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University of Southern California Dissertations and Theses
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Open quantum systems and error correction
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Open quantum systems and error correction
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OPEN QUANTUM SYSTEMS AND ERROR CORRECTION by Alireza Shabani Barzegar 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 (ELECTRICAL ENGINEERING) May 2009 Copyright 2009 Alireza Shabani Barzegar Dedication To my parents, Tayeb Shabani and Minoo Safi, and my sisters, Minoosh and Behnoosh. ii Acknowledgments I would like to gift this thesis to all those who are part of my life. My mom who taught me there is no limit for human ambitions and greatness, My dad who taught me there is no limit for diligence and responsibility, my sister Minoosh who stopped me reading story books and bought me those three books which threw me in the world of physics, and my other sister Behnoosh who was my the only childhood friend. Special thanks to my advisor Prof. Daniel Lidar who understood me, and taught me what are needed for success in my future career. Thanks for his generosity and patience. And thanks to all those who were beside me in my life, Hamidreza Shabani, Hossein Rokhsari, Hasti Manaviyat, Ali Saadatpoor, Pedram Razavi and Ali Ozhand. I am also grateful to all my colleagues, Kaveh Khodjasteh, Masoud Mohseni, Ali Rezakhani, Soraya Taghavi, Joe Geraci, Alioscia Hamma, Marcelo Sarandy, Sara Schnei- der and Lian-Au Wu who were kind with me. iii Table of Contents Dedication ii Acknowledgments iii List of Figures vii Abstract viii Chapter 1: A Short Introduction to Quantum Dynamics, Error, and Correction 1 1.1 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . 3 1.1.2 Open System . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 Quantum Coding Theory . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Decoherence Free Subspace and Subsystem . . . . . . . . . . . 13 1.3.3 Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . 13 1.3.4 Fault-Tolerant Quantum Computation . . . . . . . . . . . . . . 14 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2: Quantum Maps 18 2.1 Completely Positive Map . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Quantum subdynamics and the assignment map . . . . . . . . . . . . . 23 2.3 Linear Quantum Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Initial system-bath states yielding a CP map . . . . . . . . . . . . . . . 32 2.5 Positivity Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.6 Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 3: Quantum Dynamical Equation 44 3.1 Markovian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Non-Markovian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Quantum measurements approach to open system dynamics . . 48 iv 3.2.2 Derivation of a post-Markovian master equation . . . . . . . . . 50 Chapter 4: Decoherence free Subspaces and Subsystems 59 4.1 Loss of information through a decoherence process . . . . . . . . . . . 59 4.2 Review of Previous Conditions for Decoherence-Free Subspaces and Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.1 Decoherence-Free Subspaces . . . . . . . . . . . . . . . . . . . 61 4.2.2 Noiseless Subsystems . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Generalized Conditions for Decoherence-Free Subspaces and Subsystems 67 4.3.1 Decoherence-Free Subspaces . . . . . . . . . . . . . . . . . . . 68 4.3.2 Noiseless Subsystems . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Performance of Quantum Algorithms over Imperfectly Initialized DFSs 84 4.5 Decoherence Free Subspaces and Subsystems in non-Markovian Dynam- ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.1 Decoherence Free Subspaces . . . . . . . . . . . . . . . . . . . 87 4.5.2 Decoherence Free Subsystems . . . . . . . . . . . . . . . . . . 89 4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7 Proofs of Theorems and Corollaries . . . . . . . . . . . . . . . . . . . 92 4.7.1 CP Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7.2 Markovian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 95 4.7.3 Non-Markovian Dynamics . . . . . . . . . . . . . . . . . . . . 99 Chapter 5: Optimal Quantum Error Correction 103 5.1 Optimal Quantum Error Correction . . . . . . . . . . . . . . . . . . . . 103 5.2 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Direct fidelity maximization . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Robust error correction . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.5 Indirect fidelity maximization . . . . . . . . . . . . . . . . . . . . . . 110 5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Chapter 6: Linear Quantum Error Correction 124 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2 CP maps and fault tolerant quantum error correction . . . . . . . . . . . 125 6.2.1 CP maps: pro and con . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2 (In)validity of the CP map model in FT-QEC . . . . . . . . . . 126 6.3 Linear Quantum Error Correction . . . . . . . . . . . . . . . . . . . . 129 6.3.1 CP-recoverable linear noise maps . . . . . . . . . . . . . . . . 129 6.3.2 Non-CP-recoverable linear noise maps . . . . . . . . . . . . . . 133 6.3.3 The physical case: Hermitian maps . . . . . . . . . . . . . . . 134 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 v Chapter 7: QEC in the Presence of Correlated Errors 142 Chapter 8: Conclusion 154 References 156 vi List of Figures 1.1 3-Qubit Concatenated Code . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Open Quantum System . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Measurement approach to open system dynamics . . . . . . . . . . . . . 51 5.1 Standard recovery for 3,5, and 7-qubit codes . . . . . . . . . . . . . . . 114 5.2 Comparison of optimal and robust recovery standard qubit encoding . . 115 5.3 Standard 7-qubit code: recovery at 0.75 & average-case recovery . . . . 116 7.1 Correlated Error Models . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2 Performance difference for local and non-local environments . . . . . . 150 7.3 Performance in the presence of 2-body and 3-body interactions . . . . . .152 vii Abstract Quantum effects can be harnessed to manipulate information in a desired way. Quan- tum systems which are designed for this purpose are suffering from harming interaction with their surrounding environment or inaccuracy in control forces. Engineering differ- ent methods to combat errors in quantum devices are highly demanding. In this thesis, I focus on realistic formulations of quantum error correction methods. A realistic for- mulation is the one that incorporates experimental challenges. This thesis is presented in two sections of open quantum system and quantum error correction. Chapters 2 and 3 cover the material on open quantum system theory. It is essential to first study a noise process then to contemplate methods to cancel its effect. In the second chap- ter, I present the non-completely positive formulation of quantum maps. Most of these results are published in [Shabani and Lidar, 2009b,a], except a subsection on geometric characterization of positivity domain of a quantum map. The real-time formulation of the dynamics is the topic of the third chapter. After introducing the concept of Marko- vian regime, A new post-Markovian quantum master equation is derived, published in [Shabani and Lidar, 2005a]. The section of quantum error correction is presented in three chapters of 4, 5, 6 and 7. In chapter 4, we introduce a generalized theory of decoherence-free subspaces and subsystems (DFSs), which do not require accurate initialization (published in [Sha- bani and Lidar, 2005b]). In Chapter 5, we present a semidefinite program optimization viii approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the encoding, recovery, or both, and is amenable to approximations that sig- nificantly improve computational cost while retaining fidelity (see [Kosut et al., 2008] for a published version). Chapter 6 is devoted to a theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This is a complementary to the second chapter which is published in [Shabani and Lidar, 2007]. In the last chapter 7 before the conclusion, a formulation for evaluating the performance of quantum error correcting codes for a general error model is presented, also published in [Shabani, 2005]. In this formulation, the correlation between errors is quantified by a Hamiltonian description of the noise process. In particular, we con- sider Calderbank-Shor-Steane codes and observe a better performance in the presence of correlated errors depending on the timing of the error recovery. ix Chapter 1 A Short Introduction to Quantum Dynamics, Error , and Correction Natural sciences can be classified based on the typical length scale of the phenomena which are their subject of study. For example, biology is the science of objects with dimensions from micro- to centi-meters, or cosmology deals with events of the length scale about 10 18 10 26 m. Progress in different branches of science has two comple- mentary directions: toward a more comprehensive picture and toward a more detailed picture. A comprehensive picture satisfies the scientific desire to explain a large group of phenomena by a simple theory while a detailed picture is a hallmark to examine the validity of this simple theory. One of the most successful theories of the history of science is Quantum Theory which has had an extraordinary influence on all branches of science, and also has been the central pillar of technology in the modern era. The emergence of quantum physics was after a series of experiments revealing atomic scale phenomena which were not justifiable by the existing theories of the time. Now after a century from its birth, the primary formulation of the quantum theory is amazingly good at explaining the observed and predicting physical events which usually happen at length scales of micro-meter and below. Electronic computers are the most important invention after turning the knowledge of quantum physics into technology. Vacuum tubes in early types of digital comput- ers and transistors in todays microchips are the products of engineering the electronic properties of matter. Electronic computers were designed to replace the mechanical 1 machines that were mostly used for data processing tasks. Over the years, by progress in miniaturization of computers, the size of components has been shrunk and the process of computing has sped up. In 1965, Gordon Moore, co-founder of Intel, made an observa- tion that the size of transistors in integrated circuits is halved every year [Moore, 1997]. This trend has more or less continued during the next years while the role of quantum effects has become more and more important. It should be noticed that although the design of an integrated circuit requires a quantum mechanical analysis, each transistor is acting as a switch and thus operates as a classical system. By entering the regime in which quantum effects are significant, fabrication of an integrated circuit becomes more challenging and it may slow down Moore’s rate of miniaturization. Several candidates have been proposed to surrogate the silicon based computers such as DNA computers and optical computers [Adleman, 1994, McAulay, 1991]. Quan- tum computer is exceptional among these candidates for it exploits new computational resources that quantum mechanics endows. What does this mean? What are those exceptional resources? Consider the simulation of the dynamics of 100 interacting elec- trons on a digital computer. If we ignore the spatial degrees of freedom and just consider the spin character of the electrons, the total Hilbert space of the system has dimension 2 100 . It is easy to see that the memory required to store the state of the system is beyond imagination and the simulation on an electronic computer is impossible. Nevertheless, while we cannot simulate this system, the same evolution is being run in nature. The conclusion is that there are hidden resources in nature which can be discovered and employed for computational tasks. Richard Feynman was the first to bring this issue to our attention and suggested to perform computation in the way nature does it [Feynman, 2 1982]. This idea in addition to ideas for using quantum mechanics in cryptography sys- tems prepared the ground for establishment of a new field of research: Quantum Infor- mation which is divided into two branches of quantum computation and quantum com- munication. The former is about performing computation on a machine whose dynamics is ruled by quantum mechanics. The latter is the science of communicating information through quantum systems which is not the subject of this thesis. 1.1 Quantum Dynamics 1.1.1 Postulates of Quantum Mechanics In this section we briefly review the postulates of quantum mechanics and the rules governing the dynamics of a system in quantum regime. Quantum mechanics consists of four postulates which construct a mathematical framework for developing physical theories. It should be emphasized that these postulates say nothing about the physics of a system, rather they are tools to formulate a physics problem in mathematical language. Postulate 1, State Space: Any system needs to be first characterized and labeled by some parameters. The space of parameters for a quantum system is a Hilbert space. More rigorously, system state is a positive operator in the Hilbert spaceH, assigned to the system, whose matrix representation has trace one. 0 ,Tr() = 1: (1.1) This state matrix is called density matrix. A special case of density matrices are the class of norm one matrices for which there exists a measurement setup for which the outcome of measurement is predictable [Peres, 1998]. A rank one density matrix can be 3 represented as an outer-product of a vectorj ih j: 1 The vectorj i is called pure state and a density matrix with rank more than one is called mixed state. Postulate 2, Time Evolution: The evolution of a closed (isolated) quantum system is described by a unitary transformation of the density matrix (t) =U(t;t 0 )(t 0 )U(t;t 0 ) y (1.2) This relation describes a point to point change of states. The corresponding dynam- ical equation is d dt =i~[H(t);] (1.3) where the HamiltonianH is a Hermitian matrix and~ is the Planck’s universal con- stant. The dynamical equation for a pure state is the Schrodinger equation dj i dt =i~H(t)j i (1.4) The unitary transformation (1.2) is a solution of the equation (1.3): U(t;t 0 ) = T [e i R t t 0 H(s)ds ], withT being the time ordering operator. Postulate 3, Measurement: Measurement is the key element in any scientific the- ory, since it is about collecting information about the system under study by a conscious observer 2 . The mathematical formulation of measurement usually takes place by assign- ing a real number to some physical observable which is the subject of the measurement. In quantum mechanics the observable is a Hermitian operator 3 O and the assigned real 1 j i is Dirac notation for a vector in a Hilbert space and in matrix languageh j is the Hermitian transpose ofj i. 2 The connection between the system and observer is usually indirect through a meter. 3 In this thesis the words “operator” and “matrix” are used interchangeably. 4 number is the expectation value of this operator quantified byTr[O]. In the ideal sce- nario to obtain this value for an operatorO, with spectral decompositionO = P i o i P i whereP i is a 1-dimensional projector ando i 2R, a number of detectors are coupled to the system each measuring the observableP i . At every measurement repetition one of the detectors is triggered and the measurement output is recorded to beo i . Activation of a detector is a random process and takes place with probabilityp i =Tr[P i ] for thei’th detector. The probability distributionp i can be obtained by repeating the experiment a large number of times. Then the expectation value of observable O is deduced to be Tr[O] = P i o i p i . A subtle fact about this ideal scenario for measurement is that by a given set of detectors any observable of the formO f i g = P i i P i ( i 2R) can be measured. This corresponds to the set of commuting operators [O f i g ;O f i g ] = 0. The discussed inherent randomness in a quantum measurement process, in absence of noise, has no classical counterpart. Another nonclassical aspect is the change of state induced by a measurement process. The pre-measurement state of the system will collapse into P i P i Tr[P i ] when the outcomei is recorded. The above postulate of projective measurement was first formulated by John von- Neuman in [von Neumann, 1955]. In practice a generalized postulate is found to be more useful. In a generalized quantum measurement, each detector is triggered with probability p i = Tr[M i ] and the post-measurement state is M i M i Tr[M i ] . Here M i is not necessarily a projector and it can be any operator; the only condition these operators need to satisfy is the completeness relation P i M i =I. Postulate 4, Composite Systems: The Hilbert space of a system, composed of n systems with Hilbert spacesH 1 ,H 2 , ... ,H n , is the tensor productH 1 H 2 ::: H n . The density matrix of each subsystem of the total composite system is obtained by tracing out the Hilbert space of the rest of system. For instance, in a bi-partite system 5 with Hilbert spaceH 1 H 2 and state 12 , the state of system 1 is found to be 1 = Tr 2 [ 12 ]. Now we can return to quantum measurement and derive the formulation of gen- eralized measurement as a result of a projective measurement on a secondary system coupled to the main system. The system S is in interaction with a probe P with Hilbert spacesH S andH P correspondingly. The system and probe are initially in a product state S jpihpj and jointly evolve into SP =U SP ( S jpihpj)U y SP with the unitary transformationU SP . Then the probe is subjected to a projective measurement described with a complete set of operatorsfP i =jii P hijg. The post-measurement system-probe state will be i SP = P i SP P i Tr[ SP P i ] = P i U SP ( S jpihpj)U y SP P i Tr[U SP ( S jpihpj)U y SP P i ] (1.5) with probability Tr[ SP P i ] = Tr[U SP ( S jpihpj)U y SP P i ]. Then the post- measurement state of the system is found to be i S =Tr P [ i SP ] =fhijU SP jpi S hpjU y SP jiig=Tr[ SP P i ] (1.6) This equation represents the structure of a collapsed state following a generalized measurement with operatorsM i =hijU SP jpi= p Tr[ SP P i ]. 1.1.2 Open System As was stated in the second postulate, the evolution of a closed quantum system is described by a unitary transformation. What if the system is not isolated and is interact- ing with its ambient environment (bath)? In this case the dynamics can be calculated by 6 applying postulates 2 and 4. The systemS and its bathB together form a closed system therefore their joint evolution is unitary: SB (t) =U SB (t) SB (t)U y SB (t) (1.7) Applying principle 4, the evolution of system only becomes S (t) =Tr B f SB (t)g =Tr B fU SB (t) SB (t)U y SB (t)g (1.8) The dynamics of an open system is not unitary which makes it irreversible and causes loss of information when the bath is much larger than the system. 1.2 Quantum Computation Quantum mechanics provides an exceptional platform to design algorithms that accom- plish a computational task on classical data. One of the first discoveries of this new type of algorithms started with Grover’s search algorithm [Grover, 1997]. Classically, search- ing an unsorted database with size N requires a linear search O(N) while Grover’s algorithm takesO( p N) time, i.e. it provides a quadratic speedup. However quantum algorithms are more interesting when they provide an exponential speedup compared to the best known classical algorithms. The first algorithm with this property is Shor’s factoring algorithm. The best known algorithm to factor an integer N takes an expo- nential time of order O(e (logN) 1 3 (loglogN) 2 3 ). In 1995, Peter Shor published a quantum algorithm for factoring an integer number outperforming its classical counterparts by taking a polynomial timeO((logN) 3 ) [Shor, 1994]. This was a major breakthrough in exposing the capabilities of quantum systems in processing classical information. The fact that factoring large numbers is practically infeasible makes it a cornerstone in design 7 of public key in cryptography systems. Thus implementing a quantum system on which Shor’s algorithm can be run has serious applications in breaking encrypted information. This algorithm drew attention to study quantum effects as new resources for information processing and also how to implement those systems. So far, the former has progressed both in finding new algorithm and designing quantum communication/cryptograhy sys- tems. On the other hand, realization of physical systems to run quantum algorithms seems to be very challenging 4 . Quantum computer (QCR) is a system whose time evolution realizes a quantum algorithm. The structure of a quantum computer is very similar to regular (classical) computers. The information is stored in a collection of subsystems, each with a two dimensional Hilbert space called quantum bit or qubitH qubit = Spanfj0i;j1ig. The total evolution of a QCR dictated by steps of a quantum algorithm can be represented by state preparation followed by a unitary transformation and then a final readout measure- ment. Most of the quantum algorithms are written such that QCRs start and evolve in a pure state. The initial state is usually considered to be the ground state of the system j i i =j00:::0i. The total unitary transformation transforms the initial state into a final statej f i that is very close to the correct answer of the problemj a i 5 . The final mea- surement finds the answer with high probability. The probabilistic nature of the quantum measurement requires repetition of the algorithm, hence it is an essential element of the quantum algorithm design to have the output very close to the correct answer. There are different models for quantum computation (QC) all equivalent to each other such as: Circuit-based QC [Nielsen and Chuang, 2000], Adiabatic QC [Farhi 4 Optical quantum communication systems has been succesfuly developed in laboratory prototypes. 5 The closeness of two pure statesj f i andj a i is measured by q 1jh f j a ij 2 . 8 et al.], and Measurement-Based QC [Jozsa, 2005] 6 . The circuit based quantum com- putation was the first model worked out which is basically a translation of the classical logical circuits with the same terminology. A d-qubit gate is a unitary transformation acting ond qubits. The total unitary of an algorithm is constructed by a serial (in time) and parallel (in space) connection of gates chosen from a finite set of transformations. Building a quantum circuit is grounded on the fact that there exists a universal set of gates such that any arbitrary unitary transformation can be realized by a finite sequence of these universal gates only. For example it can be shown that any arbitrary 1-qubit gate in addition to almost any two qubit gate, for instance CNOT, 7 form a universal set [Nielsen and Chuang, 2000]. A physical system realizing a circuit-based QCR should satisfy a set of five criteria first introduced by David Divincenzo [DiVincenzo, 2000] and then extended to seven criteria for those proposals which are based on distributed computation 1. Being scalable with well characterized qubits 2. The ability to initialize the state of qubits to a simple fiducial state, such asj00:::0i 3. Long decoherence time 8 much longer than gate operation time: The total time of running an algorithm must be less than the time that information stored in qubits is lost due to the noise imposed by surrounding environment. 4. A universal set of gates should be implementable in this system 5. A qubit specific measurement is doable 6 Quantum Random Walks which is basically the first model of a QC proposed by Feynman [Childs, 2008] and Quantum Cellular Automata [Wiesner, 2008] are in this list. 7 CNOT is a 2-qubit gate with matrix representation 2 6 6 4 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 3 7 7 5 . 8 For a definition of decoherence time refer to the open systems section. 9 6. The ability to interconvert stationary and flying qubits 7. The ability to faithfully transmit flying qubits between specific locations The two other computational models are not relevant to this thesis however here we describe them very briefly. The adiabatic QC is based on the adiabatic theorem in quantum mechanics that loosely speaking says if the system Hamiltonian is varied slowly enough and the sys- tem is initialized in gound state, it remains in the instantaneous ground state. Using this fact, a quantum algorithm can be implemented in the following simple scenario. Suppose the Hamiltonian is a function of the controllable parameter,H adiabatic () = (1)H 0 +H 1 . The parameter changes slowly enough in the interval [0; 1] such that if the initial state of the system (QCR) will be the ground state ofH 0 then the final state is the ground state ofH 1 . In the formalism of measurement based quantum computation we start with a fixed entangled state of many qubits and perform computation by applying a sequence of mea- surements. The choice of basis for later measurements depends on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. 1.3 Quantum Error Correction After over a decade of attempts to implement a quantum computer, it seems no to be an easy goal to reach. The main reason behind this difficulty is lack of isolation of the qubits from interaction with environment. This undesired interaction opens a channel to lose information into the environmental degrees of freedom, therefore it is a source of error. In addition any inaccuracy in the applied control with the purpose of either applying a gate or performing a measurement may cause some random error adding 10 to the uncertainty of the computation. Besides the attempts to improve the technology in order to better isolation of the system plus high precision quantum control, some methods for error correction have been created which are believed to be essential to any quantum computer design. These methods are classified as passive or active methods. The passive methods are those that instead of fighting the enemy (noise) they accept its dominance, therefore hide the information into a shelter and run the computation in that place. The shelter would be a subspace of the total Hilbert space, a subsystem of it, or a combination. These methods are correspondingly named decoherence free subspace or subsystem. The other class of error correction methods are active in fighting back the noise by continuously changing the dynamics of the system similar to ducking skill in boxing, or forming the battle field in a desired way. The former one is known as dynamical decoupling technique and the latter one is quantum coding. Here we briefly explain them but a more detailed story will be presented in the coming chapters, except the dynamical decoupling which is not a part of this thesis. 1.3.1 Quantum Coding Theory Exploiting codes to prevent loss of information has been one of the first approaches that human has resorted to. Nowadays, in any data processing or data transferring device the input information is first turned into code words before any action, and then the final result is achieved by decoding this information. In classical coding theory a string of data units (word) is mapped into a longer string called code word to enhance dis- tinction of different words in an event of error. A similar idea was proposed as the first method for suppressing the destructive effect of noise in quantum computers [Shor, 1995, Steane, 1996, Knill and Laflamme, 1997]. The initial information stored in an ideal system’s state ideal is mapped into the state of another system encoded with a 11 larger Hilbert space, i.e. dim(H ideal ) < dim(H ideal ). Here we give a simple example of encoding one qubit into three qubits: Example: Suppose the initial raw information is stored in a statej (a;b)i = aj0i + bj1i and the qubit is experiencing a bit flip noise channel which is given by out = (1 p) in + pX in X. Here X = j0ih1j +j1ih0j is the error opera- tor and p represents the probability of error. Then the faulty state would be out = (1p)j (a;b)ih (a;b)j +pXj (a;b)ih (a;b)jX, whereXj (a;b)i =bj0i +aj1i is the erroneous part. Now we want to encode this single bit of information into three qubits and show how this helps to recover the initial state with a high fidelity: j (a;b)i = aj0i +bj1i =) (a;b) E = aj000i +bj111i. Assume that the total noise channel acting on three qubits is a 3-fold single-qubit bit flip channel. In the limit of low error probability (p 1) we approximate this channel to the first order as out = (13p) in +3p P i X i in X i . This yields the final state out = (13p)(aj000i+ bj111i)(a h000j +b h111j) + 3p(aj001i +bj110i)(a h001j +b h110j) + 3p(aj010i + bj101i)(a h010j + b h101j) + 3p(aj100i + bj011i)(a h100j + b h011j). Since the statesfaj000i +bj111i;aj001i +bj110i;aj010i +bj101i;aj100i +bj011ig are all mutually orthogonal, we can distinguish them by projective measurement where the measurement operators project onto one of these states, and by a unitary transformation we can recover the original state (a;b) E . Knill-Laflamme Coding Theory: The first comprehensive formulation of quantum codes was presented by E. Knill and R. Laflamme [Knill and Laflamme, 1997]. In this theory the steps of noise (E) and recovery (R) are modeled by quantum channels, i.e.E() = P E E y andR() = P R R y . Theorem 1 Suppose P c is a projector to the codes subspace of the system’s Hilbert space, and the information encoded in this subspace is disturbed by the noise channel 12 E(). Then there exists a recovery quantum channelR() if and only ifP c E y E P c / P c . 1.3.2 Decoherence Free Subspace and Subsystem A simple and smart idea would be to find a subspace of the Hilbert space in which the stored information is sheltered from the noise. This idea has been pursued and developed as a main technique for information preservation in quantum computer designs. First formulations of this concept, named noiseless code or decoherence free (DF) subspace can be found in the works by P. Zanardi [Zanardi and Rasetti, 1997] and D. Lidar [Lidar et al., 1998]. As we have seen in Eq.(1.8), the dynamics of an open system, in general, is given by a non-unitary evolution. But what if there exists a set of initial states of the open system that evolve unitarily? Then we can think of them to be the safe zone of the Hilbert space to store information. Consider a two qubit example with the noise channelN () = 1 2 (I I)(I I) + 1 2 (Z Z)(Z Z). The total Hilbert space of this system has two separate DF- subspaces:fj00i;j11ig andfj10i;j01ig. The other class of DF spaces is DF-subsystem. Again assume a two qubit system with Hilbert spaceH qubit1 H qubit2 where only the second qubit is affected by a source of noise (e.g. a system-bath HamiltonianH SB =I qubit1 Q qubit2 B bath , whereI qubit1 is the identity operator.), then the registered data in the first qubit spaceH qubit1 is auto- matically protected from any loss. This was a very trivial example to sketch the idea. More rigorous formulation will be explained in chapter 4. 1.3.3 Dynamical Decoupling Dynamical decoupling, also known as Bang-Bang control, the third method of error cor- rection, is an open loop control method for noise reduction. In the standard recipe of 13 this method the system Hamiltonian would be changed by periodically running a series of pulses acting on the system only. This pulse series is designed such that approxi- mately, at the end of each period, the average system-bath Hamiltonian is in the form I S B effective . Since this method is beyond the scope of this thesis, we resort to a simple example to clarify the idea: H SB = Z B (Z =j0ih0jj1ih1j), and the corresponding unitary evolution for a time will beU free = exp(iH SB ). Consider a four-step cycle of the evolutionU 1cycle =XU free XU free where the unitariesXs are fast pulses added with the purpose of noise reduction. The identityXZX =Z helps to simplifyU 1cycle to U 1cycle =XU free XU free =Xexp(iZ B)Xexp(iZ B) =exp(iXZX B)exp(iZ B) =exp(iZ B)exp(iZ B) =I I (1.9) After time 2 the effective noise Hamiltonian reduces toI I implying that system is decoupled from bath and there is no decoherence. For a comprehensive review see [Khodjasteh, 2008]. 1.3.4 Fault-Tolerant Quantum Computation How to apply methods of noise reduction to reach an arbitrary accuracy in computation? This question is answered in the context of fault-tolerant quantum computation (FTQC) theory [Preskill, 1999, Knill et al., 1998, Aharonov and Ben-Or, b, Kitaev, 2003, Alicki et al., 2006, Aliferis et al., 2006, B.M. Terhal, 2005, Aharonov et al., 2006]. This theory tells us how to design a quantum circuit including error correcting units by which an arbitrary accuracy can be achieved if the rate of error is below a certain value called error threshold. The most studied QC architecture is a self-similar recursion of the ideal 14 . . . . . . . . . Figure 1.1: 3-Qubit Concatenated Code circuit plus error correction units. The recursion starts with encoding the information of a single qubit (C 2 ) into ann-qubit code (C 2n ): recursively mappingR :j0i!j0i logical andj1i ! j1i logical : Thereafter, at each step of the recursion, the information of a qubit is encoded in a new code word Fig. [1.1]. In the language of transformations it means that for a circuit withl levels of recursion we apply the transformationR l to data qubits. In addition, the gates have special designs that prevent rapid error propagation into a circuit. In the FTQC literature, the noise model which is properly studied is the model of locally and temporally independent stochastic errors. Consider a circuit as a graph in which an edge represents a gate. This model assumes that errors are a combination 15 of CP mapsE(l;t) which take place at locations l and time t. A norm of the map sup l;t jjE(l;t)jj is usually defined as the rate of error. The core results of this theory, the threshold theorem, states that Theorem 1 When the error rate per qubit per gate is below a certain value, indefinitely long quantum computation becomes feasible, even if all of the involved qubits are sub- ject to noise processes, and no qubit manipulation is error free. 1.4 Motivation The noise problem is recognized to be the main obstacle on the way to build a quantum computer that works! This brief introduction gave an overview of the known methods for canceling the effect of noise however the implementation of these methods in real experiments has revealed the limitation of their performance. What is needed to face real experimental challenges? The first step is to achieve a more accurate formulation of a noisy system dynamics. Theoreticians usually prefer to work with formulations which can be handled easily and those yielding an overall view of the exact solution. We should start with a simple model, but a model that is too approximated takes us away from the reality. Overcoming this issue is the main theme of research presented in this thesis. Obtaining an exact dynamical map formulation for an open quantum system, or deriving a new non-Markovian master equation capturing the essence of a dynamics with a memory. Next we apply our modeling of a noise process to design more advanced error cor- rection methods. The linear quantum map picture of error correction, a decoherence free space method adapted to incorporate imperfectness in initialization of the system state, searching for optimal encoding system and quantifying the notion of correlation of 16 error, are all projects which are conducted based on this philosophy that we need more accurate techniques for error correction. This thesis is presented in two sections: 1. Open quantum system theory, chapters 1 and 2. 2. Quantum Error Correction, chapters 3, 4, 5, and 6. 17 Chapter 2 Quantum Maps In the introduction chapter, the notion of an open quantum system was briefly introduced as a system coupled to a second system called bath. In nature there exists no ideal closed system except the whole universe and any physical object that we deal with is an open system, although under some conditions the system can be considered to be effectively isolated. To formulate the dynamics of an open system, ideally the system should be considered as a subset of the universe and its dynamics can be found by tracing out universe minus the system degrees of freedom. However human is not God 1 and has limited capabilities, therefore interaction with the universe is replaced by interaction with an effective bath. For instance, the electron in a quantum dot is interacting with the substrate lattice and also impurities. At high temperatures, we can ignore the coupling to impurities and the effective bath will be the Bosonic bath of phonons. In the other limit of low temperatures the interaction with phonons is negligible and the effective bath will be the spin bath of impurities. In this way a tractable formulation is at hand with the general form introduced in the previous chapter, Eq.(1.8). The system S is coupled to bath B, Fig.[2.1], with the Hamiltonian H SB = H S (t) I B +I S H B (t) +H int (t) resulting in the unitary transformationU SB (t) = T (e i R t 0 H SB (t 0 )dt 0 ). Then the state of the system at timet becomes S (t) =Tr B [ SB (t)] =Tr B [U SB (t) SB (0)U y SB (t)] (2.1) 1 You may ask the author in private about this word. 18 Environment (Bath) System Figure 2.1: Open Quantum System The dynamical equation of an open system is found by calculating the time derivative of Eq.(2.1) _ S (t) =Tr B [ _ U SB (t) SB (0)U y SB (t)] +Tr B [U SB (t) SB (0) _ U y SB (t)] (2.2) These equations are the beginning of the story of open quantum system theory. It is more desirable to rewrite the first equation as a map from S (0) to S (t). Why? Because we like simplicity and a formulation that can be handled. Also for the dynamical equa- tion, a differential equation form is what we need for calculations. These are two main topics which have been the subject of intense study since the birth of quantum mechan- ics [Breuer and Petruccione, 2002, Alicki and Lendi, 1987, Alicki and Fannes, 2001]. In 19 the next two chapters, “Quantum Map” and “Quantum Dynamical Equation”, the author presents a short introduction with new results he has achieved during his PhD research. 2.1 Completely Positive Map The standard noise model in the theory of open quantum systems is a mappingE in the form of the so-called Kraus operator sum representation [Kraus, 1983, Alicki and Lendi, 1987, Breuer and Petruccione, 2002], from the initial system state S (0) = Tr B [ SB (0)] (partial trace of the initial joint system-bath state) to the final system state S (t): S (t) =E[ S (0)] = X K (t) S (0)K y (t): (2.3) HereK s are called Kraus operators satisfying P K y K I (I is the identity opera- tor), which guarantees Tr[ S (t)] 1. The reason for this popular choice of noise model is its presumed generality: the well-known Kraus representation theorem [Kraus, 1983] states that, assuming factorized initial conditions SB (0) = S (0) B (0), a map has the representation (3.10) if and only if it is linear and completely positive (CP). A linear map is called CP if satisfies both 0 and I n 08n2Z + , where I n is then-dimensional identity operator. A linear transformation acting on density matrices :H!H (H is a Hilbert space) is a superoperator that may be represented by the “dynamical matrix”D() [Sudarshan et al., 1961] (or “Choi matrix” [Choi, 1975]), as follows: mn = X s;t D ms;nt st ; (2.4) where mn =hmjjni are matrix elements of in some fixed orthonormal basis for H, and D is Hermitian: (D y ) ms;nt = D nt;ms . Choi proved that the positivity of D is 20 equivalent to complete positivity of : D 0 if and only if is CP [Choi, 1975]. Complete positivity of in the caseD 0 follows from the spectral decomposition D ms;nt = X (K ) ms (K y ) tn ; 08; (2.5) where the are the eigenvalues ofD and theK can be identified with the Kraus oper- ators. The Choi matrix is further constrained if ones assumes that it is trace preserving: D ms;mt = st , or if is unital [(I) =I]:D ms;ns = mn . Consider the initial state of system and bath is a product state SB (0) = S (0) B (0). Then it is easy to construct a linear map between S (0) and S (t) where this map is a function of the unitary evolutionU SB (t) and the state of bath B (0) only. This becomes possible by the spectral decomposition of the bath state B (0) = P j ih j by which the quantum map is obtained as S (t) = X ; hb jU SB (t)j i S (0)h jU y SB (t)jb i (2.6) By defining the operatorE = p hb jU SB (t)j i, this map takes a compact form S (t) = P E S (0)E y . The condition SB (0) = S (0) B (0) is satisfied in some physical conditions. For example when a photon is sent through an optical fiber for quantum communication purposes, the system (photon) state is not correlated with the bath (fiber) state. Never- theless, there is a special but important case that two systems are always in a product state independent of the physical situation that is when the state of one of the systems is pure. Theorem 1 The state of a bipartite system 12 with one system in a pure state 1 = j ih j is a product state 12 = 1 2 . 21 Proof We choosej1i = j i in addition to a set of statesfj2i:::jNig to be a com- plete orthogonal basis ofH 1 and 1 ;:::; M a basis forH 2 . We repeatedly apply a theorem in linear algebra that any principal submatrix of a positive semi-definite (psd) matrix is psd. 1) The diagonal elements of a psd are psd, Tr(j ih j) = 1, Tr( 12 ) = 1 =) P i>1 12 (ij;ij) = 0,8i> 1. 2) If you delete the firstM rows and columns, the remaining submatrix must be psd but we have from 1) that it has trace zero therefore it must be equivalent to a zero matrix. 3) Any 22 psd matrix with a zero diagonal element has zero off-diagonal elements. Applying this to all matrices 0 @ 12 (1j; 1j) 12 (1j;ij) 12 (ij; 1j) 0 1 A , we find all the elements 12 (1j;ij) = 0. This completes the proof that 12 = 1 2 . The conditions under which one can meaningfully associate a reduced dynamics (or “subdynamics”) with the system state alone have been the subject of much research, and have been greatly clarified by Lindblad [Lindblad, 1996], who, following a lead by Pechukas [Pechukas, 1994], showed that the factorizability assumption is necessary in order for there to exist a subdynamics over the entire subsystem state space (i.e., no domain restrictions). Lindblad also emphasized that if the factorizability assumption is dropped it is still possible to have a subdynamics as an approximation involving assump- tions on the time scale and on the relaxation properties of the bath. These assumptions are weaker than those needed to make the dynamics Markovian [Alicki and Lendi, 1987, Breuer and Petruccione, 2002], but they are of a similar type. The relevance of com- plete positivity has been hotly debated [Pechukas, 1994, Alicki, 1995, Pechukas, 1995]. A number of studies have addressed the problem of dynamical maps that are not neces- sarily CP [Lindblad, 1998, Royer, 1996, Stelmachovic and Buzek, 2001, Jordan et al., 22 2004, Carteret et al., 2008], or even non-linear [Romero et al., 2004, Ziman, 2006]. Another approach has been to show that the factorization assumption can sometimes be relaxed, in the sense that in the weak-coupling limit the trace of the non-factorized part of the initial state vanishes in a short time [Tasaki et al., 2007]. That quantum subdynamics need not always exist can be understood from a simple and physically relevant example. Consider a system and bath undergoing joint periodic unitary evolution with periodT . Then, if we start from a product state S B , after timeT system and bath return to their initial product state. This CP map on the system state is the identity operatorIE 0T . Now consider two different transformations on the system: one fromt = 0 to an intermediate timet 0 < T , which we denoteE 0t 0, and the other fromt 0 to the end of the periodT , which we denoteF t 0 T . The transformation E 0t 0 is a CP map, andI =E 0T =F t 0 T E 0t 0. But since typically the joint system-bath state will not be factorized att 0 , a quantum subdynamics will not exist for all initial states, so by Lindblad’s result [Lindblad, 1996]F t 0 T is in general not a map (let alone CP). In fact,E 0t 0 need not even be invertible (it may be invertible over a restricted domain). 2.2 Quantum subdynamics and the assignment map One can formulate the problem of quantum subdynamics as follows. We wish to find a map from S (0) to S (t). For all t, the reduced system density matrix is defined as S (t) = Tr B [ SB (t)], and the evolution of SB (t) is unitary: SB (t) = U SB (t) SB (0)U y SB (t), where U SB is the unitary propagator of the joint system-bath dynamics. Thus, the following diagram holds: S (0) Tr B SB (0) U SB (t) ! SB (t) Tr B ! S (t): (2.7) 23 In order to arrive at a map from S (0) to S (t), it is thus clear that one should be able to invert the first arrow, i.e., find a map from S (0) to SB (0). Determining is the problem known as finding the “assignment map” [Pechukas, 1994, Alicki, 1995] (or “preparation map” [Stelmachovic and Buzek, 2001, Ziman, 2006], or “blow-up map” [Romero et al., 2004], or extension map [Carteret et al., 2008]). Once this is done, one can write the quantum dynamical process (QDP) for the open systemS interacting with a bathB as S (0)! S (t) = (t) S (0) = Tr B [U SB (t)( S (0))U y SB (t)]: (2.8) The unitary evolution and the partial trace are both CP maps, and hence their composi- tion is a CP map. The assignment map S (0) = SB (0), assigns to each S (0) a unique state of the total systemS +B, and may or may not be CP. The traditional factorizing assignment map, S (0) = S (0) B (0), is CP, and with it the full map (t) is also CP. Pechukas proved a fundamental result about assignment maps [Pechukas, 1994], by considering the case of a two-dimensional Hilbert space. Assume the following conditions: (0) The domain of is the set of all positive semidefinite system states S . (i) is linear-convex: [ S1 + (1) S2 ] = S1 + (1) S2 for all states S1 ; S2 ( preserves mixtures). (ii) is consistent in the sense that S = Tr B [ S ] for all S . (iii) S 0 if S 0. (iv) The state of the bath, B = Tr S [ S ], is independent of S . 24 Then must be factorizing: S = S B . Pechukas then argued that one should give up the positivity condition (iii), since factorized initial conditions are often unphys- ical, in particular outside of the realm of weak system-bath coupling. This would, of course, correspond to an abandonment of the complete positivity requirement. He argued, furthermore, that a dynamical map could be properly defined on a subset of initial system states even when system and bath are initially correlated, thus effectively abandoning condition (0). If one accepts that factorized initial conditions are indeed often unphysical, Pechukas’ result implies a trade-off between conditions (0)-(iii). In a Comment [Alicki, 1995], Alicki argued in favor of (iii) and complete positivity, and proposed to restrict the domain of the assignment map, namely to abandon condition (0). Specif- ically, he proposed a manifestly CP assignment map with a restricted domain of ini- tial states S , namely those invariant under the “instrument preparation” operation V S = P n V n S V y n (with P n V y n V n =I). Those states that do not satisfy the invariance requirement S =V S , i.e., are instantaneously perturbed by the preparation step, also violate condition (ii) [Alicki, 1995]. In a Reply [Pechukas, 1995], Pechukas argued that the invariance requirement may be physically demanding to satisfy. Lindblad generalized Pechukas’ result and proved for arbitrary finite-dimensional Hilbert spaces that conditions (0)-(iv) imply that must be factorizing [Lindblad, 1996] (in fact this is a special case of a result by Takesaki about the lack of non-trivial condi- tional expectations in non-commutative operator algebras [Takesaki, 1972], which holds also for strictly infinite systems). For another proof see the appendix of [Jordan et al., 2004]. Later Fonseca Romero et al. [Romero et al., 2004], extended Pechukas’ argument and demonstrated that it holds not just for linear but also for affine time evolutions. I.e., subject to conditions (0)-(iv), but replacing the linearity of condition (i) by an affine 25 mapping, they proved that the assignment map must be factorizing, where as usual the environment density matrix must be independent of the system density matrix. This result is quite relevant for our purposes, as we will also be dealing with affine time evolutions, but under somewhat different assumptions: initially we will impose only conditions (0)-(ii),(iv) and later we will impose only conditions (i)-(iv). The idea of abandoning some subset of the condition (0)-(iv) has been adopted in a number of recent papers, each detailing a different approach to the subdynamics prob- lem. Shortly after the Pechukas-Alicki debate, Royer [Royer, 1996], using cumulant methods, derived an exact convolutionless evolution equation for the reduced system state while allowing all operators and states to be fully time-dependent (including the bath, i.e., no equilibrium or stationarity is needed). The price paid for this generality is that the transformation depend on the initial state [Eq. (22) in [Royer, 1996])], simi- larly to the general Kraus representation derived in [Tong et al., 2004]. Thus, in Royer’s approach one abandons not only conditions (0) and (iii), but also condition (i), namely the map cannot be meaningfully applied to convex combinations of states. In a more recent development Jordan et al. [Jordan et al., 2004] emphasized the lin- earity condition (i), and proposed abandoning condition (0). By restricting the domain of initial states to the so-called “compatibility domain” (the set of states of the single system described by varying mean values of quantities for that system that are compat- ible with fixed mean values of other quantities for the combined system in describing an initial state of the combined system), they showed that it is possible to obtain linear subdynamics if system and bath are initially correlated. For states within the compati- bility domain, the map is not necessarily CP and the spectrum of the Choi matrixD [Eq. (2.5)] can be divided into positive and negative parts. Jordan et al. showed that the non-CP map can be represented in a difference form, () =E 1 ()E 2 (), where 26 E 1 () = P >0 K K y ; andE 2 () = P <0 K K y , and both mapsE 1;2 are CP. Carteret et al. [Carteret et al., 2008], like Jordan et al. [Jordan et al., 2004], aban- don condition (0) and consider the domain of states for which the assignment map is positive, by definition. They then define a (non-CP) “physically accessible map” as the composition of such an assignment map with a CP map, and analyze the structure of the accessible maps and the conditions for attaining them. 2.3 Linear Quantum Map The QDP (2.8) is a transformation from SB (0) to S (t). However, since we are not interested in the state of the bath, it is natural to ask: under which conditions is the QDP a map from S (0) to S (t) 2 ? When is this map linear? A map L :M n 7!M m (whereM n is the space ofnn matrices) is linear iff it can be represented as L () = X i E i E 0y i (2.9) where the left and right “operation elements”fE i g andfE 0 i g are, respectively, mn andnm matrices. We reproduce the proof here for convenience: Eq. (2.9) immediately implies that L is a linear map. For the other direction, let f M = P n i;j=1 jiihjj jiihjj =njihj, where jii is a column vector with 1 at positioni and 0’s elsewhere, andji =n 1=2 P i jii jii is a maximally entangled state overH H, whereH is the Hilbert space spanned by fjiig n i=1 . f M is also an nn array of nn matrices, whose (i;j)th block isjiihjj. Construct two equivalent expressions for (I L )[ f M], whereI is the (nn)(nn) 2 We use the term “map” solely to indicate a transformation between two copies of the same Hilbert space, in particularH S 7!H S 27 identity matrix. (i) (I L )[ f M] is annn array ofmm matrices, whose (i;j)th block is L [jiihjj]. (ii) Consider a singular value decomposition (SVD) [Horn and Johnson, 1999]: (I L )[ f M] =UDV = P UjihjV = P ju ihv j. HereU andV are unitary,D = diag(f g) is diagonal and 0 are the singular values of (I L )[ f M]. Divide the column (row) vectorju i (hv j) inton segments each of lengthm and define an mn (nm) matrix E (E 0 ) whose ith column (row) is the ith segment; then E jii (hijE 0y ) is theith segment ofju i (hv j). Therefore the (i;j)th block ofju ihv j becomesE jiihjjE 0y . Equating the two expressions in (i) and (ii) for the (i;j)th block of (I L )[ f M], we find L [jiihjj] = P E jiihjjE 0y . Since 0 we can redefine E as p E and E 0 as p E 0 , which we do from now on. Finally, the linearity assumption on L , together with the fact that the setfjiihjjg n i;j=1 spansM n , implies Eq. (2.9). The probability interpretation of quantum mechanics restricts linear maps to being positive maps P , i.e., maps from positive operators to positive operators. Clearly, pro- vided the assignment map assigns to every valid system density matrix a valid system- bath density matrix, quantum maps are a subset of positive maps. Unfortunately, there is no simple representation [akin to the Kraus representation or Eq. (2.9)] known for posi- tive maps, though there is a useful characterization using so-called “block-positivity” for positive maps due to Jamiołkowski [Jamiołkowski, 1972] (see [ ˙ Zyczkowski and Bengts- son, 2004] for a detailed review). Therefore it is of interest to consider a relaxed class of maps, from Hermitian operators to Hermitian operators. Such maps H are called Hermitian maps. Theorem 1 A map H :M n 7!M m is Hermitian iff it can be represented as H () = X i c i E i E y i ; c i 2R: (2.10) 28 Note that – unlike the CP maps case – thec i may be negative and hence we cannot absorb them into the operation elementsE i . Theorem 1 is not new; see, e.g., Refs. [Hill, 1973, Jordan et al., 2004, Royer, 1996]. However the proof we give next is new. It is inspired by methods developed in [ ˙ Zyczkowski and Bengtsson, 2004]. Proof Eq. (2.10) immediately implies that is a Hermitian map (we drop the subscript H for notational simplicity). For the other direction, associate a matrix L with the Hermitian map : 0 = ()() 0 m =L m n n (summation over repeated indices is implied). Hermiticity of and its image 0 implies 0 m = 0 m =L m n n =L m n n , i.e., L m n = L m n . We can use this property of L to show that if is a Hermitian map, thenI is Hermiticity preserving. ConsiderM = M n k jkihj jnihj. ThenM 0 = (I )[M] = M n k jkihj (jnihj) = M m k jkihj L m n jnihj. Assume thatM m k = M m k . This property holds forM = f M = jihj where ji = dim(H) 1=2 P i jii jii is a maximally entangled state overH H (M m k 1). ThenM 0y =M m k jihkj L m n jihnj =M m k jihkj L m n jihnj =M 0 . Therefore (I )[jihj] is Hermitian and invertible. It follows that the SVD used in the proof of the previous theorem can be replaced by standard diagonalization (U =V y ). In this case the left and right singular vectorsju i =hv j y are the eigenvectors of (I )[jihj] andc = are its eigenvalues. ThenE =E 0 in Eq. (2.9) andc 2R. By splitting the spectrum of (I )[jihj] into positive and negative eigenvalues, fc + i 0g andfc i 0g, we have as an immediate corollary a fact that was also noted in [Jordan et al., 2004,?]: Any Hermitian map can be represented as the difference of two CP maps: () = P i c + i E + i E +y i P i jc i jE i E y i . In [Shabani and Lidar, 2009b], the author and D. Lidar discuss a general formalism to represent the dynamics of an arbitrary open system by linear map. This becomes possible by splitting density matrix of a bipartite system into two parts: One is a sum- mation of the product terms ij jiihjj ' ij of which Tr[' ij ]6= 0 or' ij = 0 (fjiig are 29 orthonormal states in the system Hilbert spaceH S =N), the other one includes product terms ij jii S hjj ij with bath operator ij satisfying ij 6= 0 and Tr[ ij ] = 0: If the expansion terms of a density matrix SE (t) are of the former form only, the state SE (t) is called special linear (SL). Hereafter, we refer to a product term in the former type as a SL term and the latter would be non-SL. SE = X ij(SL) ij jiihjj ' ij + X ij(nonSL) ij jiihjj ij (2.11) We are interested in the dynamical mapping Q (t;t 0 ) : S (t 0 )! S (t): Notice that the second term in decomposition [2.11] has no contribution to the system only state S = Tr E [ SE ]: Consequently an affine form of the quantum map [2.8] is obtained Q (t;t 0 )[ S (t 0 )] = SL (t;t 0 )[ S (t 0 )] +K nonSL (2.12) Lemma 1 If SB (0) is an SL-class state then the QDP (2.8) is a linear map L : S (0)7! S (t). Proof Consider the singular value decomposition (SVD) ij = P ij jx ij ihy ij j, where ij are the singular values andjx ij i (hy ij j) are the right (left) singular vectors. Letfj k ig be an orthonormal basis for the bath Hilbert spaceH B , and define the system operators V kij h k jU SB jx ij i,W kij h k jU SB jy ij i. Since SB (0) is an SL-class state, a QDP 30 (2.8) generated by an arbitrary unitary evolutionU SB yields [recallCf(i;j)jTr[ ij ] = 1g]: S (t) = Tr B [ SB (t)] = X ij % ij Tr B [U SB jiihjj ij U y SB ] = X (i;j)2C;k; ij % ij V kij jiihjj(W kij ) y : (2.13) Now note thatP i S (0)P j = % ij jiihjj, whereP i jiihij is a projector and (i;j)2C. Therefore: L [ S (0)] X (i;j)2C;k; ij V kij P i S (0)P j (W kij ) y (2.14) = X (i;j)2C;k; ij % ij V kij jiihjj(W kij ) y ; (2.15) which equals S (t) according to Eq. (2.13). This defines the linear map L = fE ijk ;E 0 ijk g, whose left and right operation elements arefE ijk p ij V kij P i g and fE 0 ijk p ij W kij P j g, respectively. In addition, the non-SL terms in Eq.(2.11) imply the independent termK nonSL = P ij ij Tr E [U SE jiihjj ij U y SE ]. Now we take a further step to argue what is found to be an affine map (2.12) is actually a linear map if the map is acting only on the space of the density matrices. This is a direct application of the result in Ref. [Jordan et al., 2004]. Theorem 1 A reduced dynamics of a system coupled to a second system (environment) is representable as a linear map L (t;t 0 ) : S (t 0 )7! S (t). Proof For the space of system operators, we can find a basis ofN 2 mutually orthogonal Hermitian matricesF (I = F 0 ) : Tr E [F F ] = N . Then the density matrix S (t 0 ) 31 can be expanded as S (t 0 ) = 1 N (I + P N 2 1 =1 Tr[ S (t 0 )F ]F ) and the system final state is found to be S (t) = 1 N ( SL (I) + N 2 1 X =1 Tr[ S (0)F ] SL (F )) +K nonSL (2.16) We simply construct the equivalent linear map L by setting L (I) = SL (I) + NK nonSL and L (F ) = SL (F ). Hermicity of this map is easy to observe for all the components are Hermitian. We need to show that S (t) = H [ S (0)] = y S (t) if S (0) = y S (0). This is now a simple calculation which uses Eqs. (2.14) and (2.15), the definitions ofV kij andW kij , ij = y ji , and the SVD of ij . Discussion.— Note that Tr[jjihij I B SB ] = % ij Tr[ ij ], so that non-SLness can also be written as Tr[jjihij I B SB ] = 0, i.e., ashjiihjj I B ; SB i = 0 (Hilbert- Schmidt inner producthA;Bi Tr[A y B]) and hence SB must lie in the hyperplane orthogonal tojiihjj I B . Thus non-SL-class states are confined to a lower-dimensional surface in the space of bipartite states, and must be sparse. Note that, conversely, the SL condition Tr[ ij ] = 1 yields Tr[jjihij I B SB ] = % ij , which is not a constraint since % ij is arbitrary. 2.4 Initial system-bath states yielding a CP map When is a quantum map completely positive (CP) [Kraus, 1983]? This is a fundamen- tal question which has been the subject of intense studies with a long history [Sudar- shan et al., 1961, Pechukas, 1994, Alicki, 1995, Lindblad, 1996, Stelmachovic and Buzek, 2001, Jordan et al., 2004, ˙ Zyczkowski and Bengtsson, 2004, Carteret et al., 2008, Rodriguez et al., 2008], also more recently in the context of non-Markovian master 32 equations [Budini, 2006, Breuer, 2007, Vacchini, 2008]. One reason that this question has attracted so much interest is the fundamental role played by CP maps in quantum information [Nielsen and Chuang, 2000] and open quantum systems theory [Breuer and Petruccione, 2002]. CP maps are the “workhorse” in these fields, and hence an under- standing of their domain of validity is essential. For this reason it is perhaps surprising that the problem of identifying the general physical conditions under which CP maps are valid has remained open since it was first posed in a vigorous debate [Pechukas, 1994, Alicki, 1995]. In particular, while sufficient conditions have been developed for complete positivity [Alicki, 1995, Rodriguez et al., 2008], it is not known which is the most general class of states for which the QDP (2.8) is always CP, for arbitraryU SB . In this section we settle this old open question. We prove that the QDP yields a CP map S (0)7! S (t) iff SB (0) has vanishing “quantum discord” [Ollivier and Zurek, 2002], i.e., is purely classicaly correlated. it was recently shown [Hayashi et al., 2003] that the QDP (2.8) with arbitrary SB (0) becomes a CP map iff a most restrictive condition is satisfied byU SB (t), namely, it must be locally unitary:U SB (t) =U S (t) U B (t), i.e., the effective system-bath interaction must vanish. If one gives up the consistency condition S = Tr B [ SB ] for all S , or gives up linearity except in the weak coupling regime, CP maps arise for more general initial states [Alicki, 1995]. In [Rodriguez et al., 2008], Rodriguez et al. showed that CP maps arise for arbi- traryU SB even for certain non-factorized initial conditions, namely provided the initial state SB (0) is invariant under the application of a complete set of orthogonal one- dimensional projections on S, i.e., the state has vanishing quantum discord. Here we show that vanishing quantum discord is not only sufficient but also necessary for the QDP to induce a CP map (Theorem 2). Before stating the result, we need one more definition about the block structure associated with a matrixA = [a ij ]: 33 Definition 1 We call two diagonal elements a i 1 i 1 and a i B i B “block-connected via the pathfi b g B1 b=2 ” if there exists a set of unequal indexesfi b g B b=1 such thatfa i b i b+1 g B1 b=1 are all non-zero, i.e., they can be connected via a path that involves only horizontal and vertical (but not diagonal) moves. The “block-index set”D () A is the set of all index pairsf(i;j)g of the elements of theth block ofA. This is just the standard notion of a block in a matrix, possibly before permutation matrices are applied to sort it into the standard block-diagonal structure. We are now ready to state our main result. Lemma 2 Let SB (0) be an SL-class state, let [ ij ] = L () (a supermatrix), and letf P (i;i)2D () jiihijg be a complete set of projectors fromH S toH S . Let C () f(i;j)2D () jTr[ ij ] = 1g and () S S (0) =p = X (i;j)2C () % ij jiihjj=p ; (2.17) where p = Tr[ S (0) ]. Let () B be a density matrix. The Hermitian map H : S (0)7! S (t) induced by the QDP (2.8) is a CP map iff ( () ) ij =f0 or () B g8(i;j)2 D () : SB (0) = X p () S () B : (2.18) Clearly, () S can be thought of as the post-measurement state arising with probability p from S (0) after the application of the projective measurement described by the set f g. Moreover, SB (0) is not merely separable: Theorem 2 The Hermitian map H : S (0)7! S (t) is a CP map iff the initial system- bath state SB (0) has vanishing quantum discord (VQD), i.e., can be written as: SB (0) = X k; k SB (0) k ; (2.19) 34 wheref k g are one-dimensional projectors onto the eigenvectors of () S , and P k k = . Proof By expanding () S as P k p k k , withp k = Tr[ S (0) k ] 0 and P k p k = 1, we obtain using Eq. (2.18): SB (0) = P () S () B = P k; p k k () B , which implies Eq. (2.19). On the other hand P k; k SB (0) k is the state after a non-selective pro- jective measurementf k g onS, so that SB (0) = P k; p k k () B . The quantum discord has a deep information-theoretic origin and interpretation, for the details of which we refer the reader to Ref. [Ollivier and Zurek, 2002]; we shall merely remark that when the discord vanishes all the information about B that exists in theS-B correlations is locally recoverable just from the state ofS, which is not the case for a general separable state ofS andB. In this sense a VQD state is “completely classical”. Proof of Lemma 2 We start with necessity; sufficiency will turn out to be trivial. Let us assume that the Hermitian map H : S (0)7! S (t) induced by the QDP, S (t) = Tr B [U SB (0)U y ]; is CP, and determine the class of allowed initial states. We start from an SL-class state since we know that in this case the QDP (2.8) is indeed equivalent to a Hermitian map. Let f M =j ih j, wherej i = 1 p d S P d S i=1 jii jii is a maximally entangled state overH S H S , and where d S = dimH S . It follows directly from Eq. (2.15) below that H [jiihjj] = Tr B [Ujiihjj ij U y ]. Thus the Choi matrix [Choi, 1975] for L is M (I H )[ f M] = 1 d S X ij jiihjj H [jiihjj] = 1 d S X ij jiihjj Tr B [Ujiihjj ij U y ]; (2.20) 35 We assume thatM is positive as this is equivalent to H being CP [Choi, 1975]. A useful fact is that a matrixA is positive iff every principal submatrix ofA is positive (a principal submatrix is the matrix obtained by deleting fromA some number of columns and rows with equal indexes). Therefore, let us focus on the pair of rows and columns (k;l) (k6=l) ofd S M, and consider the 2 2 principal submatrix P kl = 0 @ Tr B [Ujkihkj kk U y ] Tr B [Ujkihlj kl U y ] Tr B [Ujlihkj lk U y ] Tr B [Ujlihlj ll U y ] 1 A : (2.21) The submatrixP kl must be positive for anyU, and we choose to examine the caseU = 1 p 2 (I IiX A), whereA is Hermitian and unitary (henceA 2 =I), andX =jkihlj+ jlihkj+ P i6=k;l jiihij. This will allow us to find restrictions onf kl g. Note that it follows from Hermiticity ofA, kk and ll , and from y kl = lk , that Tr[A kk ]; Tr[A ll ]2R, and that Tr[A kl ] = (Tr[A lk ]) . Thus some algebra yields: P kl = 1 4 0 B B B B B B B @ t kk ia ib t kl ia t kk t kl ib ib t kl t ll ic t kl ib ic t ll 1 C C C C C C C A ; (2.22) a = Tr[A kk ]2R; b = Tr[A kl ]; c = Tr[A ll ]2R; t ij = Tr[ ij ] = 1 or 0: To proceed we require the following Lemma (proof at end): Lemma 3 If Tr[AX] = 0 for any unitary and Hermitian matrixA thenX = 0. Proposition 1 If kk = 0 or ll = 0 then kl = lk = 0. Proof Assume that ll = 0 or kk = 0, but not both, so that either (t ll = 0;t kk = 1), or (t ll = 1;t kk = 0). Construct the principal submatrix obtained by deleting rows 36 and columns 1 and 3 from P kl . This leaves a principal submatrix with eigenvalues (1 p 1 + 4jbj 2 )=8. The positivity of these requires b = Tr[A kl ] = 0, so that by Lemma 3 kl = y lk = 0. When ll = kk = 0 the same principal submatrix has eigenvaluesjbj, so that again kl = y lk = 0. Proposition 2 If all of kk ; ll ; kl 6= 0 then kk = ll = kl = lk . Proof After a couple of elementary row and column operations onP kl we obtain: P 0 kl = 0 @ 1 2 B B y 1 2 1 A ; 1 2 = 0 @ 1 1 1 1 1 A ; B = 0 @ ib ia ic ib 1 A : (2.23) Diagonalizing the two diagonal blocks 1 2 using Q = 1 p 2 (I + i y ) yields P 00 kl = Q 2 P 0 kl (Q y ) 2 , where P 00 kl = 0 @ C D D y C 1 A ; C = 0 @ 2 0 0 0 1 A ; D =i 0 @ 1 A ; = (a +b +b +c)=2; = (ab +b c)=2; = (ab +b +c)=2; = (a +b +b c)=2: Positivity of P kl implies that also P 00 kl > 0, so that we can again apply the principal submatrix method. Lete(i;j) denote the eigenvalues of theP 00 kl submatrix obtained by retaining only theith andjth rows and columns ofP 00 kl . We finde(1; 4) = 1 p 1 +jj 2 , e(2; 3) = 1 p 1 +j j 2 and e(2; 4) =jj 2 . Since all these eigenvalues must be positive we conclude that = = = 0, i.e., Tr[A kk ] = Tr[A kl ] = Tr[A lk ] = Tr[A ll ]. Applying Lemma 3 we have Tr[A( kk kl )] = 0, so that kk = kl , and similarly kl = lk = ll . 37 It is simple to check that the only permissible case not covered by Propositions 1 and 2 is when kk ; ll 6= 0 and kl = lk = 0; in this case we have no further restrictions. Lemma 4 The matrix [ ij ] can be decomposed as = L () , where ( () ) (i;j)2D () = () (a constant) or 0. Proof Every matrix is a direct sum of blocks (possibly only one). Therefore our task is to prove that the matrix elements of theth block () obey ( () ) (i;j)2D () = () or 0. Collecting the results above we see that there are only four cases: Proposition 1 =) (i) kk = kl = lk = ll = 0, (ii) kk = kl = lk = 0 and ll 6= 0; Proposition 2 =) (iii) kk = kl = lk = ll 6= 0; (iv) kk ; ll 6= 0 and kl = lk = 0. First note that if kk = 0 then by cases (i) and (ii) also kl = lk = 08l, i.e., the row and column crossing at a zero diagonal element must be zero. Now let () ij denote the 22 principal submatrixf () ii ; () ij ; () ji ; () jj g, i6= j. Assume () ij 6= 0 and consider () ij . Only case (iii) applies, so () ii = () ij = () ji = () jj . We can use this to show that any two block-connected diagonal elements are equal. Indeed, assume that () i 1 i 1 and () i B i B are both non-zero and block-connected via the pathfi b g B1 b=2 . Then by case (iii) all elements of each member of the set of principal submatricesf () i b i b+1 g B1 b=1 are equal, and since successive members always share a diagonal element, their elements are all equal, to an element we call () . We have thus shown that ( () ) (i;j)2D () = () or 0. Finally, note that case (iv) with kk 6= ll can only arise between two different blocks, since if kk 6= ll the previous argument shows that they cannot be block-connected. We are now ready to conclude the proof of Lemma 2: It follows from Lemma 4 that () ij = ( () ) ij = () or 0 for (i;j)2D () . Moreover, since SB (0) is an SL-class state, Tr[ () ] = 1. Thus the total index setD for the initial state SB (0) splits into a union of disjoint index setsD () , so that Eqs. (2.17) and (2.18) are satisfied, where S (0) = P P (i;j)2C () % ij jiihjj = P S (0) = P () S , where () S = p () S 38 and where () B () . Here is the projector onto the subspace corresponding to block (as defined above). Next we need to show that the B ’s are density matrices. From the properties of the () we already have Tr () B = 1, so what is left to prove is that () B > 0. Indeed, by definition of positivityh B jhi () j SB (0)ji () ij B i> 0 for any state ji () i in the support of and any bath statej B i. Inserting SB (0) = P p () S () B into this inequality, we findh B j () B j B i > 0;8j B i2H B . This completes the proof of necessity. Sufficiency: using the spectral decomposition B = P j j j j ih j j and definingE ij h i jUj j i :H S 7!H S , wherefj i ig is an orthonormal basis forH B , we have, using Eqs. (2.8) and (2.17): S (t) = Tr B [U SB (0)U y ] = X ij i E ij S (0)E y ij : (2.24) Now we simply note that if SB (0) satisfies Eq. (2.18) with () B > 0 (i.e., j > 0), then Eq. (2.24) is already in the form of a CP map, with operation elementsf p i E ij g ij . 2.5 Positivity Domain One fundamental issue in defining a non-CP quantum map is to find a valid domain for which the map is meaningful. One minimal condition for the validity is the domain for which the output of the map is positive that is called positivity domain. This is required to assure the positivity of the output density matrix. Here we propose a geo- metric method to characterize the positivity domain of a given linear quantum map. In Ref.[Kimura and Kossakowski, 2005], a complete geometric picture of density matrices is given by using the Bloch vector representation for an arbitrary dimensional Hilbert space N = dimfH S g: Letf i ;i = 1;:::;N 2 1g be a complete basis for the set of 39 traceless Hermitian matrices which are mutually orthogonal with respect to the Hilbert- Schmidt inner product, i.e. Tr[ i j ] = 2 ij . Then, any density matrix can be written in the form = 1 N I N + 1 2 N 2 1 X i=1 b i i (2.25) whereb i = Tr[ i ]: A Bloch vector is defined by elementsb i , b = (b 1 ;:::;b N 2 1 ). The complete knowledge of a Bloch vector yields the complete knowledge of a den- sity matrix. A Bloch spaceB(R N 2 1 ) is known to be a closed convex set completely characterized by the minimum eigenvalues of i s,m( i ). B(R N 2 1 ) = b =rn2R N 2 1 :r 2 Njm( n )j (2.26) where n is a unit vector, i.e. n2 R N 2 1 ( P N 2 1 i=1 n 2 i = 1). This is the simplest geometric picture which one may wish for. Now we apply the linear map under study to the closed convex set of density matrices to show that the positivity domain is a convex set too, and then introduce an algorithm to specify this convex set. Proposition 3 The positivity domain (PD) of a linear quantum map is a convex set. Proof Consider two points inside the PD with Bloch vectors b = (b 1 ;:::;b N 2 1 ) and b 0 = (b 0 1 ;:::;b 0 N 2 1 ). The claim is that a third point b 00 () = (b 00 1 ;:::;b 00 N 2 1 ) defined by b 00 () =b+(1)b 0 is inside the PD. This is trivial by linearity of the quantum map Linear : 40 Linear ( 1 N I N + 1 2 N 2 1 X i=1 b i i )> 0; Linear ( 1 N I N + 1 2 N 2 1 X i=1 b 0 i i )> 0 =) Linear ( 1 N I N + 1 2 N 2 1 X i=1 b i i ) + (1) Linear ( 1 N I N + 1 2 N 2 1 X i=1 b 0 i i )> 0 =) Linear ( 1 N I N + 1 2 N 2 1 X i=1 b 00 i i )> 0 (2.27) We proved that PD is convex, so in the above geometric picture the PD will be a convex set inside the convex set of density matrices. Therefore any line passing through the origin intersects the boundary of PD at two points b 1 and b 2 . These two points are on the boundary of the density matrices space, or if they are interior points then det[ Linear ( 1 N I N + 1 2 N 2 1 X i=1 b (1;2) i i )] = 0 (2.28) For each line passing through the origin, the points on the border of the PD can be found in this way. Then the algorithm would be to construct the boundary of the PD by finding the border points in all directions. Of course some resolution is needed to be chosen for the spatial angle. This yields a complete geometry of PD of a given linear quantum map. 2.6 Discussion and Outlook In this chapter, we introduced a linear map representation for dynamics of an open system. A reader may ask about the condition under which the linearity of the map holds. Our answer is that if you look at the construction of the linear map (2.15), you find the Kraus operators are functions of system-bath propagatorU SB and bath operators ij , therefore the linearity of Eq. (2.15) is conditioned by operators ij to be fixed. 41 These operators determine how quantum and classical correlations between system and bath can change the dynamical behavior of the open system. To see this, note that SB = P i % ii jiihij ii is a state with vanishing quantum discord ( SB = P i i SB i , where i = jiihij are one-dimensional projectors onH S ). Thus the “diagonal” set f ii g quantifies the purely classical correlations in the initial state. On the other hand, SB SB = P i6=j % ij jiihjj ij , where SB = P ij % ij jiihjj ij is a general SL initial state. Thus ij withi6= j are exactly the operators that appear in the non-classically correlated (non-VQD) part of SB . Thus, our formulation shows how system-bath correlations may affect the dynamics for a general system state S = P % ij jiihjj. We found the bath-only operatorsf ij g to be quantifying the role of correlations; this is so for any choice of% ij . (Such generality cannot be found in other related works) By fixing thef ij g we fix the initial system- bath correlations. Together with fixing U this completely determines the form of the linear map. In our formulation it is not possible to carry out an experiment under the “standard paradigm” where the only variable is the initial system state. The reason for this negative answer is exactly the presence of correlations. Only when these are absent is it possible to fix the initial bath state and vary only the initial system state, while still obtaining a linear (CP) map. Is this a deficiency of our formulation? We believe not: this is a natural outcome of the correlated scenario we are considering, and the linear map form of the formulation we have presented is strong enough to provide quantitative answers to the following key questions: 1. How do initial system-bath correlations affect the dynamics? 2. What class of initial system-bath states results in a CP map? Nonlinear Quantum Maps — Nevertheless, a formalism will be more appealing if it can be applied to identify a noise process. By identification we mean elaborating 42 a recipe for noise process tomography. Needless to say, any formalism would be more interesting by being general without any approximation. Such a formulation should be of the form S (final) =E(fp g)[ S (initial)] (2.29) wherefp g is the set of tunable parameters in an experiment. 43 Chapter 3 Quantum Dynamical Equation In the study of a quantum system dynamics, we are more interested in real time changes of the state rather than a point to point time variation. For it reveals the effect of funda- mental forces deriving the system. In the beginning of this section we showed that an open quantum system has a general dynamical equation of the form _ S (t) =Tr B [ _ U SB (t) SB (0)U y SB (t)] +Tr B [U SB (t) SB (0) _ U y SB (t)] (3.1) This differential equation seems to be very complicated and useless. The type of the equations that we like to work with is an integro-differential equation with the integrand being a functional of the system state at all the times ( S (t);t 0 <t<t final ). The history of derivation of such equations goes back to the early works by Enrico Fermi and Paul Dirac on the quantum theory of emission and absorption of radiation [Cohen-Tannoudji et al., 1977, Dirac, 1927]. Later on several attempts were made to derive master equa- tions for different physical systems in different approximation regimes. Here we study those type of equations which resemble a Markovian dynamics, those for which change of the system state is a functional of the state at that time only; those which do not obey this rule are known as non-Markovian or post-Markovian master equations/dynamics. 44 3.1 Markovian Dynamics As we defined above, a quantum system is said to obey a Markovian dynamics if the dynamical equation (3.1) can be well approximated by _ (t) =L t (t) (3.2) whereL t is a super-operator acting on the operator(t). In many systems for instance in typical quantum optical situations the physical con- ditions underlying the Markovian approximation are very well satisfied. A microscopic derivation of the Markovian master equation is possible on the ground of various approx- imation schemes: 1- Weak coupling between system and bath:jjH int jj<<jjH S jj;jjH B jj. 2- Singular coupling between system and bath: H SB = lim !0 H S + 2 H B + 1 H int 3- Low-density limit: The system consists of a gas of particles with low density and the master equation effectively describes the dynamics of the internal degrees of freedom. Most of physical situations fall in the first category of weak coupling. If you look at the differential equation 3.1 you find that _ S (t) depends on SB (0). This means dynam- ics has a long memory, then how does this memory become negligible in a Markovian dynamics? Actually the secret of Markovian dynamics is that the system experience the same bath at each moment. In other words the bath is so large and rigid that is not affected by interacting with the system, or SB (t) S (t) B (3.3) 45 For this purpose the bath degrees of freedom must be so large that any correlation in the system and bath state disappears on the system relaxation time scale R . Therefore, the bath time scale B should satisfy B << R . We saw before that the condition SB = S B results in a CP map. Therefore the condition (3.3) guarantees that the evolution map between system states and between any two times is described by a CP map. This restricts the allowed form of the superop- eratorL t . The solution of the Eq. (3.2) isTe R t 2 t 1 L t 0dt 0 and this transformation must be CP for any pair of times (t 1 ;t 2 ). The most general form ofL satisfying this condition was first found by Gorini, et al. [V . Gorini and Sudarshan, 1976] and separately by Lind- blad [Lindblad, 1976]. This equation is named the Lindblad equation after his general formulation L t :=i[H S (t);] + 1 2 X a a (t)([F (t);F y (t)] + [F (t);F y (t)]): (3.4) The Lindblad operatorsF ’s are bounded operators acting onH S , and thea 0 are scalars that describe decoherence rates. Example 1 An example of Markovian dynamics is a quantum system, e.g. atom or molecule, which interacts with a radiation field with an infinite number of degrees of freedom. The Hamiltonian of a quantized field can be written as H B = X ! k X =1;2 ~! k b y k ( ! k )b k ( ! k ) (3.5) and we assume the interaction Hamiltonian to be given in the dipole approximation H I = ! D: ! E: (3.6) 46 In addition, let us take the radiation field to be in an equilibrium state at temperature T , B = ! k; (1exp[~! k ])exp[~! k b y k ( ! k )b k ( ! k )] (3.7) This leads to a Lindblad equation, d dt S (t) = i ~ [H LS ; S (t)] +D( S (t)) (3.8) The Hamiltonian H LS = P ! ~S(!) ! A y (!): ! A(!) is the renormalized system Hamiltonian H S with ! A(!) P 0 () ! D( 0 ), where () is the eigenoperator ofH S . The dissipator of the quantum master equation takes the form D( S (t)) = X !>0 4! 3 3~c 3 (1 +N(!))( ! A S ! A y 1 2 ! A y : ! A; S ) + X !<0 4! 3 3~c 3 N(!)( ! A y S ! A 1 2 ! A: ! A y ; S ) (3.9) whereN(! k ) = (exp[~!k] 1) 1 . 3.2 Non-Markovian Dynamics It is possible to write down an exact dynamical equation for an open system, but the result – an integro-differential equation [Nakajima, 1958, Zwanzig, 1960] – is mostly of formal interest, as such an exact equation can almost never be solved analytically or even numerically. In contrast, when one makes the Markovian approximation, i.e., when one neglects all bath memory effects, the resulting Lindblad master equation [V . Gorini and Sudarshan, 1976, Lindblad, 1976, Alicki and Lendi, 1987] is formally solvable and 47 amenable to numerical treatment. Moreover, the desirable property of complete posi- tivity [Kraus, 1983] is maintained (see, however, [Pechukas, 1994, 1995, Alicki, 1995] for a debate on the importance of this property). A coveted goal of the theory of open quantum systems [Breuer and Petruccione, 2002, Alicki and Lendi, 1987] is a “post- Markovian” master equation that (i) generalizes the Markovian Lindblad equation so as to include bath memory effects, at the same time (ii) remains both analytically and numerically tractable, and (iii) retains complete positivity. A variety of post-Markovian master equations have been proposed and analyzed, e.g., [Breuer and Petruccione, 2002, Shibata and Takahashi, 1977, Chaturvedi and Shibata, 1979, Imamoglu, 1994, Royer, 1996, 2003, Garraway, 1997, Breuer et al., 1999, Knezevic and Ferry, 2004, Gambetta and Wiseman, 2002, Breuer, 2004b, Strunz et al., 1999, Yu et al., 2000, Barnett and Stenholm, 2001, Daffer et al., 2004, Breuer, 2004a]. However, one of the desirable properties (i)-(iii) above is typically lost: e.g., in the case of time-convolutionless mas- ter equations (e.g., [Royer, 1996, 2003]) one may lose complete positivity, while in the case of nonlocal stochastic Schrodinger equations (e.g., [Strunz et al., 1999, Yu et al., 2000]) one loses analytical solvability. In this chapter we propose a new post-Markovian master equation that satisfies all of the desirable properties (i)-(iii) above. The key idea we introduce is an interpolation between the generalized measurement interpretation of the exact Kraus operator sum map [Kraus, 1983], and the continuous measurement interpretation of Markovian-limit dynamics [Dalibard et al., 1992, Gisin and Percival, 1992, Plenio and Knight, 1998, Breuer, 2004a]. 3.2.1 Quantum measurements approach to open system dynamics Consider a quantum system S coupled to a bath B (with respective Hilbert spaces H S ;H B ), evolving unitarily under the total system-bath Hamiltonian H SB . In addi- tion system and bath start with a product state SB (0) = (0) B (0) is the initially 48 uncorrelated system-bath state, as we derived the dynamics can be rewritten in terms of an operator sum (the Kraus representations [Kraus, 1983]) (t) = X k A y k (t)(0)A k (t); (3.10) where Tr[(t)] = 1, P k A k (t)A y k (t) =I. Let us now recall how to derive the exact Eq. (1.8) from a measurement picture [Fig. 7.1a]. Imagine the bath acting as a probe coupled to the system at t = 0, with the interaction given by H SB as above. To study the state of the system a sin- gle projective measurement is performed on the bath at time t, with a complete set of projection operatorsjiihij,H B = Spanfjiig. The measurement yields the result k and collapses the state of the bath to the corresponding eigenstatejki. This hap- pens with probability p k = Tr S [hkj SB (t)jki], and the system density matrix reduces to k (t) =hkj SB (t)jki=p k =: A y k (0)A k =p k , where A k are the Kraus operators. If we repeat this process for an identical ensemble initially prepared in state SB (0) the average system density matrix becomes(t) = P k p k k (t) = Tr B [U(t) SB (0)U y (t)], which is just Eq. (1.8), thus affirming the validity of this bath-measurement interpreta- tion of open system dynamics. The corresponding map is completely positive (CP). In contrast, in the Markovian limit the most general CP system dynamics is given in the interaction picture by the Lindblad equation [V . Gorini and Sudarshan, 1976, Lindblad, 1976] @ @t =L := 1 2 X a a ([F ;F y ] + [F ;F y ]): (3.11) The Lindblad operators F ’s are bounded operators acting onH S , and the a 0 are constants that describe decoherence rates. Now let us recall how also the Lind- blad equation can be given a measurement interpretation. Expanding Eq. (3.11) to 49 first order in the short time interval yields (t +) = (I 2 P F y F )(t)(I 2 P F y F ) + P F (t)F y . To the same order we also have the normalization con- dition (I 2 P F y F )(I 2 P F y F ) + P F y F =I. Thus the Lindblad equa- tion has been recast as a Kraus operator sum (3.10), but only to first order in , the coarse-graining time scale for which the Markovian approximation is valid [Lidar et al., 2001b]. Clearly, then, we again have a measurement interpretation, wherein as before the bath functions as a probe coupled to the system while being subjected to a contin- uous series of measurements at each infinitesimal time interval [Fig. 7.1b)]. This is the well-known quantum jump process [Dalibard et al., 1992, Gisin and Percival, 1992, Plenio and Knight, 1998], wherein the measurement operators areI 2 P F y F (the “conditional” evolution) and p F (the “jump”). We have thus seen how a measurement picture leads to the two limits of exact dynamics (via an evolution of the coupled system-bath followed by a single general- ized measurement at time t), and Markovian dynamics (via a series of measurements interrupting the joint evolution after each time interval). With this in mind it is now easy to see that by relaxing the many-measurements process one is led to a less restricted approximation than the Markovian one. Here we use this observation to derive a post- Markovian master equation based on a probabilistic single-shot measurement process. 3.2.2 Derivation of a post-Markovian master equation The first stage of exerting an approximation on the exact Eq. (1.8) should be to include one extra measurement in the time interval [0;t]. Thus we consider the following pro- cess: a probe (bath) is coupled to the system att = 0, they evolve jointly for a timet 0 (0t 0 <t) such that att 0 the system state is (t 0 )(0), where (t 0 ) is a one-parameter map, at which moment the extra generalized measurement is performed on the bath. does not depend ont since the bath resets upon measurement. System and bath continue 50 time M P time M M P time P M M M M M M Exact Solution: Markovian Approximation (Quantum Trajectories): Single-Shot Measurement: M P Measurement: Preparation: Figure 3.1: Measurement approach to open system dynamics. P=preparation, M= mea- surement, time proceeds from left to right. a) Exact Kraus operator sum representation, b) Markovian approximation, c) Single-shot measurement, d) Single-shot measurement followed by Markovian dynamics. their coupled evolution betweent 0 andt, upon which the final measurement is applied. This is illustrated in Fig. 7.1c. Since this intermediate measurement determines the sys- tem statej i at t 0 , after time tt 0 the system state will be (t) = (tt 0 )(t 0 ). It is important to stress that(t 0 ) cannot be written as (t 0 )(0), since the measurement selects(t 0 ) at random. The timet 0 characterizes bath memory effects and must be determined as a function of time-scales characterizing the evolution. We do this by introducing a bath memory function (kernel) k(tt 0 ;t) that assigns weights to different measurements: (t) = 51 R t 0 k(tt 0 ;t)(tt 0 )(t 0 )dt 0 . To derive a master equation we discretize the time interval [0;t] into N equal segments of length . We then have the weighted average (t = N) = P N m=1 k(m;N)(m)((Nm)). From hereon we assume that is trace- preserving, whence k must be normalized so that R t 0 k(t 0 ;t)dt 0 = 1 (k(t 0 ;t) = 0 for t 0 = 2 [0;t]) (though an exception to this will arise below). We then have (forN 1) (N)((N 1)) = N1 X m=1 k(m; (N 1))(m) [((Nm))((Nm 1))] + N1 X m=1 [k(m;N) k(m; (N 1))](m)((Nm)) +k(N;N)(N)(0): (3.12) In order to arrive at a differential equation the term proportional to (N)(0) must be made to vanish. We therefore impose the additional constraint lim !0 k(N;N)= = 0. Taking the limits! 0, m;N !1 such thatm = t 0 andN = t, we convert the remaining terms in Eq. (3.12) into differential form by expressing [((Nm))((N 1m))]=! @(tt 0 ) @(tt 0 ) and [k(m;N)k(m; (N 1))]=! @k(t 0 ;t) @t . Eq. (3.12) then yields: @ @t = R t 0 dt 0 [k(t 0 ;t)(t 0 ) @(tt 0 ) @(tt 0 ) + @k(t 0 ;t) @t (t 0 )(tt 0 )]. We would like to arrive at a proper integro-differential equation involving, on the right-hand-side (RHS), only and not its derivative. We thus assume, only in the derivate of on the RHS, that (tt 0 ) = (tt 0 )(0). Such an assumption is equivalent to the standard procedure 52 of first-order time-dependent perturbation theory, and can, analogously, be iterated self- consistently to obtain higher-order approximations. Expressing(0) = 1 (tt 0 )(t t 0 ) we then obtain the post-Markovian dynamical equation @ @t = Z t 0 dt 0 [k(t 0 )(t 0 ) _ (tt 0 ) 1 (tt 0 ) + @k(t 0 ;t) @t (t 0 )](tt 0 ): (3.13) This new, formal master equation is the first main result of this work. Note that in this integral form the constraint lim !0 k(N;N)= = 0 can be lifted, as it cannot change the value of the integral. To make further progress we now assume a Markovian form for the superoperator: (t) = exp(Lt). HereL can be interpreted as the Lindblad generator [Eq. (3.11)]. Using this in Eq. (3.13) yields @ @t = Z t 0 dt 0 [k(t 0 ;t)L + @k(t 0 ;t) @t ] exp(Lt 0 )(tt 0 ): (3.14) This master equation is rather interesting and appears amenable to analytical treatment, an undertaking which will be the subject of a future study. To make even further progress, let us note that Eq. (3.14) automatically preserves Tr, even without requiring normalization ofk via R t 0 k(t 0 ;t)dt 0 = 1. Since the latter was needed above to ensure trace preservation, it can now be dropped. This allows us to consider memory kernels satisfyingk(t 0 ;t) =k(t 0 ). We thus arrive at our second main result: @ @t =L Z t 0 dt 0 k(t 0 ) exp(Lt 0 )(tt 0 ) =Lk(t) exp(Lt)(t); (3.15) where denotes convolution andk no longer obeys any constraints. 53 Henceforth we confine our attention for simplicity and explicitness to the new post- Markovian master equation (3.15), though some of the results below are generalizable to Eq. (3.13). While k is still unspecified, we show below that it can be determined by an appropriate quantum state tomography experiment. As we further show below, Eq. (3.15) satisfies all the conditions we stated in the introduction for a “desirable” post- Markovian master equation. Finally, note that Eq. (3.15) reduces to a purely Markovian master equation, @=@t =L(t), when k(t 0 ) = (t 0 ), as expected for a memoriless channel. Dynamical map.— We now analytically derive the dynamical map (t) : (0)7! (t) governing our master equation. We solve the integro-differential equation (3.15) by taking the Laplace transform: se (s)(0) = [ e k(s) L sL ]e (s); (3.16) where e X(s) := Lap[X(t)] is the Laplace transform of the function X(t). Now con- sider the solution of the eigenvalue equationL = . It results in a set of (com- plex) eigenvaluesf i g and corresponding right and left eigenvectorsfR i g;fL i g that fulfill the orthonormality condition Tr[L i R j ] = ij . These eigenvectors are known as the damping basis [Briegel and Englert, 1993] of the superoperatorL. Expressing the density matrix in this basis as (t) = P i Tr[L i (t)]R i = P i i (t)R i and taking the Laplace transform, allows us to use Eq. (3.16) to solve for the expansion functions i (t): se i (s) i (0) = i e k(s i )e i (s) =) i (t) = Lap 1 [ 1 s i e k(s i ) ] i (0) =: i (t) i (0): (3.17) The functions i (t) can now be computed using the residue theorem formula applied to the Bromwich integral formula for the inverse Laplace transform: iff(s) = Lap[F (t)] 54 thenF (t) = P p k Res[e st f(s);p k ], wherep k are the poles ofe st f(s) and Res[g;p] := 1 (n1)! d n1 ds n1 [(sp) n g(s)] s=p is the residue of g, with n the order of the pole p. In our casef(s) = [s i e k(s i )] 1 and so the polesp k are determined by the solutions of the equations = i e k(s i ) fors. This equation can be solved once the Lindblad generatorL (yielding the i ) and the memory kernelk(t) are specified. Then i (t) = P p (i) k Res[e st f(s);p (i) k ]. Summarizing, the dynamical map corresponding to Eq. (3.15) is (t) :X7! X i i (t)Tr[L i X]R i : (3.18) Using the orthonormality of the damping basis it follows that (t) 1 : Y 7! P i i (t) 1 Tr[L i Y ]R i . Thus is invertible with the exception of the points where i (t) = 0. For contractive (e.g., Markovian) maps this will happen att =1, though in general additional points cannot be excluded. Condition for complete positivity of .— Let us recall Choi’s theorem [Choi, 1975]: “Let : GL(n;C) 7! GL(m;C) be a linear map. Then is CP iff the matrix whose elements aref [E ij ]g 1i;jn is positive, where E ij is a matrix with (E ij ) i;j = 1 and all other elements zero.” Using this theorem the criterion for com- plete positivity of our map is equivalent to positivity of the matrix P whose (i;j)th element is [jiihjj]. Namely, P 0 , f P k k (t)Tr[L k jiihjj]R k g 1i;jn = f P k k (t)hjjL k jiiR k g 1i;jn 0, which, in turn, is equivalent to: X k k (t)L T k R k 0: (3.19) The inequality (3.19) is a necessary and sufficient condition for our map to be CP. Because the functions k (t) are given in terms of the memory kernel k(t) through Eq. (3.17), this inequality results in a condition onk(t), which can be checked in order 55 to verify that a given such kernel results in a CP map. Further note that Eq. (3.15) pre- serves the trace of(t) [i.e.,d Tr(t)=dt = 0], as is evident from TrL = 0 and a Taylor expansion of exp(Lt). Kraus representation of .— Since the matrixP is positive it can be expressed as P = P k ja k iha k j where theja k i’s are the eigenvectors ofP . One can divide the vector ja k i into n segments of length n, where n = dim[H S ], and define a matrix M k with the ith column being the ith segment ofja k i, so that the ith segment is M k jii. Then the dynamical map is reconstructed asE() = P M M y , which is the desired Kraus representation. Connection to other master equations.— We first note that our master equa- tion (3.15) is an instance of the exact Nakajima-Zwanzig (NZ) equation : = R t 0 dt 0 O(t;t 0 )(t 0 ) [Nakajima, 1958, Zwanzig, 1960], where the NZ kernel O(t;t 0 ) is, in our case, of the special time translationally-invariant formO(t 0 t). Secondly, in the particular case thatjjLjj 1=t Eq. (3.15) reduces to @ @t =L Z t 0 dt 0 k(t 0 )(tt 0 ): (3.20) This master equation was proposed intuitively in Ref. [Barnett and Stenholm, 2001], where it was studied in the case of a damped harmonic oscillator and it was shown to lead, under certain assumptions, to unphysical behavior. This issue was clarified in the recent work [Daffer et al., 2004], where it was shown that a single qubit subject to telegraph noise can be described by Eq. (3.20), and where conditions for complete positivity of (3.20) were established; our inequality (3.19) includes this as a special case. Thirdly, we can rewrite Eq. (3.15) in time-convolutionless form using the backward propagator method [Shibata and Takahashi, 1977, Chaturvedi and Shibata, 1979]: Using Eq. (3.18) we can express the formal solution of Eq. (3.15) as (t) = (t)(0). We 56 have already discussed above the invertibility of (t); assuming 1 exists Eq. (3.15) can then be rewritten in time-convolutionless form as @ @t = L Z t 0 k(t 0 ) exp(Lt 0 )(tt 0 )dt 0 1 (t) (t); (3.21) with the operator in square brackets serving as the generator of the evolution. Experimental determination of the kernel function.— Suppose one measures(t) via quantum state tomography (QST) [Nielsen and Chuang, 2000]. It then follows from Eq. (3.18) applied to(t) that i (t) = Tr[L i (t)]=Tr[L i (0)]. The coefficients i (t) are thus directly experimentally accessible, provided one first specifies a Markovian model from which the left eigenvectors L i and eigenvalues i can be computed. Inverting Eq. (3.17) then yields the kernel as k(t) = Lap 1 [(s 1=Lap[ i (t)])]e i t = i . This inversion process for k(t) is not unique in the sense that it will depend on the choice of Markovian model. It can be optimized via well-established maximum likelihood methods, e.g., [Banaszek et al., 1999], thus yielding the optimal Markovian model. Example.— As a concrete example meant to illustrate the predictions of our master equation we consider the problem of a single qubit dephasing. The Lindblad super- operator isL = (a=2)[ z ; [ z ;]], a > 0. Using the parametrization (t) = (I + vec(t) vec)=2 [with vec 2 R 3 and vec = ( x ; y ; z )], the damping basis is found to consist of the following eigenvalues and eigenoperators:f i g 3 i=0 = f0;a;a; 0g, andfR i g 3 i=0 =fL i g 3 i=0 =fI; x ; y ; z g= p 2. The Markovian solu- tion is simple exponential coherence decay: z (t) = 1 and j (t) = j (0) exp(at), j = x;y. It follows immediately from Eq. (3.17) that 0 (t) = z (t) = Lap 1 [1=s] = 1 and that x (t) = y (t) =: f(t). We further findf j (t) = j (t)g j=x;y;z . Apply- ing the criterion (3.19) readily yields the CP condition asjf(t)j 1. Let us con- sider two kernel functions: k 1 (t) = A exp( t) ) e k 1 (s) = A s+g and k 2 (t) = 57 Ae ( a)t [cos(t) sin(t)]) e k 2 (s) = A(sa) (sa+ ) 2 + 2 . Then, following the prescrip- tion of Eq. (3.17) yieldsf 1 (t) = exp[t(a + )=2][cos(!t) + sin(!t)(a + )=2!] where ! = p 4Aa ( +a) 2 =2, andf 2 (t) = 1 Aa 2 + 2 [1e t (cos t + sin t)] where = p 2 +Aa (note that the CP conditionjf 1;2 (t)j 1 imposes restrictions on the allowed values of the various parameters appearing here). In both cases we thus find damped oscillations. The difference is that in the case of k 1 we have f 1 (1) = 0, as in the Markovian case, while in the case of k 2 we have f 2 (1) = 1 Aa 2 + 2 , which cannot be mimicked by the Markovian solution. Damped oscillations with a non-zero asymptotic coherence, as in the case ofk 2 , are a feature of the exact solution of a single qubit dephasing in the presence of a boson bath, e.g., when a peaked spectral density g(!)/ exp[c(!! 0 ) 2 ] is chosen [Lidar et al., 2001b]. We thus see explicitly through the example considered here, how our new master equation (3.15) is capable of interpo- lating between exact and Markovian open system dynamics. 58 Chapter 4 Decoherence free Subspaces and Subsystems 4.1 Loss of information through a decoherence process So far we have discussed the evolution of a system in interaction with an environment. As we have pointed out in the introduction, the main obstacle on the way to build a quantum computer is the imperfect isolation of the computer which results in distur- bance of the computation. However we have not yet explained why such an interaction is harmful. This is rooted in the way that information is stored in a quantum computer. For a single qubit in a pure state, the information is determined by the relative phase between statesj0i andj1i:j i = aj0i +e i bj1i,a;b2R and the ratioa=b. Suppose we initialize a qubit with some information; what does happen when the qubit system is coupled to a huge bath? The system begins to lose energy into the bath and decay to the ground statej0i. Thus the phase is erased and the information is lost. In a more general setting, we can look at the density matrix of a qubit, at timet = 0 (0) = 0 @ a 2 abe i abe i b 2 1 A (4.1) Under a Markovian decoherence process the state at timet is found to be 59 (t) = 0 @ 1b 2 e t T 1 abe i e t T 2 abe i e t T 2 b 2 e t T 1 1 A (4.2) where T 1 and T 2 are typical decoherence time scales. Now you can recognize the importance of Divincenzo’s third criterion for a reliable quantum computer that the com- putation must be done much faster than the decoherence time scales. In recent years much effort has been expended to develop methods for tackling the deleterious interaction of controlled quantum systems with their environment. This effort has been motivated in large part by the need to overcome decoherence in quantum information processing tasks, a goal which was thought to be unattainable at first [Lan- dauer, 1995, Unruh, 1995, Haroche and Raimond, 1996]. Decoherence-free (or noise- less) subspaces [Duan and Guo, 1998, Zanardi and Rasetti, 1997, Lidar et al., 1998, 2001a] and subsystems [Knill et al., 2000, Filippo, 2000, Kempe et al., 2001, Yang and Gea-Banacloche, 2001] (DFSs) are among the methods which have been proposed to this end, and also experimentally realized in a variety of systems [Kwiat et al., 2000, Kielpinski et al., 2001, Fortunato et al., 2002, Viola et al., 2001]. In this manner of passive quantum error correction, one uses symmetries in the form of the interaction between system and environment to find a “quiet corner” in the system Hilbert space not experiencing this interaction. Of the various methods of quantum error correction, so far only DFSs have been combined with quantum algorithms in the presence of deco- herence [Mohseni et al., 2003, Ollerenshaw et al., 2003]. For a review of DFSs and a comprehensive list of references see Ref. [Lidar and Whaley, 2003]. We first review and re-examine the previous results on DFSs, in Section 4.2. We do so for both general completely positive (CP) maps and for Markovian dynamics. The definitions we give for DFSs in these two cases are slightly different, reflecting the fact that Markovian dynamics is always continuous in time, whereas CP maps can 60 also describe discrete-time evolution. In Section 4.3, we present our generalized DFS conditions for CP maps and for Markovian dynamics. We illustrate the new conditions for Markovian dynamics with an example which reveals some of the new features. In Section 4.4 we discuss the implications of our relaxed initialization condition in the con- text of quantum algorithms. Section 4.5 is devoted to a case-study of non-Markovian dynamics, intermediate between (formally exact) CP maps and (approximate) Marko- vian dynamics. A unique formulation does not exist in this case, and we consider the master equation introduced in Ref. [Shabani and Lidar, 2005a]. The analytical solvabil- ity of this equation permits a rigorous derivation of the conditions for a DFS. For clarity of presentation we defer most supporting calculations to the appendices. 4.2 Review of Previous Conditions for Decoherence- Free Subspaces and Subsystems We refer the reader to Ref. [Lidar and Whaley, 2003] for a detailed review, including many references and historical context. 4.2.1 Decoherence-Free Subspaces Consider a system with Hilbert spaceH S . In Refs. [Zanardi and Rasetti, 1997, Lidar et al., 1998, Zanardi, 1997, Lidar et al., 1999, 2001a] a subspaceH DFS H S was called decoherence-free if any state S (0) of the system initially prepared in this subspace is unitarily related to the final state S (t) of the system, i.e., S (0) =P d S (0)P d =) S (t) = U S (0)U y : (4.3) 61 Here U is unitary andP d is the projection operator ontoH DFS . Important and motivating early examples of DFSs were given in [Palma et al., 1996, Duan and Guo, 1997, 1998, Zanardi and Rossi, 1998]. An alternative definition of a DFS is as a subspace in which the state purity is always one [Zanardi and Lidar, 2004]; here we will not pursue this approach. To exploit DF-states for quantum information preservation one needs a method to experimentally verify these states [Viola and Knill, 2003], but from a theoretical stand- point one needs to first formulate the effect of the environment. In the following, we consider general CP maps and Markovian dynamics. Completely Positive Maps Consider the dynamics of an open system given by a CP map S (t) = X E (t) S (0)E y (t); X E y E = I S ; (4.4) where I S is the identity operator on the system. In [Lidar et al., 1999] a DFS-condition was derived for general CP maps of this type. We denote the subspace of states orthogonal toH DFS byH DFS ?, so thatH S = H DFS H DFS ?. According to Eq. (4) in [Lidar et al., 1999] the Kraus operators take the block-diagonal form E = 0 @ c U DFS 0 0 B 1 A ; (4.5) where the upper (lower) non-zero block acts entirely insideH DFS (H DFS ?); U DFS is a unitary matrix that is independent of the Kraus operator label ; c is a scalar 62 ( P jc j 2 = 1); and B is arbitrary, except that P B y B = I DFS ?. It is simple to verify that the DFS definition (4.3) is satisfied in this case, with U = U DFS . Theorem 1 in [Lidar et al., 1999] reads: “A subspaceH DFS is a DFS iff all Kraus operators have an identical unitary representation upon restriction to it, up to a multi- plicative constant.” This theorem is actually compatible with a more general form for the Kraus operators than Eq. (4.5), since “upon restriction to it” concerns only the upper- left block of E . We derive the most general form of E in Section 4.3 below, and find that, indeed, a more general form than Eq. (4.5) is possible: one of the off-diagonal blocks need not vanish. In other words, leakage fromH DFS ? intoH DFS is permitted. As we further show in Section 4.3, the form (4.5) in fact appears in the context of unital channels. Markovian Dynamics As we discussed before, the most general form of CP Markovian dynamics is given by the Lindblad equation [V . Gorini and Sudarshan, 1976, Lindblad, 1976, Alicki and Lendi, 1987]: @ S @t = i[H S ; S ] +L[ S ]; L = X F F y 1 2 F y F 1 2 F y F ; (4.6) where F are bounded (or unbounded, if subject to appropriate domain restrictions [Davies, 1977, Lidar et al., 2006]) operators acting onH S , and where H S may include a Lamb shift [Lidar et al., 2001b]. Given such dynamics, one restores unitarity [i.e., the DFS definition (4.3) with U generated by the Hamiltonian H S ] if the Lindblad term 63 L[ S ] can be eliminated. According to Refs. [Lidar et al., 1998, Zanardi, 1998], a nec- essary and sufficient condition for this to be the case is F jii =c jii; (4.7) whereH DFS = Spanfjiig andfc g are arbitrary complex scalars. Thus the Lindblad operators can be written in block-form as follows: F = 0 @ c I A 0 B 1 A ; (4.8) with the blocks on the diagonal corresponding once again to operators restricted toH DFS andH DFS ?. Note the appearance of the off-diagonal block A mixingH DFS andH DFS ?; its presence is permitted since the DFS condition (4.7) gives no information about matrix elements of the formhijF jj ? i, withjii2H DFS andjj ? i2H DFS ?. As observed in Refs. [Lidar et al., 1998, 1999], one should in addition require that H S does not mix DF states with non-DF ones, i.e., mixed matrix elements of the type hj ? jH S jii, withjii2H DFS andjj ? i2H DFS ?, should vanish. We show below that this condition must be made more stringent. 4.2.2 Noiseless Subsystems An important observation made in Ref. [Knill et al., 2000] is that there is no need to restrict the decoherence-free dynamics to a subspace. A more general situation is when the DF dynamics is a “subsystem”, or a factor in a tensor product decomposition of 64 subspace. Following Ref. [Knill et al., 2000], this comes about as follows. Consider the dynamics of a systemS coupled to a bathB via the Hamiltonian H = H S I B + I S H B + H I ; (4.9) where H S (H B ), the system (bath) Hamiltonian, acts on the system (bath) Hilbert space H S (H B ); I S (I B ) is the identity operator on the system (bath) Hilbert space; H I is the interaction term of Hamiltonian which can be written in general as P S B . If the system Hamiltonian H S and the system components of the interaction Hamiltonian, the S ’s, form an algebraS, it must bey-closed to preserve the unitarity of system-bath dynamics. Now, ifA is ay-closed operator algebra which includes the identity opera- tor, then a fundamental theorem of C algebras states thatA is a reducible subalgebra of the full algebra of operators [Landsman]. This theorem implies that the algebra is isomorphic to a direct sum ofd J d J complex matrix algebras, each with multiplicity n J : S = M J2J I n J M(d J ;C) (4.10) HereJ is a finite set labeling the irreducible components ofS, andM(d J ;C) denotes ad J d J complex matrix algebra. Associated with this decomposition of the algebraS is a decomposition of the system Hilbert space: H S = M J2J C n J C d J : (4.11) 65 If we encode quantum information into a subsystem (factor)C n J it is preserved, since the noise algebraS acts trivially (as I n J ). In such a caseC n J is called a decoherence- free, or noiseless subsystem (NS) [Knill et al., 2000]. Examples of this construction were given independently in Refs. [Filippo, 2000, Yang and Gea-Banacloche, 2001]. Completely Positive Maps As the Kraus operators are given by Eq. (4.4), they take the form of the decomposition (4.10): E = M J2J I n J M (d J ); (4.12) where M (d J ) is an arbitraryd J -dimensional complex matrix. Therefore a factorC n J is a NS if the Kraus operators have the representation (4.12). Markovian Dynamics The aforementioned reducibility theorem [Landsman] does not apply directly in the Markovian case, since the set of Lindblad operatorsfF g need not be closed under conjugation. Nevertheless, as shown in [Kempe et al., 2001], the concept of a subsystem applies in the Markovian case as well: the condition for a NS was found to be F P d = I n J M (d J )P d ; (4.13) with the M again being arbitrary complex matrices andP d being the projection operator onto a given subspaceC n J C d J . The NS is then a factorC n J as in Eq. (4.11), with the same tensor product structure as in Eq. (4.13). 66 4.3 Generalized Conditions for Decoherence-Free Sub- spaces and Subsystems We now proceed to re-examine the conditions for the existence of decoherence-free sub- spaces and subsystems. We will show that the conditions presented in the papers laying the general theoretical foundation [Zanardi and Rasetti, 1997, Lidar et al., 1998, Knill et al., 2000, Zanardi, 1997, Lidar et al., 1999, Zanardi, 1998, Kempe et al., 2001] , can be generalized and sharpened, both for CP maps and for Markovian dynamics. Our main new finding is that the preparation step can tolerate arbitrarily large errors. Relatedly, we consider the possibility of leakage from outside of the protected subspace/subsystem into it. Previous studies did not allow for this possibility, but we will show that it can be permitted under appropriate restrictions. In doing so we generalize the definition of a NS with respect to the original definition that relied on the algebraic isomorphism (4.10) (see Ref. [D.W. Kribs and Lesosky, 2005] for a related recent result). In the case of Markovian dynamics, our main new finding is that if one demands perfect initializa- tion into a DFS then the condition on the Hamiltonian component of the evolution is modified compared to previous studies. The derivation of these results is somewhat tedious. Hence, for clarity of presen- tation we focus on presenting our generalized conditions in this section. Mathemat- ical proofs are deferred to the appendices at the end of this chapter. We begin with the simpler case of decoherence-free subspaces and consider the case of CP maps and Markovian dynamics. We then move on to the case of decoherence-free (noiseless) subsystems. The case of non-Markovian continuous-time dynamics is treated later, in Section 4.5. 67 4.3.1 Decoherence-Free Subspaces The system density matrix S is an operator on the entire system Hilbert spaceH S , which we assume to be decomposable into a direct sum asH = H DFS H DFS ?. It is convenient for our purposes to represent the system state (and later on the Kraus and Lindblad operators) in a matrix form whose block structure corresponds to this decomposition of the Hilbert space. Thus the system density matrix takes the form S = 0 @ DFS 2 y 2 3 1 A ; (4.14) We also define a projector P DFS = I DFS 0 ; (4.15) so that DFS =P DFS S P y DFS . Finally, P d = 0 @ I DFS 0 0 0 1 A ; P d ? = 0 @ 0 0 0 I DFS 1 A (4.16) are projection operators ontoH DFS andH DFS ?, respectively. Completely Positive Maps The original concept of a DFS, Eq. (4.3), poses a practical problem: the perfect initial- ization of a quantum system inside a DFS might be challenging in many cases. There- fore we introduce a generalized definition to relax this constraint: Definition 2 Let the system Hilbert spaceH S decompose into a direct sum asH = H DFS H DFS ?, and partition the system state S accordingly into blocks, as in 68 Eq. (4.14). Assume DFS (0) =P DFS S (0)P y DFS 6= 0. ThenH DFS is called decoherence- free iff the initial and final DFS-blocks of S are unitarily related: DFS (t) = U DFS DFS (0)U y DFS ; (4.17) where U DFS is a unitary matrix acting onH DFS . Definition 3 Perfect initialization (DF subspaces): 2 = 0 and 3 = 0 in Eq. (4.14). Definition 4 Imperfect initialization (DF subspaces): 2 and/or 3 in Eq. (4.14) are non-vanishing. We prove in Appendix 4.7.1: Theorem 3 Assume imperfect initialization. Let U be unitary, c scalars satisfying P jc j 2 = 1, and B arbitrary operators onH DFS ? satisfying P B y B = I DFS ?. A necessary and sufficient condition for the existence of a DFS with respect to CP maps is that the Kraus operators have a matrix representation of the form E = 0 @ c U 0 0 B 1 A : (4.18) This form is identical to the previous result (4.5), with the important distinction that due to the new definition of a DFS, Eq. (4.17), the theorem holds not just for states initialized perfectly intoH DFS , but for arbitrary initial states. Note that unlike fault- tolerant QECC, where the initial state must be sufficiently close to a valid code state [Preskill, 1999], here the initial state can be arbitrarily far from a DFS-code state, as long as the initial projection into the DFS is non-vanishing. 69 These observations lead us to reconsider the original definition, wherein the sys- tem is initialized inside the DFS. This situation admits more general Kraus operators. Specifically, we prove Appendix 4.7.1 that: Corollary 1 Assume perfect initialization. Then the DFS condition is: E = 0 @ c U A 0 B 1 A ; (4.19) where U is unitary. Note that and due to sum rule P E y E = I) the otherwise arbitrary operators A and B satisfy the constraints (i) P A y A + B y B = I DFS ? and (ii) P c A = 0, and where additionally the scalarsc satisfy (iii) P jc j 2 = 1. In contrast to the diagonal form in the previous conditions (4.5) and (4.18), Eq. (4.19) allows for the existence of the off-diagonal term A , which permits leakage fromH DFS ? intoH DFS . This more general form of the Kraus operators imply that a larger class of noise processes allow for the existence of DFSs, as compared to the previous condition (4.5). 1 Unital Maps A unital (sometimes called bi-stochastic) channel is a CP map () = P E E y that preserves the identity operator: (I) = P E E y = I. Consider the fixed points of , i.e., Fix()f : () = g. Such states, which are invariant under , are clearly examples of DF-states of the corresponding channel. 1 We re-emphasize that Theorem 1 in [Lidar et al., 1999] is compatible with Eq. (4.19); the latter generalizes the explicit matrix representation Eq. (4) given in that paper [condition (4.5) in the present paper], but does not invalidate Theorem 1 in [Lidar et al., 1999]. 70 Recently it has been shown that the fixed point set of unital CP maps is the commu- tant of the algebra generated by Kraus operators [Kribs, 2003]. In other words, ifE is the set of all polynomials infE g, orE = AlgfE g, then Fix() =fT2B(H) : [T;E] = 0g; (4.20) whereB(H) is the (Banach) space of all bounded operators on the Hilbert spaceH. In other words, the fixed points of a unital CP map, which are DF states, can alternatively be characterized as the commutant of AlgfE g, i.e., the setfTg. It is our purpose in this subsection to show that, under our generalized definition of DFSs, this characterization of DF states is sufficient but not necessary. Consider the generalized DFS-condition (4.19) applied to unital maps. We have () = X 0 @ c I DFS A 0 B 1 A 0 @ c I DFS 0 A y B y 1 A : (4.21) Unitality, (I) = I, together with P jc j 2 = 1 implies: 0 @ I DFS + P A A y P A B y P B A y P B B y 1 A = I: (4.22) This implies the vanishing of the matrices A , so that we are left with the Kraus opera- tors in the simple block-diagonal form: E = 0 @ c I 0 0 B 1 A ; (4.23) 71 together with the additional constraint P B B y = I DFS ? (which, in the present unital case, naturally supplements the previously derived normalization constraint P B y B = I DFS ?). Thus, unitality restricts the class of Kraus operators, so that in fact we must assume the DFS-condition (4.18) rather than (4.19). This then means that we may consider the generalized DFS definition Eq. (4.17). Next, let us find the commutant of this class of Kraus operators. First, AlgfE g =f 0 @ poly(c )I 0 0 poly(B ) 1 A g; (4.24) where poly(x) denotes all possible polynomials inx. Representing an arbitrary operator T2B(H) in the form T = 0 @ L M N P 1 A ; (4.25) it is simple to derive that the commutant of AlgfE g is the space of matrices T of the form T = 0 @ L 0 0 cI 1 A ; (4.26) where L andc are arbitrary. The aforementioned theorem [Kribs, 2003] states that the fixed-point set of the channel, i.e., the DF states, coincides with this commutant. Of course, for T to be a proper quantum state it must be Hermitian and have unit trace, whencec 0 and L is Hermitian. Subject to these constraints we see that the afore- mentioned theorem [Kribs, 2003] gives a sufficient, but not necessary characterization 72 of the allowed DF states. Indeed, the form (4.26) arises as a special case of our consid- erations, where we allow for T to be a state with support inH DFS ?, but not of the most general form allowed by Eq. (4.17), which includes off-diagonal blocks. Markovian Dynamics In the case of CP maps we are only interested in the output state and the intermediate- time states are ignored. Since, as is well known, Markovian dynamics is a special case of CP maps (e.g., [Alicki and Lendi, 1987, Lidar et al., 2001b]), one may of course apply the results we have obtained above for general CP maps in the Markovian case as well, provided one is only interested in the state at the end of the Markovian channel. However, one may instead be interested in a different notion of decoherence-freeness, wherein the system remains DF throughout the entire evolution. Such a notion is more suited to experiments in which the final time is not a priori known. This is the notion we will pursue here in our treatment of continuous-time dynamics, in both the Markovian and non-Markovian cases. Thus, while we allow that the system not be fully initialized into the DFS, we require that the component that is, undergoes unitary dynamics at all times. Correspondingly, we define a DFS in the Markovian case as follows: Definition 5 Let the system Hilbert spaceH S decompose into a direct sum asH S = H DFS H DFS ?, and partition the system state S accordingly into blocks. LetP DFS be a projector ontoH DFS and assume DFS (0)P DFS S (0)P y DFS 6= 0. ThenH DFS is called decoherence-free iff DFS undergoes Schr¨ odinger-like dynamics, @ DFS @t =i[H DFS ; DFS ]; (4.27) where H DFS is a Hermitian operator. 73 Before presenting the DFS conditions, let us recall the quantum trajectories interpre- tation of Markovian dynamics [Dalibard et al., 1992, Gisin and Percival, 1992, Plenio and Knight, 1998]. Expanding Eq. (4.6) to first order in the short time-interval yields the CP map S (t +) = X =0 W (t)W y ; (4.28) where W 0 = IiH S 2 X F y F ; (4.29) W >0 = p F ; (4.30) and to the same order we also have the normalization condition X =0 W y W = I: (4.31) Thus the Lindblad equation has been recast as a Kraus operator sum (4.4), but only to first order in, the coarse-graining time scale for which the Markovian approxima- tion is valid [Lidar et al., 2001b]. This implies a measurement interpretation, wherein the system state is S (t +) = W (t)W y =p (to first-order in ) with probability p = Tr[W (t)W y ]. This happens because the bath functions as a probe coupled to the system while being subjected to a quasi-continuous series of measurements at each infinitesimal time interval [Shabani and Lidar, 2005a]. The result is the well- known quantum jump process [Dalibard et al., 1992, Gisin and Percival, 1992, Plenio 74 and Knight, 1998], wherein the measurement operators are W 0 exp(iH c ), the “conditional” evolution, generated by the non-Hermitian “Hamiltonian” H c H S i 2 X F y F ; (4.32) and p F (the “jump”). Note that H S is here meant to include all renormaliza- tion effects due to the system-bath interaction, e.g., a possible Lamb shift (see, e.g., Ref. [Lidar et al., 2001b]). By a simple algebraic rearrangement one can rewrite the Lindblad equation in the following form: _ S =i(H c S S H y c ) + X F S F y ; (4.33) where according to the above interpretation the first term generates non-unitary dynam- ics, while the second is responsible for the quantum jumps. Now recall the Markovian DFS condition derived in Refs. [Lidar et al., 1998, Zanardi and Rossi, 1998]: the Lindblad operators should have trivial action on DF-states, as in Eq. (4.7), i.e., F jii =c jii. Viewed from the perspective of the quantum-jump picture of Markovian dynamics, this implies that the jump operators do not alter a DF-state, i.e., the term P F S F y in Eq. (4.33) transforms S to P jc j 2 S and thus has trivial action. Given Eq. (4.7), the Lindblad operators can be written in block-form as follows [Eq. (4.8)]: F = 0 @ c I A 0 B 1 A ; (4.34) with the blocks on the diagonal corresponding once again to operators restricted toH DFS andH DFS ?. Note the appearance of the off-diagonal block A mixingH DFS andH DFS ?; 75 its presence is permitted since the DFS condition (4.7) gives no information about matrix elements of the formhijF jj ? i, withjii2H DFS andjj ? i2H DFS ?. As observed in [Lidar et al., 1998], one should in addition require that H S does not mix DF states with non-DF ones. It turns out that this condition is compatible with the case that the DF state is imperfectly initialized (Definition 4). In this case, as shown in Appendix 4.7.2, the following theorem holds: Theorem 4 Assume imperfect initialization. Then a subspaceH DFS of the total Hilbert spaceH is decoherence-free with respect to Markovian dynamics iff the Lindblad oper- ators F and the system Hamiltonian H S assume the block-diagonal form H S = 0 @ H DFS 0 0 H DFS ? 1 A ; F = 0 @ c I 0 0 B 1 A ; (4.35) where H DFS and H DFS ? are Hermitian,c are scalars, and B are arbitrary operators onH DFS ?. But, as is clear from the quantum jumps picture, in particular Eqs. (4.32),(4.33), there also exists a non-Hermitian term, which appears not to be addressed properly by merely restricting H S . Indeed, this is the case if one demands that the system state is perfectly initialized into the DFS (Definition 3). As shown in Appendix 4.7.2, the full condition on the Hamiltonian term then is: hij(iH S + 1 2 X F y F )jk ? i = 0; 8i;k ? ; (4.36) wherejii 2 H DFS ,jk ? i 2 H DFS ?. Applying the DFS conditions (4.8),(4.36), the Lindblad equation (4.6) reduces to the Schr¨ odinger-like equation (4.27). Combining these results, we have: 76 Theorem 5 Assume perfect initialization. Then a subspaceH DFS of the total Hilbert spaceH is decoherence-free with respect to Markovian dynamics iff the Lindblad oper- ators F and Hamiltonian H S satisfy F = 0 @ c I A 0 B 1 A (4.37) P DFS H S P y DFS = i 2 X c A : (4.38) Note that H S (which, again, includes the Lamb shift) must satisfy a more stringent constraint than previously noted due to the extra condition on its off-diagonal block. This has implications in examples of practical interest, as we next illustrate. Example (significance of the new condition on the off-diagonal blocks of H S ) We present an example meant to demonstrate how the new constraint, Eq. (4.36) [or, equivalently, Eq. (4.38)], may lead to a different prediction than the old constraint, that matrix elements of the typehj ? jH S jii, withjii2H DFS andjj ? i2H DFS ?, should vanish. Consider a system of three qubits interacting with a common bath. The system is under influence of the bath via: 1) Spontaneous emission from the highest levelj111i to the lower levels, 2) Dephasing of the first and the second qubits. For simplicity we set the system and bath Hamiltonians, H S and H B , to zero. The total Hamiltonian then contains only the system-bath interaction: H I = 1 ( z 1 + z 2 ) B + 2 [( 1 + 2 + 3 ) b y +( + 1 + + 2 + + 3 ) b]; (4.39) 77 where 1 =j001ih111j; 2 =j010ih111j; 3 =j100ih111j; (4.40) and b is a bosonic annihilation operator. The corresponding Lindblad equation may be derived, e.g., using the method devel- oped in Ref. [Lidar et al., 2001b]. It may then be shown that L[ S ] = 1 2 2 X i=1 [F i ; S F y i ] + [F i S ; F y i ]; (4.41) where the Lindblad operators are F 1 = p d 1 (u 11 K 1 +u 12 K 2 ); F 2 = p d 2 (u 21 K 1 +u 22 K 2 ): (4.42) Here K 1 = z 1 + z 2 , K 2 = 1 + 2 + 3 , andfd 1 ;d 2 g are the eigenvalues of the Hermitian matrix A = [a ij ] of coefficients in the pre-diagonalized Lindblad equation, with the diagonalizing matrix denoted U = [u ij ]. Now let us find the DFS conditions under the assumption of perfect initialization. The previously-derived Eq. (4.7) yields thatfj000i;j001ig is a DFS, since K 2 annihi- lates these states, and they are both eigenstates of K 1 with an eigenvalue of +2: F 1 j000i = 2 p d 1 u 11 j000i; F 2 j000i = 2 p d 2 u 21 j000i F 1 j001i = 2 p d 1 u 11 j001i; F 2 j001i = 2 p d 1 u 11 j001i: (4.43) 78 However, the new condition (4.36) tightens the situation. Choosing as representatives the statesj001i2H DFS andj111i2H DFS ?, we find from Eq. (4.36): h001j 2 X =1 F y F j111i = 2d 1 u 11 u 12 + 2d 2 u 21 u 22 = 0: (4.44) Sinceu 11 u 12 +u 21 u 22 = 0 (from unitarity of U), we see that the new condition imposes the extra symmetry constraint d 1 = d 2 . This example illustrate the importance of the new condition, Eq. (4.36). 4.3.2 Noiseless Subsystems We now consider again the more general setting of subsystems, rather than subspaces. Completely Positive Maps Suppose the system Hilbert space can be decomposed asH S =H NS H in H out , where H NS is the factor in which quantum information will be stored. The subspaceH out may itself have a tensor product structure, i.e., additional factors similar toH NS may be contained in it [as in Eq. (4.11)], but we shall not be interested in those other factors since the direct sum structure implies that different noiseless factors cannot be used simultaneously in a coherent manner. As in the DF subspace case considered above, we allow for the most general situation of a system that is not necessarily initially DF. To make this notion precise, let us generalize the definitions of the projectorP DFS and projection operatorsP d ;P d ? given in the DFS case, as follows: P NSin = I NS I in 0 ; (4.45) 79 P d = 0 @ I NS I in 0 0 0 1 A ; P d ? = 0 @ 0 0 0 I NS I in 1 A (4.46) There is no risk of confusion in using the DFS notation,P d , for the NS case, as the DFS case is obtained when I in is a scalar. The system density matrix takes the corresponding block form S = 0 @ NSin 0 0y out 1 A : (4.47) Definition 6 Let the system Hilbert spaceH S decompose asH S =H NS H in H out , and partition the system state S accordingly into blocks, as in Eq. (4.47). Assume NSin (0) =P NSin S (0)P y NSin 6= 0. Then the factorH NS is called a decoherence-free (or noiseless) subsystem if the following condition holds: Tr in f NSin (t)g = U NS Tr in f NSin (0)gU y NS ; (4.48) where U NS is a unitary matrix acting onH NS . Definition 7 Perfect initialization (DF subsystems): 0 = 0 and out = 0 in Eq. (4.47). Definition 8 Imperfect initialization (DF subsystems): 0 and/or out in Eq. (4.47) are non-vanishing. According to Definition 6, a quantum state encoded into theH NS factor at some time t is unitarily related to thet = 0 state. The factorH in is unimportant, and hence is traced over. Clearly, a NS reduces to a DF subspace whenH in is one-dimensional, i.e., when H in =C. We now present the necessary and sufficient conditions for a NS and later we show that the algebra-dependent definition, Eq. (4.10), is a special case of this generalized 80 form. In stating constraints on the form of the Kraus operators, below, it is understood that in addition they must satisfy the sum rule P E y E = I, which we do not specify explicitly. Theorem 6 Assume imperfect initialization. Then a subsystemH NS in the decomposi- tionH S =H NS H in H out is decoherence-free (or noiseless) with respect to CP maps iff the Kraus operators have the matrix representation E = 0 @ U C 0 0 B 1 A (4.49) Corollary 2 Assume perfect initialization. Then the Kraus operators have the relaxed form E = 0 @ U C A 0 B 1 A (4.50) We note that this result has been recently derived from an operator quantum error correction perspective in Ref. [D.W. Kribs and Lesosky, 2005]. Note again that there is a trade-off between the quality of preparation and the amount of leakage that can be tolerated, a fact that was not noted previously for subsystems, and has important experimental implications. As discussed above, the original definition of a NS was based on representation theory of the error algebra. Here we have argued in favor of a more comprehensive definition, based on the quantum channel picture. Let us now state explicitly why our result is more general. Indeed, in the algebraic approach one arrives at the representation (4.12) of the Kraus operators, namely E = L J2J I n J G ;J . However, it is clear from Eq. (4.50) that our channel-based approach leads to a form for the Kraus operators that includes this latter form as a special case, since it allows for the off-diagonal block A . 81 The representation (4.12) of the Kraus operators does agree with Eq. (4.49), but in that case we do not need to assume initialization inside the NS, so that again, our result is more general than the algebraic one. Markovian Dynamics As in the CP-map based definition of a NS, we need to trace out theH in factor, here in order to obtain the dynamical equation for the subsystem factor: @ NS @t = @Tr in fP NSin S P y NSin g @t = Tr in f @P NSin S P y NSin @t g = Tr in fP NSin ( i ~ [H S ; S ] + 1 2 X 2F S F y F y F S S F y F )P y NSin g: (4.51) Definition 9 The factorH NS is called a decoherence-free (or noiseless) subsystem under Markovian dynamics if a state subject to Eq. (4.51), undergoes continuous unitary evolution: NS =i[M; NS ]; (4.52) where M is Hermitian. Clearly, again, a NS reduces to a DF subspace whenH in is one-dimensional, i.e., whenH in =C. Our goal is to find necessary and sufficient conditions such that Eq. (4.51) leads to Eq. (4.52). In the case of perfect initialization, since it does not involveH out , Eq. (4.51) is meaningful only if the system remains in the subspaceH NS H in . An analysis of 82 Eq. (4.51) reveals that this leakage-prevention goal is achieved by imposing the con- straints stated in the following theorem, proven in Appendix 4.7.2: Theorem 7 Assume perfect initialization. Then a subsystemH NS in the decomposition H S =H NS H in H out is decoherence-free (or noiseless) with respect to Markovian dynamics iff the Lindblad operators have the matrix representation F = 0 @ I NS C A 0 B 1 A (4.53) and the system Hamiltonian (including a possible Lamb shift) has the matrix represen- tation H S = 0 @ H NS I in +I NS H in H 2 H y 2 H 3 1 A (4.54) where H in is constant along its diagonal, and where H 2 = i 2 X I NS C y A : (4.55) Eqs. (4.54),(4.55) are new additional constraints on the Lindblad operators (com- pared to Ref. [Kempe et al., 2001]) which must be satisfied in order to find a NS. If, on the other hand, we allow for imperfect initialization, we find a different set of conditions: 83 Theorem 8 Assume imperfect initialization. Then a subsystemH NS in the decomposi- tionH S =H NS H in H out is decoherence-free (or noiseless) with respect to Marko- vian dynamics iff the Lindblad operators have the matrix representation F = 0 @ I NS C in 0 0 B 1 A ; (4.56) and the system Hamiltonian (including a possible Lamb shift) has the matrix represen- tation H = 0 @ H NS I in +I NS H in 0 0 H out 1 A : (4.57) 4.4 Performance of Quantum Algorithms over Imper- fectly Initialized DFSs In this section we discuss applications of our generalized formulation of DFSs to quan- tum algorithms. As mentioned above, a major obstacle to exploiting decoherence-free methods is the unrealistic assumption of perfect initialization inside a DFS. Remov- ing this constraint enables us to perform algorithms without perfect initialization, while not suffering from information loss. We separate the role of an initialization error in the algorithm (i.e., starting from an imperfect input state), from the effect of noise in the output due to environment-induced decoherence. Thus we first quantify an error entirely due to incorrect initialization ( leak below), then compare the DFS situations prior and post this work, by relating them to leak . 84 1) Initialization error in the absence of decoherence: Assume no decoherence at all, that the initial state is actual (0)= 0 @ 1 2 y 2 3 1 A ; (4.58) while the ideal input state is fully in the DFS: ideal (0)= 0 @ 0 0 0 1 A : (4.59) Further assume that the algorithm is implemented via unitary transformations U = U DFS I DFS ?, applied toH DFS . In general this will lead to an output error in the algorithm, which can be quantified as leak jjU actual (0)U y U ideal (0)U y jj = 0 @ U DFS ( 1 )U y DFS U DFS 2 y 2 U y DFS 3 1 A ; (4.60) wherejjjj denotes an appropriate operator norm. This error appears not because of deco- herence but because of an erroneous initial state. This is a generic situation in quantum algorithms, which is not special to the DFS case: Eq. (4.58) is generic in the sense that one can view the DFS block as the computational subspace, with the other blocks rep- resenting additional levels (e.g., a qubit which is embedded in a larger Hilbert space). Methods for correcting such deviations from the ideal result exist (leakage elimination [Wu et al., 2002, Byrd et al., 2005]), but are beyond the scope of this paper. 85 2) Initialization error in the presence of decoherence: Assume that the input state is imperfectly initialized, as in Eq. (4.58), and in addition there is decoherence, i.e., actual (t) = X E (t) actual (0)E y (t); (4.61) with the Kraus operators given by Eq. (4.18) [the form compatible with decoherence- free evolution starting from actual (0)]. Prior to our work it was believed that for an imperfect initial state of the form actual (0), leakage due to the components 2 and 3 would cause non-unitary evolution of the DFS component. Thus instead of an error U DFS ( 1 )U y DFS in the DFS block of Eq. (4.60), it was believed that one hadE( 1 ) U DFS U y DFS whereE is an appropriate superoperator component. This would have led to a reduced algorithmic fidelity, 0 leak < leak . However, we now know that even for an initial state of the form actual (0), when the Kraus operators are given by Eq. (4.18) the actual algorithmic fidelity is still given by leak , since in fact the evolution of the DFS block is still unitary. The above arguments apply when imperfect initialization is unavoidable but one knows the component 1 . A worse (though perhaps more typical) scenario is one where not only is imperfect initialization unavoidable, but one does not even know the compo- nent 1 . In this case the above arguments apply in the context of algorithms that allow arbitrary input states. Almost all the important examples of quantum algorithms are now known to have a flexibility of this type: Grover’s algorithm [Grover, 1996] was the first to be generalized to allow for arbitrary input states, first pure [Biron et al., 1998, Biham et al., 1999, 2001], then mixed [Biham and Kenigsberg, 2002]; Shor’s algorithm [Shor, 1997] can run efficiently with a single pure qubit and all other qubits in an arbitrary mixed state [Parker and Plenio, 2000]; a similar result applies to a class of interesting physics problems, such as finding the spectrum of a Hamiltonian [Knill and Laflamme, 86 1998]; the Deutsch-Josza [Deutsch and Jozsa, 1992] algorithm was generalized to allow for arbitrary input states [Chi et al., 2001], and a similar result holds for an algorithm that performs the functional phase rotation (a generalized form of the conventional con- ditional phase transform) [Kim et al., 2002]. Most recently it was shown that Simon’s problem and the period-finding problem can be solved quantumly without initializing the auxiliary qubits [Chi et al., 2005]. For algorithms that do not allow arbitrary input states, one could still make use of the flexibility we have introduced into DFS state initialization, provided it is possible to apply post-selection: one modifies the output error of algorithm by observing whether the measurement outcome came from the DFS block or not (this could be done, e.g., via frequency-selective measurements, similar to the cycling transition method used in trapped-ion quantum computing [Wineland et al., 1998]). 4.5 Decoherence Free Subspaces and Subsystems in non-Markovian Dynamics 4.5.1 Decoherence Free Subspaces In chapter 3, we discussed a class of non-Markovian master equations was introduced. The following equation was derived as an analytically solvable example of this class: @ S @t =i[H S ; S ] +L Z t 0 dt 0 k(t 0 ) exp(Lt 0 ) S (tt 0 ) (4.62) 87 whereL is Lindblad super-operator andk(t) represents the memory effects of the bath. The Markovian limit is clearly recovered whenk(t)/(t). 2 Some examples of physical systems which can be described by this master equation are (i) a two-level atom coupled to a single cavity mode, wherein the memory function is exponentially decaying,k(t) =e t [Breuer and Petruccione, 2002], and (ii) a single qubit subject to telegraph noise in the particular case thatjjLjj 1=t, whence Eq. (4.62) reduces to _ S =L R t 0 dt 0 k(t 0 )(tt 0 ) [Daffer et al., 2004]. It is interesting to investigate the conditions for a DFS in the case of dynamics governed by Eq. (4.62), and to compare the results with the Markovian limit, k(t)/ (t). We defer proofs to Appendix 4.7.3 and here present only the DFS-condition, stated in the following theorem (note that, similarly to the Markovian case, we consider here a continuous-time DFS). Theorem 9 Assume imperfect initialization. Then a subspaceH DFS is decoherence free iff the system Hamiltonian H S and Lindblad operators F have the matrix representa- tion H S = 0 @ H DFS 0 0 H DFS ? 1 A ; F = 0 @ c I 0 0 B 1 A (4.63) These conditions are identical to those we found in the case of Markovian dynam- ics with imperfect initialization – cf. Theorem 4. This fact provides evidence for the robustness of decoherence-free states against variations in the nature of the decoherence process. Interestingly, the conditions under the assumption of perfect initialization differ somewhat when comparing the Markovian and non-Markovian cases: 2 We note that Ref. [Shabani and Lidar, 2005a] contains a small error: the Markovian limit is recovered for k(t) = (t) only if the lower limit in Eq. (4.62) ist. This change can easily be applied to the derivation of Ref. [Shabani and Lidar, 2005a]. 88 Corollary 3 Assume perfect initialization. Then a subspaceH DFS is decoherence free iff the system Hamiltonian H S and Lindblad operators F have the matrix representa- tion H S = 0 @ H DFS 0 0 H DFS ? 1 A ; (4.64) F = 0 @ c I A 0 B 1 A and X c A = 0: (4.65) Compared to the Markovian case (Theorem 5), the difference is that now the off- diagonal blocks of the Hamiltonian must vanish, whereas in the Markovian case we had the constraint [Eq. (4.38)]P DFS H S P y DFS = i 2 P c A . 4.5.2 Decoherence Free Subsystems We now consider the NS case. The dynamics governing a NS is derived by tracing out H in : @ NS @t = @Tr in f S g @t = Tr in f @ S @t g = Tr in fi[H S ; S ] +L Z t 0 dt 0 k(t 0 ) exp(Lt 0 ) S (tt 0 )g (4.66) 89 Theorem 10 Assume imperfect initialization. Then a subsystemH NS in the decompo- sitionH S =H NS H in H out is decoherence-free (or noiseless) with respect to non- Markovian dynamics [Eq. (4.62)] iff the Lindblad operators and the system Hamiltonian have the matrix representation F = 0 @ I NS C 0 0 B 1 A (4.67) H S = 0 @ H NS I in +I NS H in 0 0 H out 1 A : (4.68) Note that this form is, once again, identical to the Markovian case with imperfect initialization (cf. Theorem 8). However, as in the DFS case, the conditions are slightly different between Marko- vian and non-Markovian dynamics if we demand perfect initialization: Corollary 4 Assume perfect initialization. Then a subsystemH NS in the decomposi- tionH S =H NS H in H out is decoherence-free (or noiseless) with respect to non- Markovian dynamics [Eq. (4.62)] iff the Lindblad operators and the system Hamiltonian have the matrix representation F = 0 @ I NS C A 0 B 1 A ; (4.69) X (I NS C y )A = 0; (4.70) H = 0 @ H NS I in +I NS H in 0 0 H out 1 A : (4.71) 90 4.6 Summary and Conclusions We have revisited the concepts of decoherence-free subspaces and (noiseless) subsys- tems (DFSs), and introduced definitions of DFSs that generalize previous work. We have analyzed the conditions for the existence of DFSs in the case of CP maps, Marko- vian dynamics, and (for the first time) non-Markovian continuous-time dynamics. Our main finding implies significantly relaxed demands on the preparation of decoherence- free states: the initial state can be arbitrarily noisy. If, on the other hand, the initial state is perfectly prepared, then almost arbitrary leakage from outside the DFS into the DFS can be tolerated. In the case of Markovian dynamics, if one demands perfect initialization, our find- ings are of an opposite nature: we have shown that then an additional constraint must be imposed on the system Hamiltonian, which implies more stringent conditions for the possibility of manipulating a DFS than previously believed. We have presented an example to illustrate this fact. We have also shown that the notion of noiseless subsystems, as originally developed using an algebraic approach, admits a generalization when it is instead developed from a quantum channel approach. Our results have implications for experimental work on DFSs, and in particular on quantum algorithms over DFSs [Mohseni et al., 2003, Ollerenshaw et al., 2003]. It is now known that a large class of quantum algorithms can tolerate almost arbitrary preparation errors and still provide an advantage over their classical counterparts [Biron et al., 1998, Biham et al., 1999, 2001, Biham and Kenigsberg, 2002, Parker and Plenio, 2000, Knill and Laflamme, 1998, Chi et al., 2001, Kim et al., 2002, Chi et al., 2005]. The relaxed preparation conditions for DFSs presented here are naturally compatible with this approach to quantum computation in noisy systems. This should provide further impetus for the experimental exploration of quantum computation over DFSs. 91 4.7 Proofs of Theorems and Corollaries Here we present proofs of all our results above. We shorten the calculations by starting from the NS case and obtain the DFS conditions as a special case. 4.7.1 CP Maps Arbitrary Initial State Assume the system evolution due to its interaction with a bath is described by a CP map with Kraus operatorsfE g: S (t) = X E S (0)E y : (4.72) Note that here S is an operator on the entire system Hilbert spaceH S , which we assume to be decomposable asH NS H in H out . From the NS definition, Eq. (4.48), we have Tr in fU I(P NSin S (0)P y NSin )U y Ig = Tr in f X (P NSin E ) S (0)(E y P y NSin )g: (4.73) Let us represent the Kraus operators in the same block-structure matrix-form as that of the system state, i.e., corresponding to the decompositionH S =H NS H in H out , where the blocks correspond to the subspacesH NS H in (upper-left block) andH out (lower-right block). Then S = 0 @ 1 2 y 2 3 1 A ; (4.74) E = 0 @ P A D B 1 A ; (4.75) 92 with appropriate normalization constraints, considered below. Equation (4.73) simpli- fies in this matrix form as Tr in fU I 1 U y Ig = Tr in f X P 1 P y +P 2 A y + A y 2 P y + A 3 A y g; (4.76) which must hold for arbitrary S (0). To derive constraints on the various terms we therefore consider special cases, which yield necessary conditions. First, consider an initial state S (0) such that 2 = 0. Then, as the LHS of Eq. (4.76) is independent from 3 , the last term must vanish: X A 3 A y = 0 =) A = 0: (4.77) Further assume 1 =jiihij ji 0 ihi 0 j. Note that the partial matrix elementhj 0 jP ji 0 i is an operator on theH NS factor,jiihij. Then Eq. (4.76) reduces to jiihij = X ;j 0 U y hj 0 jP ji 0 i jiihij hi 0 jP y jj 0 iU : (4.78) Taking matrix elements with respect toji ? i, a state orthogonal tojii, yields: 0 = X ;j 0 jhi ? j U y hj 0 jP ji 0 i jiij 2 =) hi ? j U y hj 0 jP ji 0 i jii = 0; (4.79) which, in turn implies that U y hj 0 jP ji 0 i jii is proportional tojii, i.e., [hj 0 jP ji 0 i]jii_ Ujii: (4.80) 93 Sinceji 0 i;jj 0 i are arbitrary this condition implies that the submatrix P must be of the form P = U C . Substituting P = U C into Eq. (4.76) we have Tr in fU I 1 U y Ig = Tr in f P U C 1 U y C y g, so that Tr in f 1 g = Tr in f X I NS C 1 I NS C y g: (4.81) Now suppose 1 = P iji 0 j 0 iji 0 j 0jiihjj ji 0 ihj 0 j; then from Eq. (4.81) we find X iji 0 iji 0 i 0jiihjj = X iji 0 j 0 k 0 iji 0 j 0jiihjjhk 0 jC ji 0 ihj 0 jC y jk 0 i: (4.82) Using P k 0 jk 0 ihk 0 j = I in , Eq. (4.82) becomes X iji 0 iji 0 i 0jiihjj = X iji 0 j 0 iji 0 j 0jiihjjhj 0 j X C y C ji 0 i: (4.83) It follows that X C y C = I in : (4.84) Next consider the normalization constraint P E y E = I for the Kraus operators, together with the additional constraints we have derived (A = 0, P = U C ): X P y P + D y D = I NS I in =) I NS X C y C + X D y D = I NS I in : (4.85) But, from Eq. (4.84) we have P P y P = I NS I in . Therefore D = 0. 94 Taking all these conditions together finalizes the matrix representation of the Kraus operators as E = 0 @ U C 0 0 B 1 A : (4.86) For a scalar C we recover the DFS condition (4.18). These considerations establish the necessity of the representation (4.86); it is simple to show that this representation is also sufficient, by substitution and checking that the NS and DFS conditions are satis- fied. Therefore we have proved Theorems 3 and 6. Perfect Initialization We now prove Corollaries 1 and 2 for DF-initialized states of the form S (0) = P d S (0)P d . Thus, we have to prove that D = 0 in Eq. (4.75). When S (0) =P d S (0)P d we have that 2 = 0 and 3 = 0 and Eq. (4.76) reduces to Tr in fU I 1 U y Ig = Tr in f X P 1 P y g: (4.87) The argument leading to the vanishing of the A [Eq. (4.77)] then does not apply, and indeed the A need not vanish. However, the arguments leading to P = U C and P P y P = I NS I in do apply. Hence D = 0. 4.7.2 Markovian Dynamics Arbitrary Initial State Consider Markovian dynamics 95 @ S @t =i[H S ; S ] + X F S F y 1 2 F y F S 1 2 S F y F ; (4.88) with the following matrix representation of the various operators: S = 0 @ 1 2 y 2 3 1 A ; H S = 0 @ H 1 H 2 H y 2 H 3 1 A ; F = 0 @ P A D B 1 A : (4.89) Then we find the dynamics of the NS block to be @ NS @t = @Tr in f 1 g @t = iTr in f[H 1 ; 1 ]giTr in f(H 2 y 2 2 H y 2 )g + Tr in f X P 1 P y + A y 2 P y + P 2 A y + A 3 A y 1 2 X (P y P + D y D ) 1 + (P y A + D y B ) y 2 1 2 X 1 (P y P + D y D ) + 2 (A y P + B y D )g (4.90) The right-hand side of this equation must be independent of 2 and 3 , for any matrices 2 and 3 . Therefore the term A 3 A y implies A = 0. Collecting the remaining terms acting on y 2 from the left yields Tr in f(iH 2 D y B ) y 2 g = 0. Together we have A = 0; iH 2 + X D y B = 0: (4.91) 96 This reduces Eq. (4.90) to @ NS @t = @Tr in f 1 g @t = iTr in [H 1 ; 1 ] + Tr in X P 1 P y 1 2 Tr in X f(P y P + D y D ); 1 g (4.92) Consider the initial state 1 = NS ji 0 ihi 0 j, withji 0 i2H in : @ NS @t =i[hi 0 jH 1 ji 0 i; NS ] + X hj 0 jP ji 0 i NS hi 0 jP y jj 0 i 1 2 X f NS ; (hi 0 jP y jj 0 ihj 0 jP ji 0 i +hi 0 jD y jj 0 ihj 0 jD ji 0 i)g (4.93) Let NS =j ih j with arbitrary and applyh ? j:::j ? i, such thath ? j i = 0, to Eq. (4.93), denoting P ;i 0 ;j 0hj 0 jP ji 0 i: X jh ? jP ;i 0 ;j 0j ij 2 = 0: (4.94) Since this identity must hold for all and ? , we find that P ;i 0 ;j 0 = c ;i 0 ;j 0I NS , which implies that P = I NS C in . Moreover, by definition of a NS, there exists a Hermitian matrix H NS such that NS obeys a Schr¨ odinger equation, @ NS =@t = i[H NS ; NS ]. Therefore the non-Hermitian term P D y D in Eq. (4.92) must vanish, implying that D = 0. Combining these results with Eq. (4.91) yields @Tr in f 1 g @t =iTr in f[H 1 ; 1 ]gi[H NS ; NS ] (4.95) 97 This identity can be realized iff H 1 = H NS I in +I NS H in . Therefore the NS conditions are obtained as H = 0 @ H NS I in +I NS H in 0 0 H 3 1 A ; F = 0 @ I NS C in 0 0 B 1 A : (4.96) The DFS condition is a special case of (4.93), with dim(H in ) = 1. This concludes the proof of Theorems 4 and 8. Perfect Initialization Now consider perfect initialization: S = 0 @ 1 0 0 0 1 A : (4.97) This is just the case of an arbitrary initial state considered above, with 2 = 0 and 3 = 0 in Eq. (4.90). This then yields the dynamics of NS as being given by Eq. (4.92). Repeat- ing the derivation following Eq. (4.92) we conclude again that D = 0; P = I NS C in and H 1 = H NS I in +I NS H in . Note that Eq. (4.91) now does not apply (it was obtained assuming nonzero 2 ; 3 ), i.e., we cannot conclude that A and H 2 vanish. This implies that that @ S =@t has a 98 non-zero off-diagonal elements, which, using the master equation (4.88), we calculate to be: upper right block: i 1 H 2 + X P 1 D y 1 2 1 (P y A + D y B ) =i 1 H 2 1 2 1 X (I NS C y in )A bottom right block: X D 1 D y = 0. To prevent the appearance of corresponding off-diagonal blocks in S , we must therefore demand H 2 + i 2 X (I NS C y in )A = 0; (4.98) which is Eq. (4.55). The DFS case is obtained with dim(H in ) = 1. This concludes the proof of Theorems 5 and 7. 4.7.3 Non-Markovian Dynamics The derivation of the conditions for decoherence-freeness in the case of non-Markovian dynamics is somewhat different from the other two cases we have considered, because of the appearance of the nonlocal-in-time integral in the master equation: @ S @t =i[H S ; S ] +L Z t 0 dt 0 k(t 0 ) exp(Lt 0 ) S (tt 0 ) (4.99) 99 In order to find necessary conditions on the structure of H S andL consider the case of smallt, expand S (t) = X n=0 t n (n) S (0); k(t) = X m=0 t m k (m) (0); (4.100) and substitute into Eq. (4.99). The constant (t 0 ) term yields (1) S (0) =i[H S ; S (0)]. (4.101) The terms involvingt 1 yield, after Taylor-expanding exp(Lt 0 ): 2 (2) S (0) =i[H S ; (1) S (0)] +k(0)L S (0). (4.102) Thus the solution of Eq. (4.99) up to first and second order in time is: S (t) = S (0)it[H S ; S (0)] +O(t 2 ); (4.103) S (t) = S (0)it[H S ; S (0)] t 2 2 f[H S ; [H S ; S (0)]] +k(0)L S (0)g +O(t 3 ): (4.104) Arbitrary Initial State Consider once again the matrix representations as in Eq. (4.89). Substituting these expressions into the first order equation (4.103), the 1 (t) block yields NS (t) = NS (0)itTr in f[H 1 ; 1 (0)]gitTr in fH 2 y 2 (0) 2 (0)H y 2 g =) H 2 = 0; H 1 = H NS I in + I NS H in : (4.105) 100 Continuing to second order, Eq. (4.104), the NS block is found to be NS (t) = NS (0)it[H NS ; NS (0)] t 2 2 [H NS ; [H NS ; NS (0)]] +Tr in f2k(0) X P 1 P y + A y 2 P y + P 2 A y + A 3 A y k(0) X (P y P + D y D ) 1 + (P y A + D y B ) y 2 k(0) X 1 (P y P + D y D ) + 2 (A y P + B y D )g: (4.106) The first three terms correspond to unitary evolution, but the remaining terms are essen- tially identical to the case of Markovian dynamics and must be made to vanish, just as in Eq. (4.90). The same arguments used there apply and consequently F = 0 @ I NS C in 0 0 B 1 A : (4.107) The conditions (4.105), (4.107) are necessary and sufficient for unitary evolution of the NS block under our non-Markovian master equation. The DFS case is obtained with dim(H in ) = 1. This concludes the proof of Theorems 9 and 10. Perfect Initialization Assume S (0) = 0 @ (0) 0 0 0 1 A ; (4.108) 101 then from the first order equation (4.103), the NS block is found to satisfy NS (t) = NS (0)itTr in f[H 1 ;(0)]g =) H 1 = H NS I in + I NS H in : (4.109) To second order in time [Eq. (4.104)]: NS (t) = NS (0)it[H NS ; NS (0)] t 2 2 [H NS ; [H NS ; NS (0)]] + t 2 2 Tr in fH 2 H y 2 (0)(0)H 2 H y 2 + 2k(0) X P P y (P y P + D y D )(0)(0)(P y P + D y D )g; (4.110) which is again similar to the Markovian case. Similar logic therefore yields H 2 = D = 0, and hence F = 0 @ I NS C A 0 B 1 A : (4.111) Here we should notice that the density matrix S (0) has an off-diagonal element (0) P (P y A + D y B ) = (0) P P y A . This term must vanish, for otherwise S (t) has non-zero off-diagonal elements. Summarizing, we have F = 0 @ I NS C A 0 B 1 A ; X (I NS C y )A = 0; H = 0 @ H NS I in +I NS H in 0 0 H out 1 A : (4.112) The DFS case is obtained with dim(H in ) = 1. This concludes the proof of Corollaries 3 and 4. 102 Chapter 5 Optimal Quantum Error Correction 5.1 Optimal Quantum Error Correction Quantum error correction is essential for the scale-up of quantum information devices. A theory of quantum error correcting codes has been developed, in analogy to classical coding for noisy channels [Shor, 1995, Steane, 1996]. Recently [Reimpell and Werner, 2005, Yamamoto et al., 2005, Yamamoto and Fazel, 2006, Fletcher et al., 2007, Kosut and Lidar, 2006] did this by posing error correction design as an optimization problem to directly maximize fidelity, with the design variables being the process matrices asso- ciated with the encoding and/or recovery channels. Here we revisit the direct approach and also present an indirect approach to fidelity maximization based on minimizing the error between the actual channel and the desired channel. Both the direct and indirect approaches lead naturally to bi-convex optimization problems, specifically, two semidef- inite programs (SDPs) [Boyd and Vandenberghe, 2004] which can be iterated between recovery and encoding. For a given encoding the problem is convex in the recovery. For a given recovery, the problem is convex in the encoding. The advantage of these approaches is that noisy channels which do not satisfy the standard assumptions for perfect correction [Knill and Laflamme, 1997], can be optimized for the best possible encoding and/or recovery. There are two related problems addressed here: robustness and computational cost. Robustness— Standard error correction schemes, as well those produced by opti- mization tuned to specific errors, are often not robust to even small changes in the error 103 system. Fault tolerant correction often requires several concatenation levels to achieve a desired robust fidelity. Here we show a means to incorporate specific models of error system uncertainty, resulting in highly robust error correction. If the resulting robust fidelity levels are sufficiently good, then no further levels of concatenation are neces- sary. This assessment is not knowable without performing the robust optimization. The SDP formulation easily allows for a robust design by enumerating constraints associated with different error models. Computational cost.— The computation effort required to solve either direct SDP increases rapidly in the number of qubits used in the encoding. This can render the designer to an overnight (or fortnight) question and answer session using today’s best desktop computers. Here we propose a few approaches to alleviate this problem. As described in [Kosut and Lidar, 2006], solving the dual problem associated with the SDPs for direct recovery or encoding alleviaites some of the burden. The indirect approach presented here generally results in similar achieved fidelity but has the further advantage of additional computational reduction. We also present an approximation to optimal recovery which significantly reduces the computations while retaining excellent perfor- mance. Standard error correction model. — Subject to standard assumptions, the dynam- ics of any open quantum system can be described in terms of a completely-positive (CP) map: S ! P i A i S A y i , a result known as the Kraus Operator Sum Representation (OSR) [Nielsen and Chuang, 2000]. Here S is the initial system density matrix and theA i are called operation elements, and satisfy P i A y i A i =I (identity). The standard error correction procedure involves CP encoding (C), error (E), and recovery (R) maps (or channels): S C ! C E ! C R ! ^ S , i.e., using the OSR: ^ S = X r;e;c (R r E e C c ) S (R r E e C c ) y : (5.1) 104 The encodingfC c g m C c=1 and recoveryfR r g m R r=1 operation elements are rectangular matri- ces, respectivelyn C n S andn S n C , since they map between the system Hilbert space (of dimensionn S ) and the system+ancillae Hilbert space (of dimensionn C ). The error fE e g m E e=1 operation elements are square (n C n C ) matrices. . 5.2 Performance measures Assume that we are given the error channelE in the form of a process matrixX E , i.e., the output of a quantum process tomography experiment [Nielsen and Chuang, 2000]. The error correction objective considered here is to design the encodingC and recovery R so that, for givenE, the map S ! ^ S is as close as possible to a desiredn S n S unitaryL S . To this end we use the average fidelity: f avg = 1 n 2 s X r;e;c TrL y S R r E e C c 2 (5.2) As shown in [Nielsen and Chuang, 2000], f avg = 1 if and only if there are constants rec such that L y S R r E e C c = rec I S ; X r;e;c j rec j 2 = 1 (5.3) This suggests the indirect measure of fidelity, the “distance-like” error (using the Frobe- nius norm,kXk 2 fro = TrX y X), d ind = P r;e;c kL y S R r E e C c rec I S k 2 fro = P c kRE(I E C c ) c L S k 2 fro (5.4) with c them R m E matrix with elements rec ,E then C n C m E rectangular matrix [E 1 E m E ]; andR them R n S n C matrix obtained by stacking them R matricesR r . Hence, we have P c Tr y c c = P r;e;c j rec j 2 = 1, andR y R = P r R y r R r =I C . 105 A recovery and encoding pair,R;C achieves perfect error correction (equivalently d ind = 0; f avg = 1) if and only if forc;c 0 = 1;:::;m C , (I E C y c )E y E(I E C c 0) = y c c 0 I S (5.5) This is a generalization to non-unitary encoding of the Knill-Laflamme condition for perfect error correction with unitary encoding [Knill and Laflamme, 1997] . In this latter case,C has only a singlen C n S matrix elementC;C y C =I S whosen S columns are the codewords. Condition (5.5) then becomes, (I E C y )E y E(I E C) = I S = y ; Tr = 1; ( ism E m E ) (5.6) Asf avg andd ind are explicitly dependent on the channel elements, they are convenient for optimization. Consider then the following optimization problems. Direct Fidelity Maximization maximize f avg (R;C) subject to R y R =I C ; C y C =I S (5.7) Indirect Fidelity Maximization minimize d ind (R;C; 1 ;:::; m C ) subject to R y R =I C ; C y C =I S ; P c k c k 2 fro = 1 (5.8) HereC is them C n C n S matrix obtained by stacking them C matricesC c . The direct approach was also used in [Reimpell and Werner, 2005, Kosut and Lidar, 2006]. We now discuss methods to approximately solve (obtain local solutions) to each of these problems. 106 5.3 Direct fidelity maximization Problem (5.7) is not a convex optimization jointly inR andC . However, the individ- ual problems of optimizing over R given C, or optimizing over C given R, can each be transformed to a convex optimization, specifically, a semidefinite program (SDP) [Boyd and Vandenberghe, 2004]. As pointed out in [Reimpell and Werner, 2005], iter- ating between the two convex problems is guaranteed to increase fidelity monotoni- cally. When the iterations converge, all that can be said is that a local solution has been found to the original problem (5.7). Several variants on this formulation can be found in [Reimpell and Werner, 2005, Yamamoto et al., 2005, Yamamoto and Fazel, 2006, Fletcher et al., 2007]. Here we start with the procedure used in quantum process tomography [Nielsen and Chuang, 2000]. Expand each of the operation matricesR r and C c in a set of basis matrices, respectively,B Ri (n S n C ) andB Ci (n C n S ), that is, R r = P i x ri B Ri ; C c = P i x ci B Ci with scalar coefficientsx ri andx ci . The basis matri- ces are directly related to the observables measured in an experimental process tomog- raphy setting, and indeed, we assume that data about the channels is presented to us in the form of the observables measured and then S n C n S n C positive process matrices X R ; X C which are quadratic functions of the scalarsx ri ; x ci , i.e., (X R ) ij = P r x ri x rj and (X C ) k` = P c x ck x c` . To construct an OSR from the process matrices we proceed as follows. The recovery and encoding elements R r ;C c can be obtained from diag- onalizations of X R and X C : X R = P n S n C r=1 s Rr v Rr v y Rr and X C = P n S n C c=1 s Cr v Cr v y Cr with positive eigenvalues s Rr ;s Cc in decreasing order and with orthonormal eigen- vectors v Rr ;v Cc . This yields the expansion coefficients as x Ri = p s Rr (v Rr ) i and x Ci = p s Cr (v Cr ) i . Although the maximum number of eigenvalues isn S n C , the number 107 of non-zero eigenvalues determines both the number of encoding and recovery elements, i.e.,m R = rankX R andm C = rankX C . Hence, R r = P n S n C i=1 p s Rr (v Rr ) i B Ri r = 1;:::;m R C c = P n S n C i=1 p s Cr (v Cr ) i B Ci c = 1;:::;m C (5.9) Problem (5.7) can now be equivalently expressed in terms of the process matrices. The objective function in (5.7) in terms ofX R ;X C becomes, f avg (R;C) = TrW R (E;C)X R = TrW C (E;R)X C W R (E;C) ij = P k;` F ijk` ; W C (E;R) k;` = P i;j F ijk` F ijk` = 1 n 2 S P e (TrL y S B Ri E e B Ck )(TrL y S B Rj E e B C` ) (5.10) Sincef avg (R;C) is linear in eitherX R orX C (5.10), the non-convex quadratic equal- ity constraints (X R ) ij = P r x ri x rj and (X C ) k` = P c x ck x c` play no role except for enforcing the semidefinite (convex) constraints X R 0 and X C 0. This leads to finding a solution to (5.7) by iteratively solving the following two SDPs: Optimal Recovery maximize f avg (X R ) = TrW R (E;C)X R subject to P i;j (X R ) ij B y Ri B Rj =I C ; X R 0 (5.11) Optimal Encoding maximize f avg (X C ) = TrW C (E;R)X C subject to P i;j (X C ) ij B y Ci B Cj =I S ; X C 0 (5.12) Given an encoding C, the solution to (5.11) produces an optimal recovery R. Con- versely, given a recoveryR, the solution to (5.12) produces an optimal encodingC 1 . 1 As shown in [Kosut and Lidar, 2006], the computational complexity in solving either (5.11) or (5.12) can be reduced by appealing to Lagrange Duality Theory [Boyd and Vandenberghe, 2004, Ch.5] 108 5.4 Robust error correction A major limitation of the standard procedure of modeling the error channel as fixed, i.e., in terms of given operation elementsfE e g, is that this does not account for uncertainty in knowledge of the channel, and in most cases will hence be too conservative. For exam- ple, different runs of a tomography experiment can yield different error channelsE . Or, a physical model of the error channel might be generated by an C m E -dimensional Hamiltonian H() dependent upon an uncertain set of parameters . Not accounting for model uncertainties typically leads to non-robust error correction, in the sense that a small change in the error model can lead to poor performance of the error correction procedure. One way to account for uncertainties in terms of an OSR is to take a sam- ple from the set, say, H( ); = 1;:::;`. Tracing out the m E environmental states will result in a set of OSR models,E ; = 1;:::;`. To handle this, the objectives in the SDPs (5.11)-(5.12) need to be modified. Two possibilities are the worst-case and average-case. For the worst-case, the objectives in (5.11) and (5.12) can be replaced, respectively, by, maximize min TrW R (E ;C)X R for (5.11) maximize min TrW C (E ;R)X C for (5.12) (5.13) Similarly, the average-case objectives are, respectively, maximize P TrW R (E ;C)X R for (5.11) maximize P TrW C (E ;R)X C for (5.12) (5.14) which shows that it is more efficient to solve the associated dual problems , also SDPs, but with fewer optimization variables. 109 In the average-case the objectives can be equivalently expressed in the same form as (5.11)-(5.12) but with the error system matrixE replaced by the average error system matrix,E avg , whose elements are (E avg ) ;i =E ;i = p `m E (5.15) 5.5 Indirect fidelity maximization So far we have dealt with direct fidelity maximization. We now present our indirect approach. We restrict attention to unitary encoding (C contains the singlen C n S matrix elementC, i.e.,m C = 1). Using the constraints in (5.8) gives the distance measure (5.4) as, d ind (R;C;) =kRE(I E C) L S k 2 fro (5.16) = n S + TrE(I E CC y )E y 2Re TrRE( y CL y S ) with the single n CA n RA n S n C matrix (viz. m R = n CA n RA ). Since only the last term depends on R, minimizing d ind (R;C;) over R is equivalent to maximizing the last term overR which can be shown to be, max R y R=I C Re TrRE( y CL y S ) = Tr p E( CC y )E y (5.17) with them E m E matrix = y . Then C n RA n C optimizing recovery matrixR is given by, R = v 1 v n C [u 1 u n C ] y (5.18) 110 where thev i andu i are, respectively, the right and left singular vectors in the singular value decomposition of then C n C n RA matrixE( y CL y S ). Given (C; ), and the fact that need only be chosen so that y = , the following choice for achieves . n CA m E n RA = 1 ) = 2 4 p 0 n CA m E m E 3 5 R isn S n CA n S n CA (unitary) (5.19) n CA <m E n RA n CA =m E ) = p R isn S m E n S n CA (tall) (5.20) Result (5.19) implies that R is unitary when the number of encoding ancilla, n CA , is chosen large enough that no recovery ancilla are needed, i.e., n RA = 1. This is the case of a decoherence-free-subspace where the role of recovery is to invert the action of the noise map restricted to the code space. Related results were obtained in [Kribs and Spekkens, 2006]. When there are insufficient encoding ancilla, i.e.,n CA < m E , (5.20) reveals that additional recovery ancilla are needed so thatm E = n CA n RA . The result in (5.19)-(5.20) does not change if multiplied by a unitary. It can be shown that this unitary freedom is exactly the unitary freedom in chosing the OSR operators. Optimal recovery.— Given an encodingC, an optimal recoveryR can be obtained by first solving for which maximizes (5.17), that is, maximize Tr p E( CC y )E y subject to 0; Tr = 1 (5.21) It can be shown that the optimal is equivalently obtained by solving an associated SDP. The next step is to obtain from via (5.19)-(5.20), and finallyR from (5.18). Approximation to optimal recovery.— Assume that the errors are random unitaries, i.e., E i = p p i U i , wherefp i g are probabilities andfU i g are unitaries. In this case the 111 diagonal elements of them E m E matrix correspond to the probability of the asso- ciated error [Zanardi and Lidar, 2004]. Generalizing to arbitrary channels, we consider the approximation of setting equal to the diagonal matrix with diagonal elements, ii =kE i k 2 fro =n C (5.22) each being the average sum-square of the singular values of E i . SinceE is trace- preserving, P i ii = 1 as required. Using this approximation in (5.19)-(5.20) to cal- culateR directly is obviously very efficient, especially for large dimensions, where by comparison solving (5.21) can be computationally expensive even when is constrained to be diagonal. Optimal encoding.— Given (R;), an optimal encoding can be found by solving (5.8) for C. Replacing the non-convex equality constraint, C y C = I S , by the convex inequalityC y CI S leads to the relaxed convex optimization problem, minimize kRE(I E C) L S k 2 fro subject to C y CI S (5.23) The optimal relaxed solution, C rlx opt , can be approximated using a singular value decomposition to obtain a nearby encoding, ^ C, which satisfies ^ C y ^ C = I S . The pro- cedure is to replace the singular values with ones, i.e., C rlx opt = P i s i u i v y i ) ^ C = P i u i v y i . The unknown optimal solution, C opt , then satisfies d ind (R;C rlx opt ;) d ind (R;C opt ;) d ind (R; ^ C;). If the upper and lower bounds are close then ^ C can be considered optimal. Observe also, that unlike the optimal encoding solution obtained from (5.12), which depends on the rank ofX C , here the ancilla size can be set in advance. 112 Robust error correction.— As in the direct case [Eqs. (5.13),(5.11)], the indirect methods can also be extended to the robust worst-case and average-case. For the worst- case, Eqs. (5.16),(5.17) show that min 2Tr q E ( CC y )E y TrE (I E CC y )E y max kRE (I E C) L S k 2 fro (5.24) replace the recovery objectives in (5.21) and (5.23), respectively. For the average-case, replace the error system matrix E in (5.21) and (5.23) with the average error system matrix (5.15). 5.6 Examples We now apply the methods developed above to the goal of preserving a single qubit (n S = 2) using a q C -qubit (n C = 2 q C ) codespace. We model the environment by adding a single environment qubit and randomly selecting a 2n C 2n C time-independent HamiltonianH 2n C acting on the joint codespace-environment Hilbert space. For every value of n C the Hamiltonian is normalized by dividing it by the maximum absolute value of its eigenvalues (singular values). The unitary evolution operator generated by this Hamiltonian is then used to find the two operation elements (m E = 2) of the error channel, by tracing out the two environmental states. One can view this as either time evolution under H 2n C with fixed coupling strength between system and environment, or switch the role of time and strength. We thus refer to the free parameter (time or strength) as error magnitude in Figs. (5.1-5.3). 113 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 error magnitude channel fidelity (f avg ) 7−qubit 5−qubit 3−qubit optimal recovery at 0.75 standard recovery Figure 5.1: Lower curves: standard recovery for 3,5, and 7-qubit codes. Upper curves: optimal recovery at mean 0.75. Optimal recovery for the 3 and 5-qubit codes are obtained by solving the direct optimization problem (5.11). The 7-qubit optimal recov- ery is obtained from the indirect approach (5.19), using the diagonal approximation (5.22). Figure 5.1 shows plots of fidelity vs. error magnitude in the range [0.5,1.0] for the standard 3, 5, and 7-qubit codes (respectively, bit flip [Nielsen and Chuang, 2000], “per- fect” [Laflamme et al., 1996], and Steane [Steane, 1996]) along with an optimal recov- ery of these codes at the mean error magnitude 0.75. Clearly, optimal recovery greatly improves the performance of these codes relative to their standard recovery, designed for independent single-qubit errors. The standard recovery results do not even make the fidelity scale of Figure (5.2), where we compare optimal and average-case fidelities for the standard 3,5, and 7-qubit codes. The recoveries obtained using the average-case objective (5.24) are more robust 114 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.975 0.98 0.985 0.99 0.995 1 error magnitude channel fidelity (f avg ) optimal recovery at 0.75 average−case recovery 3−qubit (direct) 5−qubit (direct) 7−qubit (indirect−approx) worst−case recovery Figure 5.2: Comparison of optimal and robust recovery for 3,5,7 standard qubit encod- ing. The optimal recoveries are from Figure 5.1. The 3 & 5 robust recoveries are obtained from the direct method (5.11) using the average error system (5.15). The 7- qubit robust recovery is obtained from the indirect approach (5.19) using the diagonal approximation (5.22) with the average error system. than the optimal codes with the robustness increasing markedly for the 5 and 7-qubit codes. The recovery which is optimal for the worst-case objective with 3-qubits (dashed line) shows less variation then the average-case as expected, but is not nearly as good at the lower error magnitudes. In Figs. (5.1,5.2) we optimized only the recovery but used standard encoding. In the further compressed fidelity scale of Figure (5.3) we show the results of optimizing the 3 and 5-qubit encoding along with the recovery. These now outperform the 7-qubit code for which only the recovery is optimized. Although more robustness than exhibited by either the 3-qubit or 5-qubit code is undoubtedly attainable from a robust 7-qubit code, 115 0.5 0.6 0.7 0.8 0.9 1 0.998 0.9982 0.9984 0.9986 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 1 error magnitude channel fidelity (f avg ) 7−qubit recovery at 0.75 (indirect−approx) 7−qubit avg−case recovery (indirect−approx) 3−qubit optimal at 0.75 (direct) 3−qubit optimal avg−case (direct) 5−qubit optimal avg−case (indirect) Figure 5.3: Standard 7-qubit code: recovery at 0.75 & average-case recovery (same as Fig. 5.2). 3-qubit case: optimal and robust recovery/encoding by iterating the direct method (5.11),(5.12) to 5 significant digits in fidelity. 5-qubit case: robust recov- ery/encoding obtained by iterating the indirect method (5.21)-(5.23) to 6 significant digits. making this effort (possibly requiring several days of computation) would have to be determined e.g., by considering the tradeoff in resources involved in concatenation in order to meet a requisite threshold for fault-tolerant computation. 5.7 Conclusions We have presented an optimization approach to quantum error correction that yields codes which achieve robust performance. An intriguing prospect is to integrate the results found here within a complete “black-box” error correction scheme, that takes 116 quantum state or process tomography as input and iterates until it finds an optimal error correcting encoding and recovery. 117 5.8 Appendix Proof of (5.5) Condition (5.5) follows directly from (5.3) by multipling both sides by their respective conjugate (with indices c and c 0 ) which also eliminates R because R y R = I C . This immediaitely establishes that (5.5) is a necessary condition for (5.3). To prove sufficiency, first expand (5.4) to get, d ind = P c Tr (I E C y c )E y E(I E C c ) +Tr c I S 2ReTrRE( y c C c L y S ) c = y c c (5.25) In a later section of the Appendix we establish that, max R y R=I C ReTrRW = Tr p WW y (5.26) Consequently, min R y R=I C d ind = P c [Tr (I E C y c )E y E(I E C c ) +Tr c I S ] 2Tr p WW y W = P c E( y c C c L y S ) (5.27) Using (5.5) we get, Tr p WW y = P c E(I E C c C y c )E y This, together with repeated uses of (5.5) shows that min R y R=I C d ind = 0. Sinced ind is a norm, and is zero, then so is the argument of the norm zero, which by definition establishes (5.3) and thus shows sufficiency of (5.5). 118 Dual optimization for direct method The convex optimization problems (5.11) and (5.12) are both SDPs of the form, maximize TrXW subject to X 0; P ij X ij B y i B j =I m (5.28) with optimization variable X = X y 2 C nn , n = rm for some integer r, and with each basis matrixB i 2 C rm . We will refer to this as the primal problem. Accounting for the linear (matrix) equality constraint and the Hermiticity ofX, the number of real optimization variables in (5.28) isn 2 m 2 = (r 2 1)m 2 . The computational burden can be somewhat alleviated by appealing to Lagrange Duality Theory [Boyd and Vandenberghe, 2004, Ch.5] which in this case provides a a more efficient means to numerically solve the original problem. Replacing the maxi- mization in (5.28) with an equivalent minimization of the negative of the objective, gives the Lagrangian, L(X;Z;Y ) = TrXW TrXZ TrY (I m P ij X ij B y i B j ) = P ij X ij (W ji Z ji + TrYB y i B j ) TrY (5.29) whereZ =Z y 2 C nn andY =Y y 2 C mm are Lagrange multipliers associated with the (Hermitian) inequality and equality constraints, respectively. The Lagrange dual function is then, g(Z;Y ) = inf X L(X;Z;Y ) = 8 < : TrY; Z ji = TrYB y i B j W ji 1 otherwise (5.30) 119 For any Y and Z 0, g(Z;Y ) yields a lower bound on the optimal objec- tive Tr X opt W . The largest lower bound from this dual function is then maxfg(Z;Y )gZ 0. Eliminating Z, the dual optimization problem can be written equivalently as, minimize TrY subject to K(Y )W 0; K ij (Y ) = TrYB y j B i (5.31) with optimization variableY = Y y 2 C mm . The number of (real) optimization vari- ables for the dual problem is then at mostm 2 , a reduction in the number of optimization variables by a factor ofr 2 1. For this problem strong duality holds [Boyd and Vandenberghe, 2004]. Conse- quently, at optimality of (5.28) and (5.31) the primal and dual objectives are equal, i.e., TrX opt W = TrY opt . The complementary slackness condition isZ opt X opt = 0 with Z opt =K(Y opt )W . This together with the linear equality constraint in (5.28) can be used to obtain the primal solutionX opt from the dual solutionY opt . That is, solve for X opt from the set of linear equations, P ij X opt ij B y i B j = I m (K(Y opt )W )X opt = 0 (5.32) Proof of (5.17) and (5.18) Then C n C n RA matrixE( y C) has a maximum rank ofn C . Hence a singular value decomposition is of the form,E( y C) =USV y ; S = [S 0 0]; withS 0 ann C n C diagonal matrix containing then C singular values. IfV is partitioned asV = [V 1 V 2 ] withV 1 n RA n C n C then the objective function in (5.17) becomes, Re TrRE( y CU y S ) = Re TrS 0 X; X =V y 1 RU (5.33) 120 SincekXk 1, then Re Tr S 0 X Tr S 0 . Equality occurs if and only if X = I C , or equivalently, R = V 1 U y , which is precisely the result in (5.18). This also estab- lishes that the optimal objective function is Tr S 0 which, by definition, is equal to Tr p E( CC y )E y , thus establishing (5.17). Unitary freedom in (5.17) In (5.17), = y remains unchanged if is is multiplied by a unitary. This unitary freedom is exactly the unitary freedom in describing the error system OSR. To see this, recall that two error systems with OSR elementsE = [E 1 :::E m E ] andF = [F 1 :::F m E ] are equivalent if and only ifE i = P j W ij F j where them E m E matrixW is unitary [Nielsen and Chuang, 2000, Thm.8.2]. Equivalently, E =F (W I C ). Subtituitng this forE into the left hand side of (5.17) gives, Re TrRE( y CL y S ) = Re TrRF (( 0 ) y CL y S ) (5.34) with 0 =W y . Hence, ( 0 ) y 0 =W y W y = , which establishes the claim. Solving (5.21) via an SDP Problem (5.21) is of the form, maximize Tr p F ( ) subject to 0; Tr = 1 (5.35) whereF ( ) is linear in . Consider the relaxed problem, maximize TrY subject to F ( )Y 2 0; 0; Tr = 1 (5.36) 121 This is an SDP in andY with Lagrangian, L( ;Y;P;Z) = TrY TrP (F ( )Y 2 ) TrZ +(Tr 1) (5.37) The dual function is, g(P;;Z) = inf ;Y L( ;Y;P;Z) = 8 < : inf Y Tr(PY 2 Y ); Z =IA(P ) 1 otherwise (5.38) withA(P ) = @ @ TrPF ( ), which is not dependnet on becauseF ( ) is linear in . Performing the indicated inf Y givesY = (1=2)P 1 andg =( + (1=4)TrP 1 ). The dual optimization associated with (5.36) is to maximizeg, or equivalently, minimize its negative, i.e., minimize + 1 4 TrP 1 subject to P > 0; IA(P ) 0 (5.39) This is an SDP in the dual variablesP; . For this problem strong duality holds [Boyd and Vandenberghe, 2004]. Consequently, at optimality of (5.36) and (5.39) the com- plementary slackness condition isP opt (F ( opt )Y 2 opt ) = 0. SinceP opt > 0, we have Y opt = p F ( opt ). This establishes that solving the SDP (5.36) is equivalent to solving the original problem (5.35). Computational cost The main difficulty with all the optimization methods is that the number of design parameters scales exponentially with the number of qubits. Although exponential scaling at the moment seems unavoidable, the optimization approaches pre- sented have differing computational cost at the expense of small differences in fidelity. 122 Table 5.1 displays the number of optimization variables associated with each of the approaches presented along with the sizes of the matrices which require singular value and/or eigenvalue decompostion (SVD/EVD). Method Recovery Encoding Direct Primal (n S 2 1)n C 2 (n C 2 1)n S 2 Dual n C 2 n S 2 SVD/EVD Dual to Primal (n S n C ) 2 (n S n C ) 2 Primal to OSR (n S n C ) 2 (n S n C ) 2 Indirect full m E 2 = (n CA n RA ) 2 diagonal m E =n CA n RA Relaxed SDP n S n C approx. 0 SVD/EVD to OSR n C m E n C Relaxed SDP to OSR n C n S Table 5.1: Number of optimization and transformation variables. 123 Chapter 6 Linear Quantum Error Correction 6.1 Introduction The problem of the formulation and characterization of the dynamics of quantum open systems has a long and extensive history [Davies, 1976, Alicki and Lendi, 1987, Breuer and Petruccione, 2002]. This problem has become particularly relevant in the context of quantum information processing [Nielsen and Chuang, 2000], where a remarkable theory of quantum error correction (QEC) was developed in recent years to address the problem of how to process quantum information in the presence of decoherence and imperfect control [Gaitan, 2004]. A key assumption common to many previous QEC studies is that the evolution of the quantum information processor can be described by a succession of completely positive (CP) maps [Kraus, 1983], interrupted by unitary gates or measurements [Knill and Laflamme, 1997]. However, it is well known that if the initial total system state is entangled, quantum dynamics is not described by a CP map [Pechukas, 1994, Stelmachovic and Buzek, 2001, Jordan et al., 2004, Carteret et al., 2008, Rodriguez et al., 2008]. In fact, we showed very recently in Ref. [Shabani and Lidar, 2009b] that a CP map arises if and only if the initial total system state has van- ishing quantum discord [Ollivier and Zurek, 2002], i.e., is purely classically correlated. One is thus naturally led to ask whether this impacts the applicability of QEC theory under circumstances where non-classical initial state correlations play a role. Here “ini- tial state” does not refer exclusively to the “t = 0” point, but also to intermediate times where the recovery map is applied, since this map was also assumed to be CP in standard 124 quantum error correction theory [Knill and Laflamme, 1997]. Motivated by this fact we here critically revisit the CP maps assumption in QEC, and show that it can be relaxed 1 . To do so, we argue that the generic noise map describing the evolution of a quantum computer as it undergoes fault tolerant quantum error correction (FT-QEC) is indeed not a CP map, but rather such a Hermitian, linear map (Section 6.2). The reason is, essentially, that imperfect error correction results in residual non-classical correlations between the system and the bath, as the next QEC cycle is applied. To deal with this, we develop a generalized theory of QEC which we call “linear quantum error correction” (LQEC), which applies to arbitrary linear maps on the system (Section 6.3). Then we show that, fortunately, the CP-map based version of QEC theory applies without modi- fications in the physically relevant setting of Hermitian maps. However, we show that a more general scenario is also possible, where the recovery map is Hermitian but not CP. This is useful since it obviates the unrealistic assumption that the recovery ancillas enter the QEC cycle as classically correlated with the other system qubits. Our results signif- icantly extend the realm of applicability of QEC, in particular to arbitrarily correlated system-environment states. We conclude in Section 6.4. 6.2 CP maps and fault tolerant quantum error correc- tion 6.2.1 CP maps: pro and con We have already mentioned that a QDP (2.8) becomes a CP map iff the initial system- bath state has vanishing quantum discord, i.e., is purely classically correlated [Shabani 1 Note that this is issue is entirely distinct from the critique of Markovian fault tolerant QEC expressed in [Alicki et al., 2006], which was concerned with the compatibility of other assumptions of fault-tolerant QEC (specifically, fast gates and pure ancillas) with rigorous derivations of the Markovian limit. 125 and Lidar, 2009b]. The standard argument in favor of CP maps is that since the system S may be coupled with the bathB, the maps describing physical processes onS should be such that all their extensions into higher dimensional spaces should remain positive, i.e., CP I n 08n2Z + , whereI n is then-dimensional identity operator. However, one may question whether this is the right criterion for describing quantum dynamics [Pechukas, 1994, Alicki, 1995]. An alternative viewpoint is to seek a description that applies to arbitrary SB (0), as we have done above. We now argue that this viewpoint is the correct one for fault-tolerant quantum error correction (FT-QEC). 6.2.2 (In)validity of the CP map model in FT-QEC System-environment correlations impose a severe restriction on the applicability of CP maps in FT-QEC, as we now argue. The CP map model used in FT-QEC [Knill and Laflamme, 1997, Aharonov and Ben-Or, b, Knill et al., 1998, Steane, 2003, Knill, 2005, Aharonov and Ben-Or, a, Aliferis and Preskill, 2008] can be described as follows (see, e.g., Eq. (8.1) in [Aharonov and Ben-Or, a]): S (T ) = tot CP (T;t 0 )[ S (t 0 )] where tot CP (T;t 0 ) = O N i=1 U (t i ) CP (t i ;t i1 ); (6.1) where T t N is the total circuit time, and where U [ S ] = U S S U y S is a unitary map (automatically CP) that describes an ideal quantum logic gate. 2 This represents the idea used repeatedly in FT-QEC, that the noisy evolution at every time step can be decomposed into “pure noise” CP (t i ;t i1 ) followed by an instantaneous and perfect unitary gate U (t i ). More precisely, in FT-QEC one assumes that the evolution starts (t =t 0 = 0) from a product state, then undergoes a CP map CP (t 1 ;t 0 ) due to coupling 2 In this subsection we denote noise maps by their initial and final times, to distinguish them from the instantaneous unitary maps. 126 to the environment, followed by an instantaneous error correction step U (t 1 ). If the latter were perfect then the post-error-correction state would again be a product state S (t 1 ) B (t 1 ). However, FT-QEC allows for the fact that the error correction step is almost never perfect, which means that there is a residual correlation between system and bath att 1 . Hence the map that describes the evolution of the system is a CP map if and only if the residual correlation is purely classical. Otherwise it is a Hermitian map. To make this point more explicit, consider a sequence of two noise time-steps, inter- rupted by one error correction step. In the ideal scenario, where the error correction step U (t 1 ) works perfectly (i.e., reduces the system-bath correlations to purely classical), we would have (2) CP (t 2 ;t 0 ) = CP (t 2 ;t 1 ) U (t 1 ) CP (t 1 ;t 0 ); (6.2) where CP (t 2 ;t 1 ) is again a CP noise map. However, in reality U (t 1 ) works imper- fectly [system-bath correlations are not purely classical after the action of U (t 1 )], and the actual map obtained is (2) H (t 2 ;t 0 ) = H (t 2 ;t 1 ) U (t 1 ) CP (t 1 ;t 0 ); (6.3) where H (t 2 ;t 1 ) is now a Hermitian map. Note that, in fact, even the assumption that the first noise map is CP will not be true in general, due to errors in the preparation of the initial state, leading to non-classical correlations between system and bath. We conclude that in general the CP map model (6.1) should be replaced by tot H (T;t 0 ) = O N i=1 U (t i ) H (t i ;t i1 ); (6.4) where H (t i ;t i1 ) are Hermitian maps, not necessarily CP. 127 It is worth emphasizing that this distinction between purely classical and other corre- lations, and resulting difference between CP and Hermitian evolution is not a distinction that has thus far been made in FT-QEC theory. Rather, in FT-QEC one distinguishes between “good” and “bad” fault paths, where the former (latter) contain only a few (too many) errors. Quoting from [B.M. Terhal, 2005]: “There are good fault paths with so-called sparse numbers of faults which keep being corrected during the computation and which lead to (approximately) correct answers of the computation; and there are bad fault-paths which contain too many faults to be corrected and imply a crash of the quantum computer.” This leads to a splitting of the total map (6.1) into a sum over good and bad paths. One then shows that the computation can proceed robustly via the use of concatenated codes, provided the “bad” paths are appropriately bounded. In [Aharonov and Ben-Or, a](p.1272) it was pointed out that the sum over “good” paths need not be a CP map, but can be decomposed into a new sum over CP maps [Eq. (8.13) there]. This new decomposition can then be treated using standard FT-QEC techniques. However, this assumes again that the total evolution is a CP map. These observations motivate a generalized theory of QEC, which can handle non- CP noise maps. This is the subject of the next section. The main result of this theory is reassuring: in spite of the invalidity of the CP map model in FT-QEC, the CP-map based results apply because the same encoding and recovery that corrects a Hermitian map can be used to correct a closely related CP map, whose coefficients are the absolute values of the Hermitian map. This is formalized in Corollary 5. 128 6.3 Linear Quantum Error Correction Having argued that non-CP Hermitian maps arise naturally in the study of open systems, and in particular FT-QEC, we now proceed to develop the theory of Linear QEC. For generality we do this for arbitrary linear maps L () = N X i=1 E i E 0y i (6.5) We then specialize to the physically relevant case of Hermitian maps. Let us first recall the fundamental theorem of “standard” QEC (for CP noise and CP recovery maps) [Knill and Laflamme, 1997]: Let P be a projection operator onto the code space. Necessary and sufficient conditions for quantum error correction of a CP map, CP () = P i F i F y i are PF y i F j P = ij P 8i;j: (6.6) An elegant proof of this theorem and a construction of the corresponding CP recovery map was given in Refs. [Nielsen and Chuang, 2000]; we use some of their methods in the proofs of Theorems 11,12. 6.3.1 CP-recoverable linear noise maps While general (non-Hermitian) linear maps of the form (6.5) do not arise from quantum dynamical processes [Eq. (2.8)], it is still interesting from a purely mathematical stand- point to consider QEC for such maps. Moreover, we easily recover the physical setting from these general considerations. 129 Theorem 11 shows that there is a class of linear noise maps which are equivalent to certain non-trace-preserving CP noise maps when it comes to error correction using CP recovery maps. Theorem 11 Consider a general linear noise map L () = P N i=1 E i E 0y i and associate to it an “expanded” CP map ~ CP () = 1 2 P N i=1 E i E y i + 1 2 P N i=1 E 0 i E 0y i . Then any QEC codeC and corresponding CP recovery mapR for ~ CP are also a QEC code and CP recovery map for L . Proof The operation elements of ~ CP arefF i g N i=1 = f 1 p 2 E i g N i=1 andfF N+i g N i=1 = f 1 p 2 E 0 i g N i=1 , whence ~ CP () = P 2N i=1 F i F y i . The standard quantum error correction conditions (6.6) for ~ CP , where 2 0 @ y 0 1 A = y ; (6.7) become three sets of conditions in terms of theE i andE 0 i : (i)PE y i E j P = 2 ij P; (ii)PE 0y i E 0 j P = 2 0 ij P; (iii)PE y i E 0 j P = 2 ij P; (6.8) wherei;j2f1;:::;Ng and ij = ij , ij = i;N+j , 0 ij = N+i;N+j . The existence of a projectorP which satisfies Eqs. (6.8)(i)-(iii) is equivalent to the existence of a QEC code for ~ CP . Assuming that a codeC has been found (i.e.,PC =C) for ~ CP , we use this as a code for L and show that the corresponding CP recovery mapR CP is also a recovery map for L . Indeed, letG j P 2N i=1 u ij F i be new operation elements for ~ CP , whereu is the unitary matrix that diagonalizes, i.e.,u y u =d. Then ~ CP = P 2N j=1 G j G y j . Let 130 R CP =fR k g be the CP recovery map for ~ CP . Assume that is in the code space, i.e., PP =. We now show thatR CP [ L ()] =, i.e., we have CP recovery. First, R CP [ L ()] = X k R k N X i=1 F i F y N+i ! R y k = N X i=1 2N X j;j 0 =1 u ij u N+i;j 0 X k (R k G j P ) PG y j 0 R y k : (6.9) Now, note that PG y k G l P = X ij u ik u jl PF y i F j P = X ij u ik ij u jl P = d k kl P: (6.10) Then the polar decomposition yields G k P =U k (PG y k G k P ) 1=2 = p d k U k P: (6.11) The recovery operation elements are given by R k =U y k P k ; (6.12) where P k = U k PU y k . Therefore P k = G k PU y k = p d k . This allows us to calculate the action of thekth recovery operator on thelth error [Nielsen and Chuang, 2000]: R k G l P = U y k P y k G l P =U y k (U k PG y k = p d k )G l P = kl p d k P: (6.13) 131 Therefore, R CP [ L ()] = N X i=1 2N X j;j 0 =1 u ij u N+i;j 0 X k kj p d k P P p d k kj 0 = N X i=1 udu y N+i;i = N X i=1 N+i;i = 2Tr y : (6.14) Next note that, using condition (6.8)(iii) and trace preservation by L : PE 0y i E i P = 2 y ii P =) 2Tr y P =P X i E 0y i E i P =P =) Tr y = 1 2 : (6.15) Hence, finally: R CP [ L ()] = (6.16) for any in the codespace. Note that ~ CP () need not be trace preserving: Tr[ ~ CP ()] = 1 2 Tr[( P N i=1 E y i E i + P N i=1 E 0y i E 0 i )], and while P N i=1 E 0y i E i = I if L is trace preserving, we do not have conditions on P N i=1 E y i E i and P N i=1 E 0y i E 0 i . We define the class of “CP-recoverable linear noise maps”f CPR g as those L for which CP recovery is always possible. By Theorem 11 this includes all L for which P can be found satisfying conditions (6.8)(i)-(iii). However, these conditions are not necessary. 132 6.3.2 Non-CP-recoverable linear noise maps We now define “non-CP-recoverable linear noise maps”f nCPR g as those L for which non-CP-recovery is always possible. Theorem 12 shows constructively thatf nCPR g includes all linear noise maps L for which P can be found satisfying only condi- tions (6.8)(i) and (ii). Clearly,f CP gf CPR gf nCPR gf L g. Theorem 12 Let L =fE i ;E 0 i g i be a linear noise map. Then every state = PP encoded using a QEC code defined by a projectorP satisfying only Eqs. (6.8)(i) and (ii) can be recovered using a non-CP recovery map. Proof Let G k = P i u ik E i and G 0 k = P i u 0 ik E 0 i , where the unitaries u and u 0 respec- tively diagonalize the Hermitian matrices and 0 : d =u y u andd 0 =u 0y 0 u 0 . Define a recovery mapR =fR k ;R 0 k g (not necessarily CP) with operation elements R k =U y k P k ; R 0 k =U 0y k P 0 k : (6.17) HereP k = U k PU y k ,P 0 k = U 0 k PU 0y k are projection operators, andU k andU 0 k arise from the polar decomposition ofG k P andG 0 k P , i.e.,G k P = U k (PG y k G k P ) 1=2 andG 0 k P = U k (PG 0y k G 0 k P ) 1=2 . The proof is entirely analogous to the proof of Theorem 11, except that we must keep track of both the primed and unprimed operators. Following through the same calculations we thus obtainR k G l p = p d k kl p andR 0 k G 0 l p = p d 0 k kl p . Using this in the recovery map applied to the linear noise map, we find: R[(PP )] = X kl R k E l PPE 0y l R 0y k = X kl R k ( X j u lj G j )PP ( X i u 0 li G 0y i )R 0y k = F L PP/; (6.18) 133 where F L X ijkl u lj u 0 li p d k d 0 k kj ki = X kl u lk u 0 lk p d k d 0 k = Tr[u 0 d 0y du y ] = Tr[u 0 u y 0y ] (6.19) is a “correction factor” for non-CP recovery of linear noise maps, which was 1 in the case of CP recovery, above. Gathering the expressions derived in the last proof, we have the following explicit expressions for the left and right recovery operations: R k =U y k P y k = 1 p d k P X i u ik E y i ; R 0 k = 1 p d 0 k P X i u 0 ik E 0y i : (6.20) This also shows that, in general,R k need not equalR 0 k , i.e., the recovery map is linear but not necessarily CP. Note that standard QEC can also be interpreted as “error correction by inversion”, in the following sense: when the noise map is CP and recovery is also CP, recovery is the inverse of the noise map restricted to the code space (Theorem III.3 in Ref. [Knill and Laflamme, 1997]). The same is true for our LQEC results above, which relax the restriction to CP noise maps. 6.3.3 The physical case: Hermitian maps The general physical case is the case of Hermitian noise maps, to which any quantum dynamical process can be reduced. We can specialize Theorems 11 and 12 to this case. Corollary 5 Consider a Hermitian noise map H () = P N i=1 c i K i K y i and associate to it a CP map ~ CP () = P N i=1 jc i jK i K y i . Then any QEC codeC and corresponding CP recovery mapR CP for ~ CP are also a QEC code and CP recovery map for H . 134 The important conclusion we can draw from Corollary 5 is that standard QEC tech- niques apply whether the noise map is CP or, as it will almost always be due to non- classical correlations, Hermitian. This is because Corollary 5 tells us that it is safe to replace all negativec i coefficients by their absolute values, and thus replace the actual noise map by its CP counterpart. Proof We have H () = P N i=1 E i E 0y i with fE i = p c i K i g N i=1 and fE 0 i = ( p c i ) K i g N i=1 , whence we can apply the construction of Theorem 11. Indeed, the “expanded” CP map becomes ~ CP () = 1 2 P N i=1 E i E y i + 1 2 P N i=1 E 0 i E 0y i = P N i=1 jc i jK i K y i , as claimed, and hence a QEC code and CP recovery for ~ CP is also a QEC code and CP recovery for H . In particular,R CP [ H ()] =. Note that ~ CP need not be trace preserving even in the Hermitian map case: Tr[ ~ CP ()] = Tr[ P N i=1 jc i jK y i K i ], but if H is trace preserving then we only have P N i=1 c i K y i K i = I, hence cannot conclude more about Tr[ ~ CP ()]. Also note that sub- stitution ofE i = p c i K i andE 0 i = ( p c i ) K i into the QEC conditions (6.8)(i)-(iii) yields 0 ij = q c i c j q c j c i ij and ij = ( p c j ) p c j ij , i.e., unlike in the general linear maps case, the matrices 0 and in Eq. (6.7) are not independent from. In fact, we can give a direct proof of Corollary 5 which only invokes a single block of the matrix: Direct Proof The operation elements of ~ CP are fF i = p jc i jK i g N i=1 , whence ~ CP () = P N i=1 F i F y i . The standard quantum error conditions (6.6) for ~ CP is a set of conditions in terms of theF i : PF y i F j P = ij P; i;j2f1;:::;Ng: (6.21) The existence of a projectorP which satisfies Eq. (6.21) is equivalent to the existence of a QEC code for ~ CP . Assuming that a codeC has been found (i.e.,PC =C) for ~ CP , 135 we use this as a code for H and show that the corresponding CP recovery mapR CP is also a recovery map for H . Indeed, letG j P N i=1 u ij F i be new operation elements for ~ CP , i.e., ~ CP = P N j=1 G j G y j , whereu is the unitary matrix that diagonalizes the Hermitian matrix = [ ij ], i.e., u y u = d. LetR CP =fR k g be the CP recovery map for ~ CP . Assume that is in the code space, i.e., PP = . We now show that R CP [ H ()] =, i.e., we have CP recovery. First, R CP [ H ()] = X k R k N X i=1 c i jc i j F i F y i ! R y k = N X i=1 c i jc i j N X j;j 0 =1 u ij u ij 0 X k (R k G j P ) PG y j 0 R y k : (6.22) Now, note that, using Eq. (6.21): PG y k G l P = X ij u ik u jl PF y i F j P = X ij u ik ij u jl P = d k kl P: (6.23) Then the polar decomposition yieldsG k P =U k (PG y k G k P ) 1=2 = p d k U k P . The recov- ery operation elements are given by R k =U y k P k ; P k =U k PU y k : (6.24) ThereforeP k =G k PU y k = p d k . This allows us to calculate the action of thekth recovery operator on thelth error: R k G l P = U y k P y k G l P =U y k (U k PG y k = p d k )G l P = kl p d k P: (6.25) 136 Therefore, R CP [ H ()] = N X i=1 c i jc i j N X j;j 0 =1 u ij u ij 0 X k kj p d k P P p d k kj 0 = N X i=1 c i jc i j udu y ii = ( N X i=1 c i jc i j ii ): (6.26) Next note that, using condition (6.21) and trace preservation by H : PF y i F i P = ii P =) N X i=1 c i jc i j ii P = P N X i=1 c i jc i j F y i F i P =P N X i=1 c i K y i K i P =P =) N X i=1 c i jc i j ii = 1: (6.27) Hence, finally: R CP [ L ()] = (6.28) for any in the codespace. Example of CP recovery: Inverse bit-flip map Consider “diagonalizable maps”, i.e., D () P i c i K i K y i , wherec i 2C. A straight- forward calculation shows that the expanded CP map is ~ CP = P i jc i jK i K y i . Now consider as a specific instance an independent-errors inverse bit-flip map on three qubits: IPF () = c 0 +c 1 P 3 n=1 X n X n , where X n is the Pauli x matrix applied 137 to qubit n, where c 0 and c 1 are real, have opposite sign, and c 0 + 3c 1 = 1 (a Her- mitian map). Then ~ CP = jc 0 j +jc 1 j P 3 n=1 X n X n , which is a non-trace pre- serving version of the well known independent-errors CP bit-flip map. The code is C = spanfj0 L ij000i;j1 L ij111ig, and P =j0 L ih0 L j +j1 L ih1 L j, which satis- fies Eq. (6.21) with F 1 = p jc 0 jI and F 2;3;4 = p jc 1 jX 1;2;3 . Then by Corollary 5 the same code (and corresponding CP recovery map) also corrects IPF . The CP recovery mapR CP has operation elementsR 0 =P andfR n = 1 p 3 PX n g 3 n=1 ; indeed, it is easily checked thatR CP [ IPF (PP )] =PP for any state2C. Hermitian recovery maps Since Hermitian maps are the most general physical maps, it is natural to consider Hermitian recovery of Hermitian noise maps. We thus define “Hermitian recovery maps”fR H g as those Hermitian maps that correct a Hermitian noise map H , i.e., R H H ()_. The following theorem presents a possible set of Hermitian recovery maps. Corollary 6 Consider a Hermitian noise map H () = P N i=1 c i K i K y i with error oper- atorsfK i g satisfying the relationsPK y i K j P = ij P . Any Hermitian mapR H of the form P k h k R k :R y k with recovery operatorsfR k g as in Eq. (6.12) andfh k g2R corrects the noise map H . 138 Proof LetfF i = p jc i jK i g N i=1 ; we simply use the identities given in the proof of the previous theorem – specifically Eq. (6.26) – to calculateR H H () R H [ H ()] = X k h k R k N X i=1 c i jc i j F i F y i ! R y k = N X i=1 h k c i jc i j N X j;j 0 =1 u ij u ij 0 X k kj p d k P P p d k kj 0 = PP N X i=1 c i jc i j X k h k u ik u ik d k = F H PP/; (6.29) where F H N X i=1 c i jc i j udhu y ii ; (6.30) whereh diag(fh k g), andF H is a “correction factor” for Hermitian recovery of Her- mitian noise maps, which was 1 in the case of CP recovery, above. How does non-CP, Hermitian recovery arise? In standard QEC theory the recovery map is considered CP. The reason for this is that the recovery ancillas are introduced after the action of the noise channel so that they enter in a tensor product state with the encoded qubits that underwent the noise channel. The recovery map is obtained in the standard setting by first applying a unitary over the encoded qubits plus recovery ancillas, then tracing out the recovery ancillas. This is manifestly a CP map over the encoded qubits. Since we know that the recovery map experienced by the encoded qubits is CP if and only if the initial state of the encoded and recovery ancilla qubits has vanishing quantum discord [Shabani and Lidar, 2009b], it is clear how a non-CP recovery map 139 can be implemented: the recovery ancillas should have non-vanishing quantum discord with the encoded qubits. Since this will still be a QDP, the resulting recovery map will be Hermitian according. Such a situation can come about in various ways. For example, a scenario which is particularly relevant for quantum computation and communication, is one where the environment causes the recovery ancillas to become non-classically correlated with the encoded qubits before the recovery operation can be applied. This is a reasonable sce- nario since, while the recovery ancillas are presumably kept pure and isolated from the environment for as long as possible, at some point they must be brought into contact with the encoded qubits, and at this point all qubits (encoded and recovery ancillas) are susceptible to correlations mediated by the environment. 6.4 Conclusions This work aimed to fill two gaps: one in the theory of open quantum systems, and a resulting gap in the theory of quantum error correction. The first gap had to do with the type of maps that describe open systems given arbitrary initial states of the total system. In fact, it was not a priori clear that there should even be a linear map connecting the initial to the final open system state for arbitrary initial total system states. Building upon the class of “special linear states” we introduced in [Shabani and Lidar, 2009b] we showed here that in fact such a linear map description does always exist, and moreover, for quantum dynamics the map is always Hermitian. The map reduces to the com- pletely positive type if and only if the initial total system state has vanishing quantum discord [Shabani and Lidar, 2009b]; in all other cases it is Hermitian but not CP. This result, we argued, impacts the theory of quantum error correction, where previously the assumption of CP maps was taken for granted. In the second part of this work we filled 140 this gap in QEC theory, by developing a theory of Linear Quantum Error Correction (LQEC), which generalizes the CP-map-based standard theory of QEC. We showed that to every linear map L is associated a CP map which, if correctable, also provides an encoding with corresponding CP recovery map for L (Theorem 11). Moreover, it is possible to find a non-CP recovery for L within a larger class of codes (Theorem 12). From a physical standpoint this result is actually too general, since only Hermitian maps ever arise from quantum dynamics [to the extent that the standard quantum dynamical process (2.8) is valid]. Hence we specialized LQEC to the Hermitian maps case, and showed that in this case standard QEC theory for CP maps already suffices, in the sense that it is legitimate to replace a given Hermitian noise map by a corresponding CP map obtained simply by taking the absolute values of all the Hermitian map coefficients. Any QEC code which corrects this CP map will also correct the original Hermitian map (Corollary 5). Nevertheless, there is room for a genuine generalization when one consid- ers Hermitian maps, since it is also possible to perform QEC using Hermitian recovery maps (Corollary 6). We argued that, in fact, recovery maps will generically be non-CP Hermitian maps, since recovery ancillas that are introduced into a quantum circuit prior to the recovery step will become non-classically correlated with the environment and consequently with the rest of the system. An interesting open question for future studies is whether the results presented here have an impact on the threshold for fault tolerant quantum error correction. For exam- ple, note that while CP recovery perfectly returns the encoded state [Eqs. (6.16) and (6.28)], non-CP recovery only does so up to a proportionality factor which depends on the details of the noise and recovery maps [F L in Eq. (6.19) andF H in Eq. (6.30)]. This proportionality factor – assuming non-CP recovery is applied – may differ for different terms in the fault path decomposition [Aharonov and Ben-Or, a], an effect which may propagate into the value of the fault tolerance threshold. 141 Chapter 7 QEC in the Presence of Correlated Errors Introduction – The theory of fault-tolerant quantum computation (FTQC) has been developed as a realistic extension of the ideal theory of quantum computation. In an ideal prototype, all computational components are functioning with no deficiency, while in the real world deviations from the ideal design is inevitable. This can be due to imperfections in the components or interactions imposed by the environment. In FTQC, schemes of error correction are combined with the computational elements in a spe- cial design to prevent the spread of errors through a circuit. Applying error correction steps in a multi-layer circuit is a well established procedure for FTQC [Knill et al., 1998, Aharonov and Ben-Or, b, Kitaev, 2003, Alicki et al., 2006, Aliferis et al., 2006, B.M. Terhal, 2005, Aharonov et al., 2006]. Inspired by classical coding theory, a fault-tolerant form of quantum error correc- tion codes (QECC) can be achieved by concatenating single units of encoding. The performance of fault-tolerant codes has been studied for various noise models. The classical model of a Markovian independent noise has been thoroughly investigated in the literature [Knill et al., 1998, Aharonov and Ben-Or, b, Kitaev, 2003, Alicki et al., 2006]. In this picture, each individual component of the circuit is only coupled to a single designated bath, with no cross-talk between the baths. In addition, each bath is assumed to be large enough to constitute a local Markovian decoherence channel. Other possible defects coming from imperfections in computational gates are also assumed to 142 have local and instantaneous effects. All these give rise to a probabilistic description of the noise process. Upper bounding the error probability guarantees an arbitrary small likelihood of computational failure. However an analysis of quantum computer proposals reveals the inaccuracy of the above noise model. For example, ion qubits in an ion trap set up are collectively cou- pled to their vibrational modes [Garg, 1996]. In a quantum dot design, different qubits are coupled to the same lattice thus interacting with a common phonon bath [Loss and DiVincenzo, 1998]. The exchange interaction is the main candidate for implementing two-qubit gates in solid state proposals. Recently, it has been shown that many-body exchange interactions can be strong enough to act as a major source of noise [Mizel, 2004]. Equally important, collective control of the qubits may also give rise to errors due to the inaccuracy of the control field [Wu et al., 2004, Kay and Pachos, 2004]. These examples invalidate the assumption of error independence (locality in space). In addi- tion, the non-Markovian nature of noise has been observed in various systems [Breuer and Petruccione, 2002]. Therefore, the assumptions of exact locality in time and space should be relaxed to attain a more realistic model [Gottesman, 2007]. First attempts to introduce correlations into noise models were in a classical manner: multi-location joint probabilities that are stronger than the independent model [Aharonov and Ben-Or, b, Knill et al., 1998]. Recently a physical approach to the problem was taken by introduc- ing a Hamiltonian description of the noise process[Aliferis et al., 2006, Aharonov et al., 2006]. However because of the discrete nature of error correction, the noise Hamiltonian was perturbatively treated to achieve a faulty paths description of the noisy computation. In this paper, we study the performance of quantum codes in a single block of error correction towards a comprehensive analysis of the FTQC problem with a real- istic model. Our aim is to compare spatial correlations in noise without any limiting assumptions on time. Focusing on well-defined CSS codes [Calderbank and Shor, 1996, 143 Steane, 1996], we first formulate a measure for the code performance and then apply it to different decoherence scenarios. Our main finding is that a coding system can function better in the presence of error correlations. This is mainly due to the entangling power of a common bath which can help in the preservation of the code state coherence designated for recovery [Benatti et al., 2003, Benatti and Floreanini, 2006]. This is in contrast to the results from previous studies [Klesse and Frank, 2005, Novais and Baranger, 2006, Duan and Guo, 1999, Clemens et al., 2004]. Correlated Errors – First we define the notion of correlation in errors. From the physical point of view, the primary probabilistic description of the noise originates from two main assumptions on the system-bath interaction: each qubit is coupled to its own bath, and the resulting decoherence is described by a completely positive (CP) map. A CP map modeling of the noise process allows a probabilistic interpretation of the errors which may be otherwise impossible [Shabani and Lidar, 2005a]. Based on this classical randomness picture, correlations could be defined by some joint probability distribution for the multi-qubit error operators. However, a microscopic description is lacking in such a model, motivating a more physical formulation. We remark that the source of error is either a second system with a large Hilbert space or an imperfect driving field of a quantum gate. In general there are three possible scenarios as depicted in Fig. (7.1): 1) errors with no correlation which corresponds to model of separated baths, each coupled only to a single qubit, 2) short-range correlations in which all the qubits are only coupled to a common bath, and 3) The most general case with long-range correlations where qubits are interacting with each other while sharing a common environment. In the following, we locate the qubits of a code in the above scenarios and compare their performance as a function of decoherence time d . This is when we enter the 144 (b) (c) (a) Environment Qubit Coupling Figure 7.1: A sketch of (a) an uncorrelated error model, (b) a short-range correlation, (c) a long-range correlation. recovery phase. The choice of CSS code is made to obtain an analytical solution to the problem. A [n;k;d] CSS code is annqubit code, encodingk bits of information, capable of correcting errors acting on at mostt = [ d1 2 ] different qubits. Consider ann- qubit quantum code in a single step of error correction. Then qubits are sent through a noise channelN expressed as a completely positive (CP) map:N ( S ) = P E S E y : We expand this map in the basisf 1 ;:::;n = 1 1 ::: n n g; N ( S ) = X fpg=p 1 ;:::;pn; fqg=q 1 ;:::;qn e fpg;fqg fpg S fqg (7.1) 145 wheref 0 =I; 1 = x ; 2 = y ; 3 = z g: Following the matrix product structure of the basis, we introduce anwth level sub-mapN w ( S ) to be the sum of the terms in the above expansion (7.1) of which the array ( 1 + 1 ;:::;w +w ) hasw non-zero elements. The matrix representation of the superoperatorN w = [e w fpg;fqg ] has the following elements: e w fpg;fqg = 8 < : e fpg;fqg (p i +q i ) hasw non-zero elements, 0 otherwise. (7.2) The noise mapN ( S ) can be decomposed as a sum of sub-mapsN w N ( S ) = X w N w ( S ) = X w;fpg;fqg e w fpg;fqg fpg S fqg (7.3) We can separate the correctable (by means of a [n;k;d] CSS code) and uncorrectable parts of the map by splitting the noise map into submapsN c ( S ) = P w6t N w ( S ) and N ic ( S ) = P t<w N w ( S ): N ( S ) =N c ( S ) +N ic ( S ) (7.4) The correctibility of the linear mapN c follows from the fact that a Hermitian linear map can be corrected by quantum codes [Shabani and Lidar, 2007]. By a Hermitian map we mean a linear map :M u !M v (whereM m is the space ofmm matrices) with a symmetric representation () = P c F F y , where the “map operators”fF g arevu matrices and thec s are real numbers. Theorem 1 Consider a CP noise map CP () = P F F y and a code space P C . A recovery CP mapR correcting the map CP () can also correct a Hermitian map () = P c F F y : 146 Proof This is a straightforward application of theorem 2 in reference [Shabani and Lidar, 2007]. It is obvious that the correctable sum noise mapN c is a Hermitian linear map by definition, therefore one can find a recovery mapR() invertingN c that belongs to the correctibility domain of a [n;k;d] code; i.e. there exists a real numberp N such that RN c =p N I (7.5) whereI is the identity superoperator. Performance Measure – We aim to study the performance of a quantum code for different decoherence processes; therefore we need a proper measure to quantify it. As we discussed, the noise mapN is partially correctable by the recovery mapR. We introducep N in Eq. (7.5) to be the ”performance measure” of a code. Before a rigorous derivation of this measure, let us apply it to a model of independent single qubit noise channels N indp ( S ) =fE 1 ::: E n g( S ) (7.6) 147 whereE i () = (1p)+p(), withp the probability of the error channel. SupposeR is the recovery channel correcting errors in up tot different locations in the code, then we have RN indp ( S ) =RfN indp c ( S ) +N indp ic ( S )g = t X c=0 (1p) (1c) p c R X r 1 +:::+rc =nc fI r 1 ::: I rc g( S ) + n X c=t+1 (1p) (1c) p c R X r 1 +:::+rc =nc fI r 1 ::: I rc g( S ) = t X c=0 n c (1p) (1c) p c ! S + Error. By definition of the performance measure,p N indp is equal to P t c=0 n c (1p) (1c) p c : This value can be interpreted in the language of probability as the likelihood of having fewer thant individual errors [Nielsen and Chuang, 2000]. In the above definition we discardedRN ic as the error. Actually this term vanishes in an averaging procedure introduced in Ref. [Klesse and Frank, 2005]. Consider a code word S = j ih j experiencing the noiseN ( S ) and being corrected by a CSS code recovery map R() = X 2f0;1g n ;2f0;3g n jj;jjt P fg fg fg fg P (7.7) Our method is similar to the one introduced in a pertinent recent chapter [Klesse and Frank, 2005] for CSS codes, but we generalize it to an arbitrary noise model. We choose fidelity [Nielsen and Chuang, 2000] as a proper measure of code performance: F( S ;R(N ( S ))) =h jR(N ( S ))j i (7.8) 148 An analytical criterion is achieved by averaging fidelity over all the code wordsj i. A CSS quantum codejCi is spanned by code wordsjQi whereQ is a coset ofC ? 2 andC ? 1 , (C ? i is the orthogonal space toC i ;C 2 C 1 Z n 2 ). Applying the average over all pairs C 2 C 1 suppresses all the terms with e n>d fpg;fpg or e n fpg6=fqg which are of order 2 O(n) : We have verified this fact for a general noise model, justifying the bi-partitioning of the noise map in Eq. (7.4). As a consequence, the fidelity average converges to a simple form of p N =F ave (jCihCj;R(N (jCihCj))) = X n6t;fpg e n fpg;fpg (7.9) Such a code-independent measure equips us with a quantitative tool to compare different noise models. A closed form for p N can be derived for dynamics with an initial product state SB (0) = S (0) B (0) and propagatorU d as function of d , (the decoherence time before recovery operation): p N ( d ) = 1 2 2n X jjt Tr B [Tr S (U y d fg )Tr S ( fg U d ) B (0)] (7.10) In the following we apply this measure on different system-bath interaction config- urations. Local vs. Non-local Environments – Here we compare system-bath configurations (a) and (b) in Fig. (7.1): Each qubit is coupled to its own bath versus all qubits coupled to the same bath. We consider a spin star model which consists of a central spin- 1 2 particle representing a qubit, surrounded byN localized spin- 1 2 particles acting as a spin bath for the qubit [Breuer, 2004a, Prokof’ev and Stamp, 2000]. In the local model the central spin is coupled to the bath spins i ; (i = 1;:::;N) via a dephasing interaction of the form H hf = N X i=1 g i z Z i (7.11) 149 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 !0.4 !0.2 0 0.2 0.4 Fidelity Difference N=7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 !0.4 !0.2 0 0.2 0.4 N=196 Fidelity Difference g! d Different Size Baths Same Size Baths Figure 7.2: Performance difference of a [7,1,3] CSS code for two scenarios of local and non-local environments as a function of decoherence time d . A negative sign p local N (g d ) p nonlocal N (g d ) demonstrates better performance for the non-local error.[N =f7; 196g, = 0:01] where z and Z i are the z components of and i respectively. At low temperatures and in the presence of a weak magnetic field, this hyperfine contact type of interac- tion is the dominant source of electron spin decoherence in quantum dots due to the interaction with nuclei [Khaetskii et al., 2002, Saykin et al., 2002]. Two different configurations of local and non-local decoherence for ann qubit code are modeled as H local = P n m=1 P N i=1 g im m z Z m i andH nonlocal = P n m=1 P N i=1 g im m z Z i . As an exam- ple we consider a [7; 1; 3] code surrounded byN =f7; 196g bath spins [Breuer, 2004a, Prokof’ev and Stamp, 2000]. The bath state, B in Eq.(7.10), is a thermal state of the spin bath at temperatureT = 1mK with an internal HamiltonianH B = P N i=1 Z i : In addition, for simplicity we assume a symmetric couplingg im =g. The difference of the performance of local and non-local modelsp local N (g d )p nonlocal N (g d ) is plotted in Fig. (7.2). We observe that the performance strongly depends on the decoherence time d . 150 Consequently, choosing the correct time for recovery can improve the functioning of the code in the presence of correlations. This is in contrast to a classical system in which a common source of error acts simultaneously on different parts, and therefore leads to a higher risk of computational failure. While in the quantum case, the indirect coupling of the qubits through a shared common bath or direct coupling by means of many-body interactions imposes additional dynamics on the code coherence that changes the per- formance of the code. Furthermore, these results reveal that engineering the qubits of a code block in the same location (non-local bath) may have an advantage over spatially separating them (local bath), as suggested in [Aharonov et al., 2006]. Another interesting comparison can be made between local and non-local baths of the same size, with the corresponding Hamiltonians H local = P n m=1 P N=n i=1 g im m z Z m i andH nonlocal = P n m=1 P N i=1 g im m z Z i . The simulation results are plotted in Fig. (7.2) which demonstrates the time dependent, alternative behavior of the code performance difference. Two-Body vs. Many-Body Interactions – We now introduce many-body interac- tions as a counterpart to multivariate joint probability in classical descriptions of correla- tions. In a recent study [Aharonov et al., 2006], the fault-tolerant quantum computation in the presence of many body interactions has been investigated. Following their formu- lation, we include additional termsH ij which act simultaneously on qubit pairs<ij > and on the bath. As an example consider the noise Hamiltonian H 3body =g n X j=1 N X i=1 j z Z i +g 0 n X j;k=1 N X i=1 j z k z Z i (7.12) in whichg andg 0 determines the strength of 2-body and 3-body interactions. The ratiog 0 =g, in some specific configurations of spin- 1 2 particles, can rise up to %16 [Mizel, 2004]. We compare two cases of 2-body only interactions, with g 0 = 0, and 3-body 151 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 !0.2 0 0.2 0.4 0.6 0.8 1 1.2 g! d Fidelity 2!body 3!body p 3!body !p 2!body Figure 7.3: A [7,1,3] code performance in the presence of 2-body and 3-body interac- tions. The performance differencep 2body N ( d )p 3body N ( d ) determines how the effect of 3-body interaction varies by decoherence time d . (N = 7) interactions withg 0 = 0:1g. As shown in Fig. 7.3, for short recovery times they behave similarly, while at longer times an alternative performance of two cases emerges. Faulty Gate – A faulty gate is another main source of errors in computation. In many proposals for quantum computers, a magnetic field or laser pulse is an ingredient of the control field which can act locally [Nielsen and Chuang, 2000] or globally (non- local) [Wu et al., 2004, Kay and Pachos, 2004]. Obviously any inaccuracy in manipulat- ing the control field causes inaccuracy in the computation. Here we compare fault toler- ability of ann qubit code driven by an imperfect magnetic field acting locally (H ml = g P n i=1 B i Z i ) or globally (H mg =gB 0 P n i=1 Z i ). Because of the classical nature of the error, we assume a random process for the control error, i.e. B i = B ideal +W i , where the random variableW i has a probability distributionp W i (w) =p(w). For a time r the total rotation generated by this field isU ml (w; r ) = exp[i r g P n i=1 B i (w)Z i ] for local 152 control, andU mg (w; r ) = exp[i r gB 0 (w) P n i=1 Z i ] for global control. The distance of two unitary operationsU andV , acting on a Hilbert spaceH, can be measured by the average fidelity: Ave: j i2H h jV y Uj i = 1 d 2 jTr(V y U)j 2 , withd = dim(H). To quanti- tatively compare a noisy gate with the ideal one, we use this distance averaged over the random variableW i . For the local control this value becomes Flocal ( r g) = ( Z 1 1 cos( r g!)p(!)d!) n (7.13) While for the global one it is Fglobal ( r g) = Z 1 1 (cos( r g!)) n p(!)d! (7.14) Applying the Hlder inequality [Hardy et al., 1934], we find that Flocal ( r g) Fglobal ( r g) (7.15) This indicates the higher fidelity of the global control in compare to the local one. Conclusion – In summary, we have studied quantum error correction in the presence of correlated errors. A microscopic description of the noise process enables us to classify correlations based on their physical relevance in quantum computer designs. Roughly speaking, we find that it is not necessarily true that correlations in errors imply additional flaws in computation, but on the contrary these correlations may positively affect the performance. Furthermore, the results presented in this paper emphasize that, besides choosing a proper code, the pre-recovery decoherence time is a main factor to achieve higher fidelity, since the code performance can drastically vary with the timing of the recovery operation. 153 Chapter 8 Conclusion In this thesis a theoretical study of the noise process in quantum computers and the methods to reduce its disturbing effect were presented. At the time we started research in the field of quantum information, almost after one decade of the first publications proposing the idea of quantum computers, a serious gap was evident between the the- oretical progress in this field and the experimental achievements. We believed that the main reason behind this gap was the imprecise modeling of the decoherence process, the main obstacle for implementing quantum computers, in theoretical studies which results in inadequate experimental results. To alleviate this problem we revisited the fundamental concepts and formulations resulting from the 70 years old theory of open quantum systems [Weisskopf and Wigner, 1930]. The early formulation of quantum coding theory was based on the complete positivity assumption of noise dynamical map that was strongly challenged in 90s [Pechukas, 1994] and then in a recent series of papers [Stelmachovic and Buzek, 2001, Jordan et al., 2004, ˙ Zyczkowski and Bengts- son, 2004, Carteret et al., 2008]. In this thesis we have developed a general framework for non-completely positivity maps. In this framework we introduced the correspond- ing error coding theory which can pave the way toward a more realistic formulation of fault-tolerant quantum computation. 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Abstract (if available)
Abstract
Quantum effects can be harnessed to manipulate information in a desired way. Quantum systems which are designed for this purpose are suffering from harming interaction with their surrounding environment or inaccuracy in control forces. Engineering different methods to combat errors in quantum devices are highly demanding. In this thesis, I focus on realistic formulations of quantum error correction methods. A realistic formulation is the one that incorporates experimental challenges. This thesis is presented in two sections of open quantum system and quantum error correction. Chapters 2 and 3 cover the material on open quantum system theory. It is essential to first study a noise process then to contemplate methods to cancel its effect. In the second chapter, I present the non-completely positive formulation of quantum maps. Most of these results are published in [Shabani and Lidar, 2009b,a], except a subsection on geometric characterization of positivity domain of a quantum map. The real-time formulation of the dynamics is the topic of the third chapter. After introducing the concept of Markovian regime, A new post-Markovian quantum master equation is derived, published in [Shabani and Lidar, 2005a].
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Creator
Shabani Barzegar, Alireza
(author)
Core Title
Open quantum systems and error correction
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
04/07/2009
Defense Date
02/26/2009
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
decoherence,OAI-PMH Harvest,open quantum system,quantum computation,quantum error correction
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Lidar, Daniel A. (
committee chair
), Brun, Todd A. (
committee member
), Jonckheere, Edmond A. (
committee member
), Zanardi, Paolo (
committee member
)
Creator Email
arshabani@yahoo.com,shabanib@usc.edu
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https://doi.org/10.25549/usctheses-m2060
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UC1289400
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etd-Barzegar-2701.pdf
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208072
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Dissertation
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Shabani Barzegar, Alireza
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Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
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Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
decoherence
open quantum system
quantum computation
quantum error correction