ αβ [δ]. While the list above is exhaus- tive, it is not necessarily optimal in the number of operation used. Now, we can cancel the first order errors inM by simply multiplying the required generatorsE ± αβ [cδ] with properly chosenc’s such that the unitary[E (1) ] −1 (Eq.(8.10)) is generated. Applying this operator immediately afterM will then give an approximation forU that is good up to the second order inǫ. The sequence given in Eq. (8.13) is actually not generated exactly this way but is tailored so that all pulse widths are an integer multiple of a basic pulse widths and only rotation angles of±θ,π,π/2 are used in its construction and no NOOP operation is used. This simplifies the implementation requirements. For the purpose of completeness we note that it is possible to generate E + 00 [cδ] = exp(−icδB 0 ⊗I S ) for anyc> 0, by canceling any two generators such asE + αβ ( c 2 δ)E − αβ ( c 2 δ). 8.4 Discussion 8.4.1 Higher order sequences Suppose M (1) = M[E (1) ] −1 is the 1 st order approximation for the unitary U that we constructed in the previous subsection that uses switching times of the order ofO(δ): M (1) =UE (8.14) whereE =E (2) +O(ǫ 3 ) and E (2) ({θ i },δ) = exp(−iδ 2 X α,β,γ g αβγ ({θ i })B α B β ⊗S γ ) (8.15) 141 The leading order error unitary operatorE (2) consists of a sum of products of bath and system operators and the coefficientsg αβγ play the same role as the coefficientsf αβ for the leading order error unitary operatorE (1) in subsection 8.2 and Eq. (8.10). Note that we are not providing a simple method of computing this expansion (see Addendum). To obtain the second order canceling sequences, we constructing palindromic sequences from the first order self correcting sequences. The palindromic sequences are obtained by repeating the time-reversal of a sequence once it is applied. By sym- metry arguments (See Sec.9.1) one can show that such sequences are self-correcting operators of order O(ǫ 3 ). The recursive construction and analysis of such “designer” self correcting sequences for higher orders is the subject of a future project. 8.4.2 Improvement at What Cost? Consider the sequence given by Eq. (8.13) where a sequence of length 16δ has an effective error given byO(ǫ 2 ), while a single operation of lengthδ carries an error of the orderO(ǫ). To reach the improvement given by Eq. (8.13), one might use a single operations of lengthδǫ, with the error given byO(ǫ 2 ). A crude way of comparing the costs of these two operations is to compare the pulse power requirements: given by R T 0 kH S (t)k 2 dt for a control sequence given byH S (t) of durationT . For the sequence (8.13), the energy costE C ∝ 1/δ and for the single pulse achieving the same errorǫ 2 , this energy cost is given byE S ∝ δ (δǫ) 2 = 1 δǫ 2 . Therefore we expect the sequence given by Eq. (8.13) to be cost-effective as long asǫ is sufficiently smaller than 1, which is also a condition for the validity of the series expansions that we have used throughout this chapter. Besides the energy requirements, we must also consider the effect of timing and synchronization errors. To obtain a simple model for such errors let us assume that the switching times of our control apparatus carry a relative random error: Instead of a 142 desired pulse widthδ we end up with a pulse widthδ(1+d) whered is a random variable such thathdi = 0 andhd 2 i = r 2 . On a single quantum operation the error caused by this pulse-width error is given by O(dδkH S k) up to the first order in d, where H S is the control Hamiltonian for the desired quantum operation. For a single qubit (or any finite quantum system) the maximum norm ofδH S will beO(1). Therefor the standard deviation of the error phasee caused by the timing error is given by he 2 i =γr 2 (8.16) for some γ = O(1) and does not depend on the actual pulse width δ. For a self- correcting sequence of order n and consisting of s(n) pulse segments, to provide a decisive improvement over a single pulse in the presence of random timing errors we requires(n)r≪ ǫ n . If this cannot be satisfied we will have to use an active error cor- rection technique such as quantum error correcting codes for correcting random pulse shape errors. 8.4.3 Pulse Shape Considerations We have implicitly assumed a rather special way of implementing quantum operations given in Eq. (8.4) by assuming the system control HamiltonianH s to be fixed during the application of the pulse, and have ignored the possible effects of “pulse shaping”. Also we have assumed perfect control over the rectangular temporal profile of H S . These assumptions directly affect the definition of the error Hamiltonian and subsequently the error structure constants (Eqs.(8.8,8.10)). Notice however that, the definition of the error Hamiltonian for an operation is universal but the assumption of existence of a time- scaleδ associated with each pulse is not. In simple settings that such an “effective pulse width”δ, can be defined, we can still use the results of the rectangular pulse methods, as 143 the effects of the time-varying control Hamiltonian will be of the second order inδkH s k. This approximation will fail for higher order corrections though. In practice, one may rederive the structure constants associated with a given realistic pulsing scheme in a similar manner and use the basic ideas described here to obtain a tailored self-correcting pulse sequence for the given scheme. The whole idea of a sequence of self-correcting operations can also be thought of as an elaborately shaped piece-wise rectangular pulse. In this sense, we expect possible improvements on our method, if arbitrary pulse shapes are allowed. Introduction of any degrees of freedom such as the pulse shapes and defining costs associated with sudden changes of the pulse Hamiltonians is an interesting optimization problem. 8.4.4 Using Composite Pulse Techniques The final result of a self-correcting operation is an operator which differs from the ideal intended operation by a unitary error operator such that the effects of the undesired couplings are canceled out to some order. Nonetheless, we have not corrected possible systematic errors in the pulse widths. To do that one may combine the output of the self-correcting pulse sequence with a length-compensating composite pulse technique. This should be done before first level self-correcting operations are constructed. In this scenario, a composite pulse sequence using pulses of width δ(1 +ξ), where ξ is an unknown number is used to form a pulse sequence equivalent to that of a single pulse of widthδ(1 +ξ m ). This can be done with a pulse overhead that scales smaller than polynomial in m [KB04]. This enables us to fully correct systematic errors in a rectangular pulse model up to any order efficiently. 144 8.5 Addendum: Calculation of higher order correction terms Let us consider the following operator product C = exp(iH s )exp[−i(H s 1 +δ 1 H e )]···exp[−i(H sn +δ n H e )] = exp(iE) (8.17) where exp(iH s ) = U † is the inverse of some target ideal unitary operator generated by switching onH s for duration of one andexp[−i(H s 1 +δ i H e )] represent the propagators for a pulse ideally generatingexp(iH s i ) by switching on a control Hamiltonian for dura- tion ofδ i and in the presence of an undesired small coupling ofH e with a norm bounded byǫ/δ. We also require the product of the ideal propagators to be the same: exp(−iH s ) = exp(−iH s 1 )···exp(−iH sn ) We are interested in computing the departure of C from identity up to order ǫ n . An apparently straightforward method is to use the Baker-Campbell-Hausdorff (BCH)[SK89] expansion for obtaining the product of exponentials as an exponential [or taking an algebraic logarithm]. There exist recursive methods of generating the BCH expansions to all orders but restricting the order of terms to be less than or equal ton in the powers ofH e results in infinite sums. For example, the first order correction toE S in powers ofH e in C 1 = exp(iH s )exp[−i(H s +δH e )] = exp(iE 1 ) (8.18) will involve an infinite series of commutators of H s , and will become intractable for higher orders. However note the following: If the leading order correction to E in 145 Eq.(8.17) is known up to order ǫ n , instead of finding E, one might as well find the operatorF defined by C = exp(iE) =I +iF +O(ǫ n+1 ). ClearlyF andE will coincide in the leading order inǫ n . Let us first focus on evaluation ofC 1 up to any ordern. We will use this expansion as a building block for the general expansion ofC in Eq.(8.17). We need to findF (n) 1 such that: C 1 = exp(iH s )exp[−i(H s +δH e )] =I +iF (n) 1 +O(ǫ n+1 ). This can be done explicitly and efficiently using the interaction picture of the Hamil- tonian H s and treating H e in theory [Sak85]. We obtain the following Dyson series expansion forC 1 : C 1 = I−i Z δ 0 dt 1 ˜ H e (t 1 )+(−i) 2 Z δ 0 dt 1 Z t 1 0 dt 2 ˜ H e (t 1 ) ˜ H e (t 2 ) ··· +(−i) n Z δ 0 dt 1 Z t 1 0 dt 2 ··· Z t n−1 0 dt n ˜ H e (t 1 ) ˜ H e (t 2 )··· ˜ H e (t n )+···, where ˜ H e (t) is the interaction picture error Hamiltonian given by the linear adjoint map Ad itH S given ˜ H e (t) = exp(itH s )H e exp(−itH s ) = Ad itH S (H e ). (8.19) 146 To obtain F (n) 1 we simply truncate the Dyson series at order n. Notice that the interaction-picture-transformed error Hamiltonian ˜ H e does not involve an infinite sum- mation ifH s is restricted to be a single or two-qubit operator (or fromsu(N) in general). In the single qubit case one can always expandH e as in Eq.(8.11) and write ˜ H e (t) = exp(it 3 X i=1 h i S i )( 3 X α=0 B α ⊗S α )exp(−it 3 X i=1 h i S i ) whereS α =I,X,Y,Z are the Pauli operators. We can find ˜ H e exactly in this case since Ad itHs (S α ) can be found exactly as rotations. A similar construction can be used for more than one qubit. 147 Chapter 9: Speculations and Outlook In the previous chapters we laid down a mathematical framework for analysis of dynam- ical decoupling pulse sequences. This framework, naturally leads to important questions on the nature and limitations of dynamical decoupling as means of quantum state preser- vation. In this chapter we will discuss some of these problems as speculations and open questions. 9.1 Higher Order Decoupling The universal dynamical decoupling sequences defined in the previous chapters do work beyond a single application: a basic cycle such asDD C [τ] can be repeated indefinitely so that the interactions with the environment are averaged out to zero over a period longer than4τ. The caveat, of course, is the fact that the cancelations are not exact and result in undecoupled error terms as described in Section 3.2. When the basic cycle is repeated the undecoupled terms remain in the effective Hamiltonian and eventually will affect the state of the system in an undesirable way. We presented a systematic treatment of the undecoupled error terms in dynamical decoupling in chapter 7 as well. Furthermore we constructed efficient concatenated sequences that perform better than what can be obtained by a periodic application of pulse sequences. However one is tempted to go beyond concatenated sequences. The theory presented in the previous chapters only describes decoupling at the lead- ing order level of the Magnus expansion. Even in concatenated sequences, we only require the leading order corrections. A natural question is whether higher order cancel- ing sequences exist. Let us first give an example of such sequences: Consider a dynami- cal decoupling sequence, such as − → s =DD C [τ] and let ← − s , denote the same sequence, only 148 applied in reverse chronological order. Now consider the combined sequence − → s ← − s . The effective Hamiltonian corresponding to this combined sequence is the same as DD C [τ] up to the first order in the Magnus expansion (Eq.(3.8)) and only consists of pure bath terms. Furthermore, the sequence is palindromic, i.e. it is invariant under time reversal. LetU − → s ← − s (H) = exp(−iTH eff ), denote the propagator for evolution under this sequence and the original HamiltonianH. We thus have: (U − → s ← − s (H)) † =U − → s ← − s (−H) (9.1) Now expand H eff in powers of components of H and focus on the terms with even powers of H. Due to the time-reversal symmetry in Eq.(9.1) these terms have to be zero. This covers the terms in the 2nd order of the Magnus expansion in Eq. (3.9). Easy enough! We were able to construct a sequence only twice longer than the original sequence − → s , that canceled the effective Hamiltonian to the next order. Nonetheless for reasons that we shall explain shortly, we are propose the following conjecture: Conjecture: There does not exist a fixed sequence ofO(n) pulses (ideal system only unitary operators) forming an evolution of durationT , with a coupling constant given byJ =kHk to obtain an effective HamiltonianH eff such thatTkH eff k = Poly n (JT) for a general a priori unknown HamiltonianH where Poly n represents a polynomial of degree at mostn. As the time of writing, we are only aware of the aforementioned palindromic sequences capable of “correcting” an arbitrary Hamiltonian up to the second order. An interesting recent result for the special case of spin-boson model of interaction of a spin quan- tum system with a bosonic environment with a cut-off, where higher order commuta- tors automatically cancel, is the sequence given in [Uhr07]. For this specific case this 149 sequence is a counterexample to our conjecture for the general case. Consider the Mag- nus expansion for a given sequence consisting of N ≥ n segments. By requiring the cancelation of terms up to ordern we need to look at the Magnus terms of up to ordern (Sec. 3.2). These terms are sums of commutators of at mostn unknown operators. In a generic case, the number of such terms grows exponentially 1 . To cancel all these terms we need to satisfy an exponential number of simultaneous equations. In the generic case this will require an exponential number of unknown parameters which in this case are the segment durations and pulse types. Thus if the equations are indeed generic we will require an exponential number of pulsesN = O(e n ) to cancel up to ordern. A future project is to evaluate this conjecture explicitly. 9.2 Dynamical Decoupling of Extended Systems Our discussion of dynamical decoupling in the previous chapters was focused on the single qubit case as the canonical form and possibly the most important case. How- ever, when decoupling an extended system such as a group of qubits, certain interesting problems arise. If we ignore the interqubit couplings and treat all qubits as one, we can use the same sequences as the single qubit case for decoupling but they will have to be applied in par- allel and collectively to all qubits. This is normally an available option in considerations of quantum computing specially with respect to spin based systems. In these cases a magnetic field that is applied on all of the qubits and can be used to generate collective pulses and will serve as the decoupling field. In some cases, we are interested in using dynamical decoupling to not only decouple the qubits from the environment but also to decouple them from each other. Various 1 This becomes transparent in the Dyson series formalism described in Sec. 3.5. 150 such decoupling procedures exist [Leu02, Woc06] and are intimately related to quantum simulation of arbitrary interaction Hamiltonians on a given fixed lattice using single qubit operations. So far they only apply to first order decoupling. To go to higher decoupling efficiencies one can use a concatenation scheme. However, if the goal is the decoupling of a single spin, one is tempted to consider methods that are more efficient in the number of pulses and the number of sites they are being acted on. In this sense we expect more efficient decoupling schemes to exist in examples such as a qubit on a spin chain with local control access to the nearby sites. Ultimately, one can devise a purification scheme, where various Hamiltonian terms can be efficiently decoupled (or recoupled) without requiring access to a growing portion of the system. The detailed discussion and an answer to this question will be addressed in a future project. 9.3 Combining Error Correcting Codes and DD for QC An important question, not directly addressed by dynamical decoupling is the question of fault tolerant quantum computation. Suppose we are interested in applying a unitary quantum operationU and use dynamical decoupling for error correction. If U can be performed instantaneously then there is hardly any requirement for dynamical decou- pling and if for some reason there should be a delay after or before application of U (such as waiting for an output from another part of the system as it is often the case in quantum error correction circuits), we might simply replace the delay with a sequence of dynamical decoupling operations to correct for the errors due to the free evolution. Unfortunately, U cannot be applied instantaneously and we are usually interested in correcting the operator U itself. An important example is operation of fault-tolerant circuits where the encoded operations are produced using Hamiltonians that are not fault-tolerant themselves. In these cases we cannot simply interrupt the evolution of the 151 system and apply decoupling as the effects can become uncontrollable. However one can envisage a solution using a combination of error correcting codes and dynamical decoupling procedures. Consider the case in which operators P i span a representation of a stabilizer sub- groupS of the real Pauli groupG of n qubits consisting of all products of the Pauli matrices and identity acting on individual qubits. The normalizerN(S) is a subgroup ofG in which all of the elements commute withS. Similar to the procedure for quantum error correcting codes (See Chapter 2): If the generators ofN(S)−S, defined as above, represent the algebrasu(2) ⊗m , and there exists encoded states|ψ i i that are “stabilized” byP i :P i |ψ j i =|ψ j i, then this representation produces an encoding ofm logical qubits into n physical qubits. It can be shown that Concatenating DD pulse sequences pro- duced byP i , ideally remove the effect of the error operators that anti-commute with at least one element ofS on the encoded qubit. In the above case, the error operator gen- erators, produce the Hamiltonians responsible for leakage from the encoded states. DD if acting fast enough can reverse and cancel the action of these Hamiltonians. This idea can be extended further to include encoded quantum operations. In this case, one has to decompose the desired encoded unitary operation into a product of operators (in an almost arbitrary manner) and then instead of applying them in sequence, apply them along the various levels of concatenated dynamical decoupling instead of the free evolution periods. This idea forms the basis of a future research project. 152 9.4 Epilogue The quest of low level error correction techniques is a fundamental component of build- ing quantum information processing devices. While theoretical quantum error correc- tion is in an advanced stage of development and quantum error thresholds are improv- ing on a yearly basis, comparatively little progress has been made in realizing quantum operations that correspond to the theoretical models employed in the high level error correction theory. Low level error correction methods such as dynamical decoupling, composite pulse techniques, and self-correcting operators will have to fill this void. Eventually a working quantum computer will consist of various types of quantum sys- tems, chosen by their function and a “coherent” error correction of such a computer will indeed involve both high and low level modes of quantum error correction. 153 References [AAK06] D. Aharonov and J. Preskill A. Kitaev. Fault-tolerant quantum computa- tion with long-range correlated noise. Phys. Rev. Lett., 96:50504, 2006. [ABC + 01] G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger. Stabilizing Distinguishable Qubits Against Spontaneous Decay by Detected-Jump Correcting Quantum Codes. Phys. Rev. Lett., 86:4402, 2001. [ABC + 03] G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger. Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation. Phys. Rev. A, 68(1):012316, Jul 2003. [ABO96] D. Aharonov and M. Ben-Or. Polynomial Simulations of Decohered Quantum Computers. In Proceedings of 37th Conference on Foundations of Computer Science (FOCS), page 46, Los Alamitos, CA, 1996. IEEE Comput. Soc. Press. [ABO97] D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of 29th Annual ACM Symposium on Theory of Computing (STOC), page 46, New York, NY, 1997. ACM. [AGP06] P. Aliferis, D. Gottesman, and J. Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput., 6:97, 2006. [AHH + 02] R. Alicki, M. Horodecki, P. Horodecki, , and R. Horodecki. Dynamical description of quantum computing: Generic nonlocality of quantum noise. Phys. Rev. A, 65:062101, 2002. [Ali07] R. Alicki. Comment on” Resilient Quantum Computation in Correlated Environments: A Quantum Phase Transition Perspective” and” Fault- tolerant Quantum Computation with Longe-range Correlated Noise, 2007. quant-ph/0702050. 154 [ALZ06] R. Alicki, D.A. Lidar, and P. Zanardi. Internal Consistency of Fault- Tolerant Quantum Error Correction in Light of Rigorous Derivations of the Quantum Markovian Limit. Phys. Rev. A, 73:52311, 2006. [ASW01] G.S. Agarwal, M.O. Scully, and H. Walther. Inhibition of Decoherence due to Decay in a Continuum. Phys. Rev. Lett., 86:4271, 2001. [A VK02] L. Glazman A. V . Khaetskii, D. Loss. Electron spin decoherence in quan- tum dots due to interaction with nuclei. Phys. Rev. Lett., 88:186802, 2002. [BB91] A Barchielli and V P Belavkin. Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A, 24(7):1495, 1991. [BBBK00] A. Beige, D. Braun, B.Tregenna, and P.L. Knight. Quantum Computing Using Dissipation. Phys. Rev. Lett., 85:1762, 2000. [BEL00] G. Burkard, H.-A. Engel, and D. Loss. Spintronics and Quantum Dots for Quantum Computing and Quantum Communication. Fortschr. Phys., 48:965, 2000. [Bha97] R. Bhatia. Matrix Analysis. Number 169 in Graduate Texts in Mathemat- ics. Springer-Verlag, New York, 1997. [BHV06] S. Bravyi, M. B. Hastings, and F. Verstraete. Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order. Phys. Rev. Lett., 97(5):50401, 2006. [BKLW00] D. Bacon, J. Kempe, D.A. Lidar, and K.B. Whaley. Universal Fault- Tolerant Computation on Decoherence-Free Subspaces. Phys. Rev. Lett., 85:1758, 2000. [BL01] M.S. Byrd and D.A. Lidar. Bang-Bang Operations from a Geometric Per- spective. Quant. Inf. Proc., 1:19, 2001. [BL02] M.S. Byrd and D.A. Lidar. Comprehensive Encoding and Decoupling Solution to Problems of Decoherence and Design in Solid-State Quantum Computing. Phys. Rev. Lett., 89:47901, 2002. [BL03] M.S. Byrd and D.A. Lidar. Combined error correction techniques for quantum computing architectures. Journal of Modern Optics, 50(8):1285, 2003. [BLWZ05] M.S. Byrd, D.A. Lidar, L.-A. Wu, and P. Zanardi. Efficient Universal Leakage Elimination. Phys. Rev. A, 71:52301, 2005. 155 [BMR + 02] P. O. Boykin, T. Mor, V . Roychowdhury, F. Vatan, and R. Vrijen. Algo- rithmic cooling and scalable NMR quantum computers. Proc. Natl. Acad. Sci. U. S. A., 99(6):3388, 2002. [BP02] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Oxford, 2002. [BPF + 02] N. Boulant, M.A. Pravia, E.M. Fortunato, T.F. Havel, and D.G. Cory. Experimental Concatenation of Quantum Error Correction with Decou- pling. Quant. Inf. Proc., 1:135, 2002. [BPM + 97] D. Bouwmeester, J.-W Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger. Experimental quantum teleportation. Nature, 390:575, 1997. [BT05] G. Burkard B.M. Terhal. Fault-Tolerant Quantum Computation For Local Non-Markovian Noise. Phys. Rev. A, 71:12336, 2005. [Cal91] P.T. Callaghan. Principles of Nuclear Magnetic Resonance Microscopy. Oxford University Press, Oxford, 1991. [Car93] H. Carmichael. An open systems approach to quantum optics. Number m18 in Lecture notes in physics. Springer-Verlag, Berlin, 1993. [CC00] A.M. Childs and I.L. Chuang. Universal quantum computation with two- level trapped ions. Phys. Rev. A, 63(1):12306, 2000. [CGEL06] W. A. Coish, Vitaly N. Golovach, J. Carlos Egues, and Daniel Loss. Mea- surement, control, and decay of quantum-dot spins. Physica Status Solidi (b), 243(14):3658, 2006. [CHN03] A. M. Childs, H. L. Haselgrove, and M. A. Nielsen. Lower bounds on the complexity of simulating quantum gates. Phys. Rev. A, 68(5):052311, 2003. [CJ00] H K Cummins and J A Jones. Use of composite rotations to correct sys- tematic errors in nmr quantum computation. New Journal of Physics, 2:6, 2000. [CLK + 00] D.G. Cory, R. Laflamme, E. Knill, L. Viola, T.F. Havel, N. Boulant, G. Boutis, E. Fortunato, S. Lloyd, R. Martinez, C. Negrevergne, M. Pravia, Y . Sharf, G. Teklemariam, Y .S. Weinstein, and W.H. Zurek. NMR Based Quantum Information Processing: Achievements and Prospects. Fortschr. Phys., 48:875, 2000. [CP54] H.Y . Carr and E.M. Purcell. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev., 94:630, 1954. 156 [CPM + 98] D.G. Cory, M.D. Price, W. Maas, E. Knill, R. Laflamme, W.H. Zurek, T.F. Havel, and S.S. Somaroo. Experimental Quantum Error Correction. Phys. Rev. Lett., 81:2152, 1998. [CPZ96] J.I. Cirac, T. Pellizzari, and P. Zoller. Enforcing Coherent Evolution in Dissipative Quantum Dynamics. Science, 273:1207, 1996. [CRSS97] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane. Quantum Error Correction and Orthogonal Geometry. Phys. Rev. Lett., 78:405, 1997. [CTPG86] H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer. Iterative maps for bistable excitation of two-level systems. Phys. Rev. Lett., 56(18):1905– 1908, 1986. [CZ95] J.I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett., 74:4091, 1995. [DBE95] D. Deutsch, A. Barenco, and A. Ekert. Universality in quantum computa- tion. Proc. Roy. Soc. London Ser. A, 449:669, 1995. [DCM92] J. Dalibard, Y . Castin, and K. Molmer. Wave-function approach to dis- sipative processes in quantum optics. Phys. Rev. Lett., 68(5):580–583, 1992. [DG98] L.-M Duan and G.-C. Guo. Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment. Phys. Rev. A, 57:737, 1998. [DiV95] D.P. DiVincenzo. Two-bit gates are universal for quantum computation. Phys. Rev. A, 51(2):1015, 1995. [DiV00] D. P. DiVincenzo. The Physical Implementation of Quantum Computa- tion. Fortschr. Phys., 48:771, 2000. [DN05] Christopher M Dawson and Michael A Nielsen. The Solovay-Kitaev algo- rithm, 2005. quant-ph/0505030. [DNBT02a] J.L. Dodd, M.A. Nielsen, M.J. Bremner, and R.T. Thew. Universal quan- tum computation and simulation using any entangling Hamiltonian and local unitaries. Phys. Rev. A, 65(4):40301, 2002. [DNBT02b] J.L. Dodd, M.A. Nielsen, M.J. Bremner, and R.T. Thew. Universal quan- tum computation and simulation using any entangling Hamiltonian and local unitaries. Phys. Rev. A, 65:040301, 2002. 157 [DP01] G. M. D’Ariano and P. Lo Presti. Tomography of Quantum Operations. Phys. Rev. Lett., 86:4195, 2001. [Fey86] R.P. Feynman. Quantum mechanical computers. Found. Phys., 16:507, 1986. [FKSS06] J. Fern, J. Kempe, S. Simic, and S. Sastry. Generalized Performance of Concatenated Quantum Codes – A Dynamical Systems Approach. IEEE Trans. on Automatic Control, 51:448, 2006. [FLP04] P. Facchi, D.A. Lidar, and S. Pascazio. Unification of Dynamical Decou- pling and the Quantum Zeno Effect. Phys. Rev. A, 69:032314, 2004. [Fre98] R. Freeman. Spin Choreography: Basic Steps in High Resolution NMR. Oxford University Press, Oxford, 1998. [FRS + 03] M. Friesen, P. Rugheimer, D.E. Savage, M.G. Lagally, D.W. van der Weide, R. Joynt, and M.A. Eriksson. Practical design and simulation of silicon-based quantum dot qubits. Phys. Rev. B, 67:121301(R), 2003. [FTP + 05] P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. A. Lidar. Control of decoherence: Analysis and comparison of three different strate- gies. Phys. Rev. A, 71(2):022302, 2005. [GBD99] D. Loss G. Burkard and D.P. DiVincenzo. Coupled quantum dots as quan- tum gates. Phys. Rev. B, 59:2070, 1999. [GJK + 96] D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H.D. Zeh, editors. Decoherence and the Appearance of a Classical World in Quan- tum Theory. Springer-Verlag, Berlin, 1996. [Got96] D. Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A, 54:1862, 1996. [Got97] D. Gottesman. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, Pasadena, CA, 1997. [Got98] D. Gottesman. Theory of fault-tolerant quantum computation. Phys. Rev. A, 57:127, 1998. [Got00] D. Gottesman. Fault-Tolerant Quantum Computation with Local Gates. J. Mod. Optics, 47:333, 2000. [GRL + 03] S. Guide, M. Riebe, G.P.T. Lancaster, C. Becher, J. Eschner, H. Haeffner, F. Schmidt-Kaler, I.L. Chuang, and R. Blatt. Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature, 421(6918):48, 2003. 158 [Gro96] L.K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on the Theory of Com- puting, page 212. ACM, New York, NY, 1996. [GZ00] C.W. Gardiner and P. Zoller. Quantum Noise, volume 56 of Springer Series in Synergetics. Springer, Berlin, 2000. [Hae76] U. Haeberlen. High Resolution NMR in Solids. Advances in Magnetic Resonance Series, Supplement 1. Academic Press, New York, 1976. [HRK00] M. Motzkus H. Rabitz, R. de Vivie-Riedle and K. Kompa. Whither the Future of Controlling Quantum Phenomena. Science, 288:824, 2000. [HS05] N. Hatano and M. Suzuki. Finding Exponential Product Formulas of Higher Orders, 2005. math-ph/0506007. [HW68] U. Haeberlen and J.S. Waugh. Coherent Averaging Effects in Magnetic Resonance. Phys. Rev., 175:453, 1968. [IAB + 99] A. Imamo¯ glu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, M. Sher- win D. Loss, and A. Small. Quantum information processing using quan- tum dot spins and cavity-QED. Phys. Rev. Lett., 83:4204, 1999. [IAM02] M. Rosen I. A. Merkulov, Al. L. Efros. Electron spin relaxation by nuclei in semiconductor quantum dots. Phys. Rev. B, 65:205309, 2002. [Ise02] A. Iserles. Expansions that Grow on Trees. University of Cambridge, Department of Applied Mathematics and Theoretical Physics, 2002. [KA05] O. Kern and G. Alber. Controlling Quantum Systems by Embedded Dynamical Decoupling Schemes. Phys. Rev. Lett., 95:250501, 2005. [Kan98] B.E. Kane. A silicon-based nuclear spin quantum computer. Nature, 393:133, 1998. [KB04] I.L. Chuang K.R. Brown, A.W. Harrow. Arbitrarily accurate composite pulse sequences. Phys. Rev. A, 70:52318, 2004. [KBAW00] P.G. Kwiat, A.J. Berglund, J.B. Altepeter, and A.G. White. Experimental Verification of Decoherence-Free Subspaces. Science, 290:498, 2000. [KBDW01] J. Kempe, D. Bacon, D.P. DiVincenzo, and K.B. Whaley. Encoded Uni- versality from a Single Physical Interaction. Quantum Inf. Comput., 1:33, 2001. [KBG01] N. Khaneja, R. Brockett, and S. J. Glaser. Time optimal control in spin systems. Phys. Rev. A, 63:0323308, 2001. 159 [KBLW01a] J. Kempe, D. Bacon, D.A. Lidar, and K.B. Whaley. Theory of decoherence-free fault-tolerant universal quantum computation. Phys. Rev. A, 63(4):42307, 2001. [KBLW01b] J. Kempe, D. Bacon, D.A. Lidar, and K.B. Whaley. Theory of Decoherence-Free, Fault-Tolerant, Universal Quantum Computation. Phys. Rev. A, 63:42307, 2001. [KCL98] E. Knill, I. Chuang, and R. Laflamme. Effective Pure States for Bulk Quantum Computation. Phys. Rev. A, 57:3348, 1998. [KG01] N. Khaneja and S. J. Glaser. Cartan decomposition ofSU(2 n ) and control of spin systems. J. Chem. Phys., 267:11, 2001. [KL97a] E. Knill and R. Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55:900, 1997. [KL97b] E. Knill and R. Laflamme. Theory of quantum error-correcting codes. pra, 55:900, 1997. [KL02] K. Khodjasteh and D.A. Lidar. Universal Fault-Tolerant Quantum Compu- tation in the Presence of Spontaneous Emission and Collective Dephasing. Phys. Rev. Lett., 89:197904, 2002. [KL03] K. Khodjasteh and D.A. Lidar. Quantum computing in the presence of spontaneous emission by a combined dynamical decoupling and quantum- error-correction strategy. Phys. Rev. A, 68:22322, 2003. [KL05a] K. Khodjasteh and D.A. Lidar. Fault-Tolerant Quantum Dynamical Decoupling. Phys. Rev. Lett., 95:180501, 2005. [KL05b] K. Khodjasteh and D.A. Lidar. Fault Tolerant Quantum Dynamical Decoupling. Phys. Rev. Lett., 95:180501, 2005. [KL06] K. Khodjasteh and D.A. Lidar. Performance of Deterministic Dynamical Decoupling Schemes: Concatenated and Periodic Pulse Sequences, 2006. eprint quant-ph/0607086, to appear in Phys. Rev. A. [KLM01] E. Knill, R. Laflamme, and G.J. Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409:46, 2001. [KLV00] E. Knill, R. Laflamme, and L. Viola. Theory of Quantum Error Correction for General Noise. Phys. Rev. Lett., 84:2525, 2000. [KLZ98a] E. Knill, R. Laflamme, and W. Zurek. Resilient Quantum Computation. Science, 279:342, 1998. 160 [KLZ98b] E. Knill, R. Laflamme, and W. Zurek. Resilient quantum computation: Error models and thresholds. Proc. Roy. Soc. London Ser. A, 454:365, 1998. [KMR + 01] D. Kielpinski, V . Meyer, M.A. Rowe, C.A. Sackett, W.M. Itano, C. Mon- roe, and D.J. Wineland. A Decoherence-Free Quantum Memory Using Trapped Ions. Science, 291:1013, 2001. [Kni04] E. Knill. Quantum Computing with Very Noisy Devices, 2004. quant- ph/0410199. [Kni05] E. Knill. Scalable quantum computing in the presence of large detected- error rates. Phys. Rev. A, 71(4):42322, 2005. [Lan96] R. Landauer. The physical nature of information. Phys. Lett. A, 217:188, 1996. [LAW02] D.A. Lidar L.-A. Wu, M.S. Byrd. Efficient Universal Leakage Elimination for Physical and Encoded Qubits. Phys. Rev. Lett., 89:127901, 2002. [LBKW01] D.A. Lidar, D. Bacon, J. Kempe, and K.B. Whaley. Decoherence-Free Subspaces for Multiple-Qubit Errors: (II) Universal, Fault-Tolerant Quan- tum Computation. Phys. Rev. A, 63:22307, 2001. [LBW99] D.A. Lidar, D. Bacon, and K.B. Whaley. Concatenating Decoherence Free Subspaces with Quantum Error Correcting Codes. Phys. Rev. Lett., 82:4556, 1999. [LCW98] D.A. Lidar, I.L. Chuang, and K.B. Whaley. Decoherence free subspaces for quantum computation. Phys. Rev. Lett., 81:2594, 1998. [LD98] D. Loss and D.P. DiVincenzo. Quantum Computation with Quantum Dots. Phys. Rev. A, 57:120, 1998. [Leu02] D.W. Leung. Simulation and reversal of n-qubit Hamiltonians using Hadamard matrices. Journal of Modern Optics, 49(8):1199, 2002. [Lev01] J. Levy. Quantum-information processing with ferroelectrically coupled quantum dots. Phys. Rev. A, 64:52306, 2001. [LF81] M. H. Levitt and R. Freeman. Composite pulse decoupling. J. Mag. Res., 43:502–507, 1981. [Lin76] G. Lindblad. On the Generators of Quantum Dynamical Semigroups. Commun. Math. Phys., 48:119, 1976. 161 [LW02] D.A. Lidar and L.-A. Wu. Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling. Phys. Rev. Lett., 88:17905, 2002. [LW03a] D.A. Lidar and K.B. Whaley. Decoherence-Free Subspaces and Sub- systems. In F. Benatti and R. Floreanini, editors, Irreversible Quantum Dynamics, volume 622 of Lecture Notes in Physics, page 83. Springer, Berlin, 2003. eprint quant-ph/0301032. [LW03b] D.A. Lidar and L.-A Wu. Encoded recoupling and decoupling: An alter- native to quantum error correcting codes applied to trapped ion quantum computation. Phys. Rev. A, 67:32313, 2003. [Mag54] W. Magnus. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7:649, 1954. [MG58] S. Meiboom and D. Gill. Compensation for pulse imperfections in Carr– Purcell NMR experiments. Review of Scientific Instruments, 29:688, 1958. [MO01] P. C. Moan and J. A. Oteo. Convergence of the exponential Lie series. J. Math. Phys., 42:501, 2001. [MPG01] D. Mozyrsky, V . Privman, and M.L. Glasser. Indirect Interaction of Solid-State Qubits via Two-Dimensional Electron Gas. Phys. Rev. Lett., 86:5112, 2001. [MW83] K.F. Milfield and R.E. Wyatt. Study, Extension, and Application of Flo- quet Theory for Quantum Molecular Systems in an Oscillating Field. Phys. Rev. A, 27:72, 1983. [MZ96] H. Mabuchi and P. Zoller. Inversion of Quantum Jumps in Quantum Opti- cal Systems under Continuous Observation. Phys. Rev. Lett., 76:3108, 1996. [NC00] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK, 2000. [NS06] B. Nachtergaele and R. Sims. Lieb-Robinson Bounds and the Expo- nential Clustering Theorem. Communications in Mathematical Physics, 265(1):119, 2006. [Osb06] T. Osborne. Efficient approximation of 1D Quantum Spin Chains. Phys. Rev. Lett., 97:157202, 2006. 162 [P. 05] P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, D.A. Lidar. Control of decoherence: analysis and comparison of three different strate- gies. Phys. Rev. A, 71:022302, 2005. [PD99] P.M. Platzman and M.I. Dykman. Quantum Computing with Electrons Floating on Liquid Helium. Science, 284:1967, 1999. [PK98] M.B. Plenio and P.L. Knight. The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys., 70:101, 1998. [PK02] D. Petrosyan and G. Kurizki. Scalable Solid-State Quantum Processor Using Subradiant Two-Atom States. Phys. Rev. Lett., 89:207902, 2002. [PR06] A. Pechen and H. Rabitz. Teaching the environment to control quantum systems. Physical Review A (Atomic, Molecular, and Optical Physics), 73(6):062102, 2006. [Pre99] J. Preskill. Fault-Tolerant Quantum Computation. In H.K. Lo, S. Popescu and T.P. Spiller, editor, Introduction to Quantum Computation and Infor- mation. World Scientific, Singapore, 1999. [PS00] N.V . Prokof’ev and P.C.E. Stamp. Theory of the spin bath. Rep. Prog. Phys., 63:669, 2000. [PS06] L. P. Pryadko and P. Sengupta. Quantum kinetics of an open system in the presence of periodic refocusing fields. Phys. Rev. B, 73:085321, 2006. [PVK97] M.B. Plenio, V . Vedral, and P.L. Knight. Quantum error correction in the presence of spontaneous emission. Phys. Rev. A, 55:67, 1997. [QJ99] L. Quiroga and N.F. Johnson. Entangled Bell and GHZ states of excitons in coupled quantum dots. Phys. Rev. Lett., 83:2270, 1999. [RDM02] B. Rahn, A.C. Doherty, and H. Mabuchi. Exact performance of concate- nated quantum codes. Phys. Rev. A, 66:032304, 2002. [Rei00] M.W. Reinsch. A simple expression for the terms in the Baker-Campbell- Hausdorff series. J. Math. Phys., 41:2434, 2000. [Rei04] B.W. Reichardt. Improved ancilla preparation scheme increases fault- tolerant threshold, 2004. quant-ph/0406025. [RR96] V . Ramakrishna and H. Rabitz. Relation between quantum computing and quantum controllability. J. Chem. Phys., 54:1715, 1996. [Sak85] J.J. Sakurai. Modern Quantum Mechanics. Addison Wesley, Reading, Mass., 1985. 163 [SCCA05] K.M. Svore, A.W. Cross, I.L. Chuang, and A.V . Aho. Pseudothreshold or threshold?–more realistic threshold estimates for fault-tolerant quantum computing, 2005. quant-ph/0508176. [SDS03] R. de Sousa and S. Das Sarma. Theory of nuclear-induced spectral dif- fusion: Spin decoherence of phosphorus donors in Si and GaAs quantum dots. Phys. Rev. B, 68:115322, 2003. [SDT06] K.M. Svore, D.P. DiVincenzo, and B.M. Terhal. Noise threshold for a fault-tolerant two-dimensional lattice architecture, 2006. quant- ph/0604090. [Sho94] P.W. Shor. Algorithms for Quantum Computation: Discrete Log and Fac- toring. In S. Goldwasser, editor, Proceedings of the 35th Annual Sympo- sium on the Foundations of Computer Science, page 124, Los Alamitos, CA, 1994. IEEE Computer Society. [Sho95] P.W. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 52:2493, 1995. [SK89] J.A. Oteo S. Klarsfeld. The Baker-Campbell-Hausdorff formula and the convergence of the Magnus expansion. J. Phys. A, 22:4565, 1989. [SL04] K. Shiokawa and D.A. Lidar. Dynamical Decoupling Using Slow Pulses: Efficient Suppression of 1/f Noise. Phys. Rev. A, 69:030302(R), 2004. eprint quant-ph/0211081. [SM99] A. Sørensen and K. Mølmer. Quantum Computation with Ions in Thermal Motion. Phys. Rev. Lett., 82:1971, 1999. [SM00] A. Sørensen and K. Mølmer. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A, 62:22311, 2000. [SMR61] E.C.G. Sudarshan, P.M. Mathews, and J. Rau. Stochastic Dynamics of Quantum-Mechanical Systems. Phys. Rev., 121:920, 1961. [SP05] P. Sengupta and L. P. Pryadko. Scalable Design of Tailored Soft Pulses for Coherent Control. Phys. Rev. Lett., 95:037202, 2005. [SS07] G. Smith and J.A. Smolin. Degenerate Quantum Codes for Pauli Chan- nels. Phys. Rev. Lett., 98(3):30501, 2007. [SSW05] R. de Sousa, N. Shenvi, and K. B. Whaley. Qubit coherence control in a nuclear spin bath. Phys. Rev. B, 72(4):045330, 2005. 164 [Ste97] A. Steane. The ion trap quantum information processor. Applied Physics B: Lasers and Optics, 64(6):623, 1997. [Ste99] A. M. Steane. Quantum Error Correction. In H.K. Lo, S. Popescu and T.P. Spiller, editor, Introduction to Quantum Computation and Information, pages 184–212. World Scientific, Singapore, 1999. [Ste03] A.M. Steane. Overhead and noise threshold of fault-tolerant quantum error correction. Phys. Rev. A, 68:42322, 2003. [STH + 99] S. Somaroo, C.-H. Tseng, T. F. Havel, R. Laflamme, and D. G. Cory. Quantum simulations on a quantum computer. Phys. Rev. Lett., 82:5381– 5384, 1999. [Suz76] M. Suzuki. Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many- body problems. Commun. Math. Phys., 51:183, 1976. [SV05] L. F. Santos and L. Viola. Dynamical control of qubit coherence: Random versus deterministic schemes. Phys. Rev. A, 72(6):062303, 2005. [SV06] L. F. Santos and L. Viola. Enhanced convergence and robust performance of randomized dynamical decoupling, 2006. [SZ97] M.O. Scully and M.S. Zubairy. Quantum Optics. Cambridge University Press, Cambridge, 1997. [Tho86] R.C. Thompson. Proof of a conjectured exponential formula. Linear and Multilinear Algebra, 19:187, 1986. [Uhr07] G.S. Uhrig. Keeping a Quantum Bit Alive by Optimized π-Pulse Sequences. Phys. Rev. Lett., 98(10):100504, 2007. [V AC + 02] D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret. Manipulating the Quantum State of an Electrical Cir- cuit. Science, 296:886, 2002. [VC04] L. M. K. Vandersypen and I. L. Chuang. NMR techniques for quantum control and computation. Rev. Mod. Phys., 76:1037, 2004. [Vio04] L. Viola. Advances in decoherence control. J. Mod. Optics, 51:2357, 2004. [VK03] L. Viola and E. Knill. Robust dynamical decoupling with bounded con- trols. Phys. Rev. Lett., 90:037901, 2003. 165 [VK05] L. Viola and E. Knill. Random Decoupling Schemes for Quantum Dynam- ical Control and Error Suppression. Phys. Rev. Lett., 94:060502, 2005. [VKL99] L. Viola, E. Knill, and S. Lloyd. Dynamical decoupling of open quantum systems. Phys. Rev. Lett., 82:2417, 1999. [VL] L. Viola and S. Lloyd. Decoherence control in quantum information pro- cessing: simple models. quant-ph/9809058. [VL98] L. Viola and S. Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A, 58:2733, 1998. [VRJ01] N. Vats, T. Rudolph, and S. John. Quantum information processing in localized modes of light within a photonic band-gap material. J. Mod. Optics, 48:1495, 2001. [VS06] L. Viola and L. F. Santos. Randomized dynamical decoupling techniques for coherent quantum control, 2006. quant-ph/0602175. [VYW + 00] R. Vrijen, E. Yablonovitch, K. Wang, H.W. Jiang, A. Balandin, V . Roy- chowdhury, T. Mor, and D. DiVincenzo. Electron-spin-resonance transis- tors for quantum computing in silicon-germanium heterostructures. Phys. Rev. A, 62:012306, 2000. [Wei95] S. Weinberg. The quantum theory of fields. Vol. 1: Foundations. Cam- bridge, New York: Cambridge University Press, 1995. [WJB] P. Wocjan, D. Janzing, and Th. Beth. Simulating Arbitrary Pair- Interactions by a Given Hamiltonian: Graph-Theoretical Bounds on the Time Complexity. [WL02a] L.-A. Wu and D.A. Lidar. Creating Decoherence-Free Subspaces Using Strong and Fast Pulses. Phys. Rev. Lett., 88:207902, 2002. [WL02b] L.-A. Wu and D.A. Lidar. Power of Anisotropic Exchange Interactions: Universality and Efficient Codes for Quantum Computing. Phys. Rev. A, 65:42318, 2002. [WL02c] L.-A. Wu and D.A. Lidar. Qubits as Parafermions. J. Math. Phys., 43:4506, 2002. [Woc06] P. Wocjan. Efficient decoupling schemes with bounded controls based on eulerian orthogonal arrays. Phys. Rev. A, 73(6):062317, 2006. [WS07] W. M. Witzel and S. Das Sarma. Multiple-pulse coherence enhancement of solid state spin qubits. Phys. Rev. Lett., 98(7):077601, 2007. 166 [WZ82] W.K. Wootters and W.H. Zurek. A single quantum cannot be cloned. Nature, 299:802, 1982. [WZ83] J.A. Wheeler and W.H. Zurek, editors. Quantum Theory and Measure- ment. Princeton University, Princeton, NJ, 1983. [YLS07] W. Yao, R. Liu, and L. J. Sham. Restoring coherence lost to a slow inter- acting mesoscopic spin bath. Phys. Rev. Lett., 98(7):077602, 2007. [YTN02] J.Q. You, J.S. Tsai, and F. Nori. Scalable Quantum Computing with Josephson Charge Qubits. Phys. Rev. Lett., 89:197902, 2002. [Zan98] P. Zanardi. Dissipation and decoherence in a quantum register. Phys. Rev. A, 57:3276, 1998. [Zan99] P. Zanardi. Symmetrizing evolutions. Phys. Lett. A, 258:77, 1999. [ZDS + 07] W. Zhang, VV Dobrovitski, L.F. Santos, L. Viola, and BN Harmon. Dynamical control of electron spin coherence in a quantum dot, 2007. Arxiv preprint cond-mat/0701507. [ZG00] S.-B. Zheng and G.-C Guo. Efficient Scheme for Two-Atom Entangle- ment and Quantum Information Processing in Cavity QED. Phys. Rev. Lett., 85:2392, 2000. [ZLL04] P. Zanardi, D. Lidar, and S. Lloyd. Quantum tensor product structures are observable-induced. Phys. Rev. Lett., 92:060402, 2004. [ZR97a] P. Zanardi and M. Rasetti. Error Avoiding Quantum Codes. Mod. Phys. Lett. B, 11:1085, 1997. [ZR97b] P. Zanardi and M. Rasetti. Noiseless Quantum Codes. Phys. Rev. Lett., 79:3306, 1997. [Zur91] W.H. Zurek. Decoherence and the transition from quantum to classical. Physics Today, 44:36, Oct. 1991. 167
Abstract (if available)
Abstract
Quantum information theory is based on the premise of manipulating quantum systems. Decoherence and noisy control directly limit this manipulation. Quantum error correction theory aims to understand the sources of errors in manipulation of quantum systems and to remedy the problems caused by the errors in an efficient manner. In this thesis I focus on error correction mechanisms that are based on a realistic and physical picture of the interactions of the quantum system with the environment. In chapters 1, 2, and 3, I provide a brief introduction to quantum information processing, quantum error correction, and dynamical decoupling. In chapters 4 and 5, I consider error correction of a set of qubits in the presence of spontaneous emission as the main source of errors. These results have been published in [KL:02] and [KL:03]. The quantum trajectories picture is used for describing the error processes. Two error correction schemes are provided in this scenario and are both built on simple quantum error detecting codes for detecting quantum jump errors. The qubit number overhead in this encoding is reduced in the first method [KL:02] by exploiting the symmetry of the conditional dynamics that can be used to create a decoherence free subspace. In the second method [KL:03], the conditional dynamics is canceled by applying parallel population swapping operations on the qubits. For both methods, I describe means of integrating the proposed error correction schemes with various proposals to achieve fault tolerant quantum computation. Chapters 6 and 7 are based on dynamical decoupling: a method for removal of undesired interaction terms from a Hamiltonian evolution by application of fixed unitary quantum operators. These results have been published in [KL:05] and [KL:06]. I describe general concatenated pulse sequences that are constructed recursively from simple dynamical decoupling pulse sequences.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Towards optimized dynamical error control and algorithms for quantum information processing
PDF
Topics in quantum information and the theory of open quantum systems
PDF
Applications of quantum error-correcting codes to quantum information processing
PDF
Open quantum systems and error correction
PDF
Towards robust dynamical decoupling and high fidelity adiabatic quantum computation
PDF
Demonstration of error suppression and algorithmic quantum speedup on noisy-intermediate scale quantum computers
PDF
Topics in quantum cryptography, quantum error correction, and channel simulation
PDF
Quantum steganography and quantum error-correction
PDF
Error correction and quantumness testing of quantum annealing devices
PDF
Quantum feedback control for measurement and error correction
PDF
Applications and error correction for adiabatic quantum optimization
PDF
Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions
PDF
Characterization and suppression of noise in superconducting quantum systems
PDF
Quantum error correction and fault-tolerant quantum computation
PDF
Towards efficient fault-tolerant quantum computation
PDF
Protecting Hamiltonian-based quantum computation using error suppression and error correction
PDF
Quantum computation and optimized error correction
PDF
Quantum information and the orbital angular momentum of light in a turbulent atmosphere
PDF
Quantum coding with entanglement
PDF
Error suppression in quantum annealing
Asset Metadata
Creator
Khodjasteh Lakelayeh, Kaveh
(author)
Core Title
Dynamical error suppression for quantum information
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
07/27/2007
Defense Date
06/12/2007
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
dynamical decoupling,OAI-PMH Harvest,quantum,quantum control,quantum error correcting coes,quantum error corrrection,quantum information,quantum open systems
Language
English
Advisor
Lidar, Daniel A. (
committee chair
), Brun, Todd A. (
committee member
), Däppen, Werner (
committee member
), Haas, Stephan (
committee member
), Lu, Jia Grace (
committee member
)
Creator Email
khodjast@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m706
Unique identifier
UC1340386
Identifier
etd-KhodjastehLakelayeh-20070727 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-549970 (legacy record id),usctheses-m706 (legacy record id)
Legacy Identifier
etd-KhodjastehLakelayeh-20070727.pdf
Dmrecord
549970
Document Type
Thesis
Rights
Khodjasteh Lakelayeh, Kaveh
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
dynamical decoupling
quantum
quantum control
quantum error correcting coes
quantum error corrrection
quantum information
quantum open systems