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Quantum computation and optimized error correction

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Quantum computation and optimized error correction

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Content QUANTUM COMPUTATION AND OPTIMIZED ERROR CORRECTION by Soraya Taghavi A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2010 Copyright 2010 Soraya Taghavi Acknowledgments I would like to express my most profound gratitude to my advisor Professor Daniel Lidar for providing a perfect research environment for me, and for his patience and guidance during my Ph.D. study. Working with Professor Lidar and learning from him was a very valuable experience for me. I would also like to give my special thanks to Professors Todd Brun and Stephan Haas for supervising my Ph.D. dissertation. It is impossible to thank all my teachers by mentioning their names here. All of my achievements during my studies are directly influenced by them, and I am grateful to all of them. I would like to especially extend my appreciation to Professors Jin Ma and Jianfeng Zhang of the Department of Mathematics for their support during my study at USC. I would also like to take the opportunity to thank Professors Edmond Jonckheere, Urbashi Mitra , Paolo Zanardi, and Robert Scholtz for all that I learned from them. I would also like to deeply thank my friends and colleagues Wan Jung Kuo, Kristen Pudenz, Alireza Shabani, Masoud Mohseni, Kaveh Khodjasteh and especially Ali Reza- khani for their friendship and thoughtful comments and suggestions. I should also thank Diane Demetras and Amy Yung for their help with all my administrative problems. Last, but far from least, my deepest thanks goes to my family for their support and encouragement during my course of study. I would like to especially thank my sister ii Ailar for her love and endless support. None of my achievements would be possible without them. iii Table of Contents Acknowledgments ii Abstract vi Chapter 1: Universal Quantum Robots 1 1.1 Introduction . . . . .... ... .... ... .... .... ... ... 1 1.1.1 Example . .... ... .... ... .... .... ... ... 2 1.1.2 Benioff’s Model . . . . .... ... .... .... ... ... 4 1.2 Preliminary Concepts... ... .... ... .... .... ... ... 6 1.2.1 Lie Algebra .... ... .... ... .... .... ... ... 6 1.2.2 Quantum Simulations . .... ... .... .... ... ... 9 1.2.3 Majorization: . . . . . . .... ... .... .... ... ... 11 1.3 Universality . . . . .... ... .... ... .... .... ... ... 14 1.3.1 step one (single-body Hamiltonians) . .... .... ... ... 15 1.3.2 Step Two (All Hamiltonians) . . . . . .... .... ... ... 17 1.4 Discussion . . . . . .... ... .... ... .... .... ... ... 19 Chapter 2: Optimized Quantum Error Correction 20 2.1 Introduction . . . . .... ... .... ... .... .... ... ... 20 2.2 Quantum Error Correction . . . .... ... .... .... ... ... 23 2.2.1 Open quantum systems and Standard error correction model . . 23 2.2.2 Optimization problems . .... ... .... .... ... ... 27 2.3 Distance Minimization . . . . . .... ... .... .... ... ... 30 2.3.1 Optimal solutions . . . . .... ... .... .... ... ... 30 2.3.2 Alternative algorithm for recovery optimization . . . . . . . . . 34 2.3.3 Robust error correction . .... ... .... .... ... ... 36 2.4 Optimized distribution of ancillas . . . . . . .... .... ... ... 37 2.4.1 Rank minimization of Γ .... ... .... .... ... ... 38 2.4.2 Numerical result for randomly generated error maps . . . . . . 39 2.5 Examples . . . . . .... ... .... ... .... .... ... ... 41 2.5.1 3-qubit bit-flip errors . . .... ... .... .... ... ... 41 2.5.2 Bit-Phase flip error . . . .... ... .... .... ... ... 45 iv 2.6 Conclusion . . . . .... ... .... ... .... .... ... ... 47 Chapter 3: Optimized Entanglement Assisted Error Correction 49 3.1 Introduction . . . . .... ... .... ... .... .... ... ... 49 3.2 Problem Formulation . . . . . . .... ... .... .... ... ... 50 3.2.1 Optimized Encoding Operator . . . . .... .... ... ... 53 3.2.2 Optimized Recovery operator . . . . .... .... ... ... 55 3.3 Random Unitary Channels . . . .... ... .... .... ... ... 57 3.3.1 Example 1 .... ... .... ... .... .... ... ... 59 3.3.2 Example 2 .... ... .... ... .... .... ... ... 60 3.3.3 Depolarizing Channel . .... ... .... .... ... ... 62 3.4 Conclusion . . . . .... ... .... ... .... .... ... ... 63 Chapter 4: Error Protection for Linear Maps 65 4.1 Introduction . . . . .... ... .... ... .... .... ... ... 65 4.2 Linear maps . . . . .... ... .... ... .... .... ... ... 66 4.3 Noiseless Subsystems . . . . . . .... ... .... .... ... ... 67 4.4 Proofs of the theorems . . . . . .... ... .... .... ... ... 70 4.4.1 Sufficiency .... ... .... ... .... .... ... ... 70 4.4.2 Necessity . .... ... .... ... .... .... ... ... 72 4.5 Conclusion . . . . .... ... .... ... .... .... ... ... 75 Chapter 5: Conclusion 76 References 78 Appendices 80 Appendix A .... ... .... ... .... ... .... .... ... ... 81 Appendix B .... ... .... ... .... ... .... .... ... ... 83 v Abstract Two subjects in the area of quantum computation are considered here. In the first chapter I present a universal model for a quantum Robot. Chapters two, three, and four are dedicated to the problem of quantum error correction/protection. A quantum robot is described as a quantum system that moves in, and interacts with, an external environment of quantum systems. Such environments consist of arbitrary numbers and types of particles in two or three dimensional space lattices. I find a set of universal operations that enables the quantum robot to simulate arbitrary quantum dynamics. A computational approach to the quantum error correction problem is presented in chapters two and three. I develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. This theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. I demonstrate via numerical examples that such optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. In the setting of a known noise channel the recovery ancillas are redundant for optimized quantum error correction. I show this using a general rank minimization heuristic and supporting numerical calculations. vi Therefore, one can further improve the fidelity by utilizing all the available ancillas in the encoding block. However, this conclusion breaks down in the presence of an initial entanglement between the encoding and recovery ancillas. Such entanglement assisted error correc- tion procedures are studied in chapter three. I show how entanglement can increase fidelity in the optimized setting by improving the function of the recovery ancillas. In the last chapter quantum error protection methods, decoherence-free subspaces and subsystems, are studied in the framework of linear maps. This framework provides the most general description of open quantum system dynamics. vii Chapter 1 Universal Quantum Robots 1.1 Introduction We all are interested in quantum computers, because we think they solve certain prob- lems faster than classical ones. However, there has been no comparable development of a quantum mechanical description of robots, which can also be considered among the most important achievements in science and technology. Quantum robots can, at least in principle, carry out experiments on the atomic level. Paul Benioff originated the idea of quantum robots [Benioff, 1998, 1997]. Benioff’s model assumes that quantum robots have on board quantum computers. In fact, he con- siders quantum computers as part of a larger system and emphasizes their interactions with the environment. Therefore, in Benioff’s model, interactions between quantum computers and external systems are essential to the overall system dynamics, despite the fact that in quantum computers these interaction should be avoided. Here a quantum robot is considered as a quantum system that moves in, and interacts with, an external environment of quantum systems. Such environments consist of arbi- trary numbers and types of particles in a two or three dimensional space lattice. While state preparation and measurement are vital elements in the simulation of a quantum system, our focus is on the simulation of this system’s evolution. The simulation goal is 1 to approximate to arbitrary accuracy evolution according to some other fixed Hamilto- nians. If this is possible for some desired Hamiltonian, we say that the Hamiltonian can be simulated. The idea in this chapter is to introduce a set of basic operators for the robot and use a small number of composition laws to build up a library of Hamiltonian evolutions we can simulate. We also prove that this set of basic operators is universal. 1.1.1 Example Consider a system of k particles in an n×n 2-D lattice. The Robot’s task is to change the particles’ position in the lattice as desired. Assume particles do not interact with each other, and therefore any change in the system’s state is due to their nearest neighbor interaction with the robot. The Hilbert space associated with this system is H =H R ⊗H P 1 ⊗ ...⊗H P k , and the Robot’s task is described as: |X iP 1 ,Y iP 1 |X iP 2 ,Y iP 2 ...|X iP k ,Y iP k |ϕ iR →|X fP 1 ,Y fP 1 |X fP 2 ,Y fP 2 ...|X fP k ,Y fP k |ϕ fR Here|X iP l ,Y iP l = s,r c l,s,r |s, r is the initial state of the lth particle, where|s, r is a site in the lattice and X iP l is the initial position in the X direction. |X fP l ,Y fP l = s,r c l,s,r |s, r is the final state of the lth particle. |ϕ iR and|ϕ fR are, respectively, the initial and final states of the robot. In this specific task, the system’s initial and final states are assumed unentangled, although they might become entangled during 2 this process. This is based on the simplifying assumption that the state of the particles includes only their positions. I also assume time and space are discrete. To characterize the robot’s task, I define some basic operators that can be performed at each time step. When the robot is in the site|s, r, it can apply the following basic Hamiltonians on each particle: H Rx = |s, rs+1,r| +|s+1,rs, r| (1.1) H Mx = i|s, rs+1,r|− i|s+1,rs, r| H Ry = |s, rs, r+1| +|s, r+1s, r| H My = i|s, rs, r+1|− i|s, r+1s, r| H Rx and H Mx are, in fact, the Pauli matrices σ x and σ y on the site|s, r and|s+1,r [Dodd et al., 2002], but I use indices R and M, because X and Y are already reserved for the directions. While the robot interacts with sites (s, r) and (s+1,r), it can perform the following Hamiltonian on a particle by using the basic Hamiltonians: h = klmn c klmn H k Rx H l Mx ⊗ H m Rx H n Mx , (1.2) where the indices k,l,m and n are either zero or one, so that either σ x or σ y or both are applied at each site, as controlled by the coefficient c klmn . For simplicity of notation the Hamiltonian h does not include the site index (s, r). In order to simulate any Hamilto- nian of the form J⊗J , we need to have in addition at least one non-zero coupling term, a term not of the form I⊗h or h ⊗I. This means that the robot should also perform an entangling Hamiltonian h⊗ h , at least once, on each particle. In this case, horizontal interaction, h and h could be either H Mx or H Rx . The same argument holds in the case of vertical interaction with H My and H Ry [Mackay et al., 2002]. 3 1.1.2 Benioff’s Model In Benioff’s model, Quantum robots are considered to be mobile systems that have a quantum computer on board. Other systems such as a memory system m, an output system o , and a control system c are also present in the quantum robots. The on board quantum computer can be represented either as a network of quantum gates or as a quantum Turing machine [Benioff, 1997]. The dynamics of the quantum robot (QR) and its interaction with the environment is described in terms of tasks. These tasks consist of alternating computation and action phases. The purpose of each computation phase is to determine what is to be done in the next action phase. The computation depends on the state of m, o, and the local environment as input. During the computation phase the robot does not move, and the goal is to put o in one of the basis states each of which specifies an action. In an action phase, the robot carries out the action determined by the previous com- putation phase. The action is determined by the state of o , but may include local mea- surements of the environment state. This action is independent of the state of the on board quantum computer and the m. The state of o does not change during the action. The control qubit regulates which type of phase is active. The computation [action] phase is active if c is in the state|1,[|0]. Thus the last step in a computation [action] phase is the change|1→|0,[|0→|1]. Time is assumed discrete and a unitary step operator T is associated with each task. T describes the task dynamics for one time step, and each task can be described by separating T into operators T a and T c for action and computation phases, respectively, so that action and computation takes place in parallel at each time step: T = T a + T c . 4 Consider B = B qc ⊗ B anc ⊗ B ext as a reference basis for the quantum robot and environment. Here B ={|b qc } is the reference basis for the quantum computer. For the ancillary system B anc ={|l 1 o |i c } where{|l 1 o } is a finite basis for the output system and{|i c } with i=0 and 1 is the basis for the control qubit. Regarding the requirement that T c does not change the environment state or the quantum robot’s location this restriction holds: T c = x,E P e x,E T c P e x,E P c 0 (1.3) where P e x,E =|x.Ex,E| is the projection operator for the QR at site x and the envi- ronment in the state|E, and P c 0 is the projection operator for control in the state|0. This condition shows that T c is inactive when c is in the state|1. T a depends on the output but does not change the state of the output. Then the requirements on T a is expressed as: T a = x ,x l 1 P qr x P o l 1 T a P o l 1 P qr x P c 1 (1.4) where P c 1 is the projection operator for output in the state |l 1 , P qr x is the projection operator for the quantum robot at lattice location x [Benioff, 1998, 1997]. Benioff finishes the argument here, and does not develop his model further. There- fore, the procedure of performing these operators and their exact relations with tasks are not clear in his model. Our model for quantum robots is, however, based on the Hamiltonians that they can perform on lattices, and the goal is to show that such robots are universal. In the next section we provide a summary of some preliminary concepts that are used in our universality proof. 5 1.2 Preliminary Concepts 1.2.1 Lie Algebra The approach here is largely along the traditional Cartan-Weyl roots. Let’s start by briefly introducing this concept. Cartan-Weyl basis A. Root vectors: Consider X υ (υ =1, ..., n) as the generators of an algebra, and A and X as arbitrary linear combinations of them, A = a ν X ν and X = b µ Xµ. Suppose [A, X]= rX This equation has the form of eigenvalue equation where r is the eigenvalue and X is the corresponding eigenvector. We can choose A such that only r =0 is l-fold degenerate, and the rest of the roots are nondegenerate. l is called the rank of the semisimple lie algebra, and l linearly independent eigenvectors H i associated with r =0 span an l-dimensional subspace which is frequently referred to as the Cartan subalgebra. This equation does not have more than R roots for an R-element Lie algebra. The remaining eigenvectors , E α , associated to R− l distinct roots will span a (R− l)-dimensional subspace of the R- dimensional vector space: [A, H i ]= 0 (i=1, ..., l) [A, E α ]= αE α 6 We can write the standard form (frequently referred to as the Cartan-Weyl basis) of the commutation relations for a semisimple Lie algebra as [Wybourne, 1974, Cohn, R. N., 1998] [H i ,H j ]= 0 (i, j =1, ..., l) (1.5) [H i ,E α ]= α i E α [E α ,E β ]= N αβ E α+β (if α + β=0) [E α ,E −α ]= α i H i B. Simple roots α is positive if its first non-zero coordinate in some arbitrary basis is positive. In general half of the nonnull roots are positive. A root is Simple if it is positive and cannot be decomposed into a sum of two positive roots. All simple roots are linearly independent, and every positive root can be represented in the form α k α α where k α are nonnegative integers, and α’s are the simple roots. For a semisimple Lie algebra of rank l there are just l simple roots, which form a basis for the l-dimensional space of the root vectors [Wybourne, 1974]. C. Cartan Matrix If S =(α 1 ,α 2 , ..., α l ) is the system of simple roots, then the elements of the Cartan matrix of the given Lie algebra are A ij = 2(α i ,α j ) (α i ,α i ) , (1.6) 7 where (, ) is the inner product. The diagonal elements will always be 2, and the off- diagonal elements are restricted to the values 0,−1,−2 and−3. The complete set of roots of a Lie algebra may be determined from the system of simple roots and the Cartan matrix, that is, the sequence (k 1 , ..., k l ) such that α∈S k i α i are roots can be understood from the Cartan matrix. In practice we only need to deter- mine the positive roots. If β = k i α i is a root, we define its level as|β| = |k i |. The level is always a positive integer, and the simple roots are all of level 1. From equation number 1.5, each E γ is the commutator of E α and E β , that|γ| > |β| and|γ| > |α|. Thus given E α ’s associated with simple roots, we can generate the rest of them by sim- ply finding the commutators [Wybourne, 1974, Cohn, R. N., 1998]. SU(n) The Cartan-Weyl basis for su(n) constructs the complexification of the so called A n algebra. This algebra is associated with all n× n complex matrixes. The root vector diagram for A l is constructed by taking l +1 mutually orthogonal unit vectors and forming all possible root vectors of the form e i − e j (i, j =1, ..., N) Here e i is a vector that has 1 in the i’s position and zero elsewhere. By definition e i − e j is positive ifi<j . Obviously the roots e i − e i+1 cannot be decomposed into a sum of two positive roots, and as a result they are simple roots. It is easy to show that the generator associated with the root e i − e j is e ij , and e ii+1 are associated with simple roots, and e i+1i are associated with their inverse [Wybourne, 1974]. The Cartan matrix associated with SU(n) is: 8 A l :                 2 −12 ... 00 −12 −1 ... 00 0 −12 ... 00 . . ... . 000 ... 2 −1 000 ... −12                 (1.7) There are l(l+1) vectors, which together with l null vectors correspond to an algebra of order l(l+1), namely, that of su(l+1). I should emphasize again that A n is the com- plexification of su(n). To generate the algebra of SU(n) we need to use anti-hermitian generators [Cohn, R. N., 1998]. 1.2.2 Quantum Simulations The following composition laws enable us to build a library of Hamiltonians that can be performed given a set of basic operators. A: Conjugation by a unitary operator Suppose in addition to Hamiltonian H, we are given the ability to perform some uni- tary operation, U, and its inverse. U † . Then by performing the sequence of operations Ue −iHt U † = e −iUHU † t , we can simulate the evolution according to the conjugate Hamil- tonian UHU † . B: simulating linear combinations If we can simulate two different Hamiltonians H 1 and H 2 , we can also simulate their sum for a small time , due to the identity: 9 e −i(H 1 +H 2 ) = e −iH 1 e −iH 2 + O( 2 ) (1.8) Each simulation step thus contributes an error O( 2 ) and there are t/ such steps for a total error O(t) [Nielsen et al., 2002]. Higher order approximations are also possible [Bremner et al., 2005, 2004]. For example, identities such as e −i(H 1 +H 2 ) = e −iH 1 /2 e −iH 2 e −iH 1 /2 + O( 3 ) (1.9) yield a cumulative error which is O(t 2 ). But the higher-order approximations require the use of somewhat more complicated gate sequence for each time step. In application this additional complication must be balanced against the improvement in accuracy to achieve optimal results. In addition, given the ability to simulate H, it is always possible to simulate−H, using single-body unitary operations [Bremner et al., 2005]. Up U p HU † p = Dtr(H)I ⇒ Up=I U p HU † p = Dtr(H)I− H where I is the identity matrix, D is the dimension of the space, and U p is a unitary irreducible representation of the group U(n). Physically, the term Dtr(H)I is an unim- portant rescaling of the energy and can be neglected. In addition, by appropriate timing, we can exactly simulate evolution according to cH for any c≥ 0. Therefore, if we can simulate the whole set of Hamiltonians, H, we can also simulate arbitrary linear combination of any of the elements of these Hamilto- nians [Bremner et al., 2005, Nielsen et al., 2002, Dodd et al., 2002]. 10 C: Simulating commutators of Hamiltonians If we can perform H 1 and H 2 and their negations on the system, we can also perform [H 1 ,H 2 ] as e −iH 1 e −iH 2 e iH 1 e −iH 2 = e −i(i[H 1 ,H 2 ]) 2 + O( 3 ) (1.10) summary: A and B may be summarized in a single equation. Given the ability to perform the evolution according to the Hamiltonian H, and the ability to perform unitaries U j it is possible to simulate evolution according to a Hamiltonian of the form j c j U j HU † j (1.11) where c j is an arbitrary real number. 1.2.3 Majorization: Consider two sets of real numbers, (x 1 , ..., x n ), and (y 1 , ..., y n ). The mathematical the- ory of Majorization has been developed to give an answer to the question of which of these sets is more mixed (i.e. disordered) than the other. Assume x=(x 1 , ..., x n ) and y =(y 1 , ..., y n ) are two n dimensional real vectors. To make the definition I introduce the notation↓ to denote the components of the vector rearranged into non-increasing order, so x ↓ =(x ↓ 1 , ..., x ↓ n ), where x ↓ 1 ≥ ...≥ x ↓ n .We say x is majorized by y, written x≺ y,if k j=1 x ↓ j ≤ k j=1 x ↓ j (1.12) 11 for k =1, ...n− 1 and with the inequality holding for equality when k = n. Operator Majorization: Consider Hermitian matrices A and B, and let λ A and λ B denote the corresponding vectors of eigenvalues. Then A is majorized by B, A≺ B,if λ A is majorized by λ B . for example    10 01    ≺    11 11    This is because their eigenvalue spectra satisfy the majorization criterion, (1, 1) ≺ (2, 0).The following theorem connects this concept with quantum simulation: Uhlman theorem: A≺ B if and only if there exist unitary operators U i and proba- bilities p i such that A = i p i U i BU † i (1.13) Thus A≺ B if and only if A can be obtained from B by mixing together operators unitarily equivalent to B. The important point about Uhlman’s theorem is that the pro- cedure for finding p n and U n is constructive, and there are at most n 2 operators U n . Here is the proof of the first direction that is more useful for us [Nielsen et al., 2002]. Proof: Suppose A ≺ B. Let Λ(A) and Λ(B) are diagonal matrices whose entries are the eigenvalues of A and B, respectively, arranged into the decreasing order. We know that there exist unitary U and V such that A = UΛ(A)U † and B = V Λ(A)V † (1.14) 12 Since A ≺ B there exist P i and p i such that λ A = i p i P i λ B , where P i is a permutation matrix and p i is a probability which implies that Λ(A)= i p i P i Λ(B)P † i Comparing with (1.14), A = i p i UP i V † BV P † i U † If we consider U i ≡ UP i V † , which is a product of unitaries, and therefore is a unitary, A = i p i U i BU † i . Another important point in Uhlman’s theorem is that it can be applied to any trace- less Hermitian operators A and B. Namely, if A and B are two traceless Hermitian operators, assuming B=0, A≺ cB for some positive constant c [Nielsen et al., 2002]: The case A =0 follows by noting that 0 ≺ cB for allc> 0. Assume A =0 and choose c≡ max k=1,...n−1 k j=1 λ ↓ j (A) k j=1 λ ↓ j (B) where n is the dimension of the space. A and B are traceless and not equal to zero, so thatc> 0 for k =1, ..., n− 1 k j=1 λ ↓ j (A)= k j=1 λ ↓ j (A) k j=1 λ ↓ j (B) k j=1 λ ↓ j (B)≤ c k j=1 λ ↓ j (B) 13 Summary: If A and B are two traceless Hermitian operators in n dimensions, assuming B =0, there is an algorithm to find a set of at most n 2 unitary operators, U i , and constant c i > 0 such that: A = i p i U i BU † i (1.15) It means that if we can simulate some coupling H α , we can also simulate every other coupling on the same set of systems. 1.3 Universality There is a so called infinitesimal approach to the construction of universality. From the physical point of view, it can be considered as using a Hamiltonian approach with Hermitian operators, instead of unitary operators. The infinitesimal approach uses the Lie algebra su(n) of the Lie group SU(n) of unitary matrices together with the set of elements A k of this algebra represented as some anti Hermitian matrices A k =−A † k . It is also possible to write iA k ≡ H k for some Hermitian matrix H k . This argument is valid according to the lemma “If the elements A k generate the full Lie algebra of su(n) by commutators, then it is possible to use the set of unitaries U τ k =exp(−iH k τ) as a universal set of gates” [Vlasov, 2003, DiVincenzo, 1995]. This proof of universality is completed in two steps: 1. We show that the basic Hamiltonians enable the Robot to perform all single-body Hamiltonians. In other words the robot can perform J⊗ I and I⊗ J on the system of robot and a particle, where J is an arbitrary n× n unitary matrix. 14 2. Assuming the additional ability to perform an entangling Hamiltonian h R ⊗ h p , we can simulate J⊗ J on the system of robot and particle. This is enough to perform any general n k+1 × n k+1 Hamiltonian on the robot and particles. 1.3.1 step one (single-body Hamiltonians) Consider the basic Hamiltonians of the robot introduced in (1.1). The aim is to show that these operators are the generators of the Algebra su(n). In the first step we only consider single-body interactions in one direction. So the basic Hamiltonians acting on the joint robot-particle Hilbert space are I⊗H iR and I⊗H iM . Recall that H iR and H iM act on the ith particle, moving it in the x and y directions, respectively. To simplify the notation, I omit the subscript x and y. The element e mn is defined as an n× n matrix that satisfies: (e mn ) ij = δ mi δ nj Then: H iR = e ii+1 + e i+1i r =1, ..., n− 1 (1.16) H iM = ie ii+1 − ie i+1i i=1, ..., n− 1 e i,i+1 and e i+1,i are the generators that correspond to the simple roots and their inverses respectively. (Here I use the notation E i for e ii+1 ,E −i for e i+1i , and E α for a generator that corresponds to a general root that may or may not be simple.) H iR = E i + E −i ,H iM = iE i − iE −i (1.17) 15 A general generator may be written as H = α (k α E α + k −α E −a ) k α = r α + ir α (1.18) where α covers all positive roots. These operators, E α , are independent and using the fact that E † i = E −i and a simple induction we find that E † α = E −α . Suppose H is Hermitian: H † = α k ∗ α E −α + k ∗ −α E α H = H † ⇒ k α = k ∗ −α H = α r α (E α + E −α )+ ir α (E α − E −α )⇒ H = α r α H αR + r α H αM (1.19) But H α is among the commutators of H i , Using (1.5): [H αR ,H βR ]=[E α ,E β ]+[E −α ,E β ]+[E α ,E −β ]+[E −α ,E −β ] (1.20) = N αβ E α+β + N −α,β E β−a + N α,−β E α−β + N −α,−β E −α−β Recognizing the antisymmetry of the N αβ , the following equations hold: N α,β =−N −α,−β =−N β,α thus: [H αR ,H βR ]= N α,β (E α+β − E −α−β )+ N α,−β (E α−β − E β−α ) (1.21) = −iN α,β H α+βM − iN α,−β H α,−βM 16 Similar calculations results in: [H αM ,H βM ]= −iN α,β H α+βR + iN α,−β H α−βR (1.22) [H αR ,H βM ]= iN α,β H α+βM + iN −α,β H β−αR thus: H α+βR = i 2N α,β ([H αR ,H βR ]− [H αM ,H βM ]) (1.23) H α+βM = i N α,β ([H αM ,H βM ]− [H αR ,H βM ]) + H α+βR Therefore, it follows by a simple induction that we can generate the whole Hermitian matrices algebra by H iR and H iM . Namely, these matrices are the generators of the Lie algebra su(n). 1.3.2 Step Two (All Hamiltonians) Using the result of the last section and the universality lemma, we can perform the following single-body operators on the system of the robot and one particle in each direction: U 1 = J r ⊗ I p U 2 = I r ⊗ J p where J is an arbitrary unitary operator. The Robot can also perform an interaction Hamiltonian h⊗ h on each particle and itself at least once. The aim is to show that by 17 adding this interaction Hamiltonian to our set, we can also apply the following Hamil- tonians in each direction H x = J rx ⊗ J px (1.24) H y = J ry ⊗ J py where J rx and J ry are arbitrary unitary operators that act on the Robot in the x and y directions respectively. J px and J py are also arbitrary unitaries but act on the particle. I omit the subscript x and y for simplicity of the notation. In the interaction term of the form h⊗ h , h and h could be either H iR or H iM . Considering that both H iR and H iM are non-zero, traceless and Hermitian operators with the eigenvalues 0, 1 and−1, using the corollary to Uhlmann’s theorem as explained in (1.15) and considering (1.11), we can simulate any Hamiltonian of the form J ⊗ J . Namely, an arbitrary two-body Hamiltonian on the robot and a particle can be formed as a real combination of this interaction term, together with terms of the form J r ⊗ I p and I r ⊗ J p . The robot can perform this Hamiltonian on the system of itself and each particle in x and y directions [Nielsen et al., 2002]. This two-body Hamiltonian can be associated to a graph, whose vertices correspond to the particles and the robot in each direction, and whose edges connect vertices repre- senting systems that are coupled by the Hamiltonian. In our model the robot can turn on and off each of these interactions. Hence, it is possible to simulate arbitrary dynamics on the robot and any of the particles in each direction. Finally, an arbitrary interaction between two particles may be effected by performing one or two swaps between the robot and one of those particles, applying the desired interaction, and swapping back. Using this method, we can apply any two-body Hamiltonian between any pair in the system, and then the problem is reduced to the case 18 already solved [Nielsen and Chuang, 2000, Bremner et al., 2005]. We can perform any general n k+1 × n k+1 Hamiltonian on the robot and particles, and therefore our model of a quantum robot is universal. 1.4 Discussion I have shown that our model of a quantum robot can perform any desired Hamiltonian on the system. This model is probably the most restricted universal one. Therefore, many other models that may be considered include the capability of this one, and consequently are universal. An example of such a more general model is a robot that interacts with a square of four sites instead of two sites at each time step. In this work, a set of basic operators are considered that enables the robot to perform the tasks. However, in practice some other operators may be considered as the building blocks of the algorithms. In fact, depending on which task is desired, and the system that is considered as the physical realization, this set may be chosen differently in order to optimize the result. In many cases it is easy to show that the new set is equivalent to our basic set of operators. Finally, Quantum Robots are interesting subjects to study not only because they may perform some tasks faster than classical ones. They are also useful test beds for studying quantum system control, because the dependence of task dynamics on the environmental states for some tasks is similar to a feedback loop [Benioff, 1997]. They also provide a well defined platform for investigation of many interesting questions. For example, what physical properties must a quantum system have to be aware of environment or to have characteristics of intelligence? 19 Chapter 2 Optimized Quantum Error Correction 2.1 Introduction Quantum error correction is essential for the scale-up of quantum information devices. The theory of quantum error correction was developed in analogy to classical coding for noisy channels [Shor, 1995, Gottesman, 1996, Steane, 1996, Laflamme et al., 1996, Knill and Laflamme, 1997, Nielsen and Chuang, 2000]. These initial efforts focused on finding conditions and procedures for perfect recovery of quantum states passing through noisy channels. Recently, several authors considered error correction design as an optimization problem, with fidelity as the optimization target [Reimpell and Werner, 2005a, Yamamoto et al., 2005, Kosut and Lidar, 2009, Kosut et al., 2008]. In this work we further develop the theory of optimal quantum error correction. As in [Reimpell and Werner, 2005a, Yamamoto et al., 2005, Kosut and Lidar, 2009, Kosut et al., 2008], we consider the scenario where one has knowledge of the noise channel, and find corre- spondingly optimal codes. That is, we assume that one has already performed a channel identification procedure, e.g., via quantum process tomography [Poyatos et al., 1997]. We show how, armed with a knowledge of the channel, one can design highly robust error correction procedures, whose fidelity is always at least as good as that of the “agnostic” codes of standard error correction [Shor, 1995, Gottesman, 1996, Steane, 1996, Laflamme et al., 1996, Knill and Laflamme, 1997, Nielsen and Chuang, 2000]. 20 Our optimization procedure is directly applicable in a setting where one knows the type of channel (say phase flip) but one does not know the strength of the channel (say what the probability of a phase flip is). This is a physically relevant setting, since one can often guess the type of channel on the basis of physical considerations (e.g,, if only elastic scattering is involved without energy relaxation then the phase flip channel is a strong suspect), but it is much harder to know the strength of the channel as this involves detailed knowledge of the parameters of the physical process. In such a setting our average fidelity approach is appropriate. More specifically, we present an indirect approach to fidelity maximization based on minimizing the error between the actual channel and the desired channel. This approach leads naturally to bi-convex optimization problems, namely, two semidefinite programs (SDPs) [Boyd and Vandenberghe, 2004] which can be iterated between the recovery and encoding. For a given encoding the problem is convex in the recovery. For a given recov- ery, the problem is convex in the encoding. An important advantage of this approach is that noisy channels, which do not satisfy the standard assumptions for perfect cor- rection [Shor, 1995, Gottesman, 1996, Steane, 1996, Laflamme et al., 1996, Knill and Laflamme, 1997, Nielsen and Chuang, 2000], can be optimized for the best possible encoding and recovery. The conventional fidelity optimization targets are the encoding and recovery opera- tors. An important way in which the present work differs from previous studies is in the fact that we further add the distribution of the ancillas in the encoding and recovery to the optimization problem. This way, we utilize all possible degrees of freedom for opti- mization. As a consequence, we find a rather surprising result: in the optimized error correction procedure the fidelity is indifferent to the existence of the recovery ancillas. 21 This result paves the way toward a more efficient utilization of the ancillas. Namely, we can use all the available ancilla qubits in the encoding to increase the fidelity. Standard error correction schemes, as well as those produced by the aforementioned optimization methods which are tuned to specific errors, are often not robust to even small changes in the error channel. These errors can be mitigated by fault-tolerant methods which rely on several levels of code concatenation [Gaitan, 2004]. However, our method naturally enjoys a desirable robustness against error variations. We show a means to incorporate specific models of error channel uncertainty, resulting in highly robust error correction. Nevertheless, concatenated fault tolerant quantum error correc- tion still enjoys a certain important advantage over the procedures we derive in this work, namely, it is robust also against imperfections in the encoding and recovery procedures, while we assume these to be perfectly executed. Since the number of optimization variables scales exponentially with the number of qubits used in the encoding and recovery operations, the computational effort required to solve any of the semidefinite program optimization (SDP) problems is similarly bur- dened. In order to reduce this effort we propose an approach based on optimization via the constrained least squares method. This alternative approach for solving the distance minimization problem does not utilize semidefinite programming, and is significantly faster in our numerical simulations. Surprisingly, this method returns the exact same result as the SDP approach. 22 2.2 Quantum Error Correction 2.2.1 Open quantum systems and Standard error correction model The dynamics of a closed quantum system are described by a unitary transform. A nat- ural way to describe the dynamics of an open quantum system is to regard it as arising from an interaction between the system of interest and an environment, which together form a closed quantum system. This composite system evolves through a unitary trans- formation:E(ρ tot )= Uρ tot U † To obtain the state of our system of interest, we perform a partial trace over the environment: E(ρ tot )= tr env [Uρ tot U † ] (2.1) Therefore, subject to the assumption that the initial system-bath state is classically correlated [Shabani and Lidar, 2009b], the dynamics of an open quantum system can be represented in an elegant form known as the Kraus operator sum representation (OSR). In this representation, the noiseE is described in terms of a completely-positive (CP) map: ρ→ i A i ρA † i [Nielsen and Chuang, 2000]. Here ρ is the initial system density matrix and the operators A i , known as Kraus operators, or operation elements, satisfy the normalization relation i A † i A i = I (identity). The standard error correction procedure involves CP encoding (C), error (E), and recovery (R) maps (or channels): ρ S C → ρ C E → σ C R → ˆ ρ S , as shown pictorially in the block diagram of Figure 2.1. ρ S ρ C σ C ˆ ρ S CE R Figure 2.1: Standard representation of error correction. 23 Here ρ S is the n S ×n S system state, ρ C is the n C ×n C encoded state, σ C is the n C ×n C perturbed encoded state, and ˆ ρ S is the n S × n S recovered system state. Using the OSR: ˆ ρ S = r,e,c (R r E e C c )ρ S (R r E e C c ) † . (2.2) The encoding{C c } m C c=1 and recovery{R r } m R r=1 operation elements are rectangular matri- ces, respectively n C ×n S and n S ×n C , since they map between the system Hilbert space (of dimension n S ) and the system/ancillas Hilbert space, the codespace, of dimension n C . The error operation elements{E e } m E e=1 are square (n C ×n C ) matrices, and represent the effects of noise on the codespace. The number of elements, m C ,m E ,m R depend on the manner of implementation and basis representation [Nielsen and Chuang, 2000]. More specifically, any OSR can be equivalently expressed, and consequently physically implemented, as a unitary with ancilla states [Nielsen and Chuang, 2000, §8.23]. An example of this representation of the standard error correction model of Figure 2.1 is shown in the block diagram of Figure 2.2. bath ρ C σ C |0 RA ρ S U C U E U R |0 CA ρ B ˆ ρ S Figure 2.2: System-ancilla-bath representation of standard encoding-error-recovery model of error correction. In this case the encoding operationC is implemented by a unitary operator U C acting on the (tensor) product of the system state, ρ S , and the encoding ancillas’ state, |0 CA , producing the encoded state ρ C = U C (|0 CA 0 CA |⊗ ρ S )U † C . If the encoding ancillas’ 24 state has dimension n CA , then the resulting codespace has dimension n C = n S n CA . If, as is customary, we take |0 CA as the n CA -column vector with a one in the first element and zeros elsewhere (i.e., it is a tensor product of log 2 n CA encoding ancillas, each in the state|0=(1, 0) t ), then the OSR forC has the single (m C =1) n C ×n S matrix element C whose columns are the first n S columns of U C , thus forming a set of orthonormal codewords, i.e., U C =[C ···],C is n C × n S (2.3) More generally n C <n S n CA , i.e., not all the ancilla are used to define the codespace, in which case there is more than one OSR matrix element inC, i.e., U C =[C ···],C =       C 1 . . . C m C       ,C c is n C × n S (2.4) For the errors, E, the ancillas’ states are not implemented by design, but rather, engendered by interaction with the bath, a term used to generically describe the physical environment. The error operation is thus equivalent to the unitary U E operating on the tensor product of ρ C , the encoded state, and ρ B , the bath state. The number of bath states may be very large, in principle infinite dimensional. However, it is always possible to representE with a finite number of OSR elements with m E ≤ n C 2 [Nielsen and Chuang, 2000, Thm.8.3]. Finally, the recovery operationR can be implemented via the unitary U R operating on the (tensor) product of the perturbed encoded state, σ C , and the (additional) recovery ancillas’ state |0 RA .If |0 RA is an n RA -column vector with a one in the first element 25 Table 2.1: Definitions of some frequently used symbols. Symbol Definition n S dimension of the system space n CA dimension of the encoding ancillas space n C dimension of the (system + encoding ancillas) space, i.e.,n C =n S ×n CA n RA dimension of the recovery ancillas space m E number of operation elements for error map m R number of operation elements for recovery map and zeros elsewhere, then the OSR{R r } m R r=1 forR has m R = n CA n RA elements which consist of the first n C columns of U R , i.e., U R =[R···],R =       R 1 . . . R m R       ,R r is n S × n C (2.5) The “real” error system is unlikely to be accurately represented by the idealized system shown in either Figure 2.1 or 2.2. The reason is that the bath is always active, so that the control “knobs” which implement the encoding and/or recovery can in general not be separated from their interactions with the bath. The model represented in Figures 2.1 and 2.2 assumes that the encoding and recov- ery operations can be implemented much faster than relevant timescales associated with the bath. For a detailed discussion of the validity of such a Markovian model see [Alicki et al., 2006]. Nevertheless, we will assume the model of Figures 2.1 and 2.2 for the remainder of this work, as complications associated with the bath being “on” during encoding and recovery are likely to be dealt with via fault tolerance methods [Aliferis et al., 2006], which require a base level of encoding of the type we find here. Table 2.1 provides definitions of some frequently used symbols. 26 2.2.2 Optimization problems Assume that we are given the OSR elements of the error channel E. This could be obtained, for example, from the output of a quantum process tomography experiment [Poyatos et al., 1997]. The error correction objective considered here is to design the encodingC and the recoveryR so that, for a given error operationE, the map ρ S → ˆ ρ S is as close as possible to a desired n S × n S unitary logic gate L S . Common measures of performance between two quantum channels are typically based on fidelity or distance [Nielsen and Chuang, 2000], [Gilchrist et al., 2005], [Kretschmann and Werner, 2004], [Kosut et al., 2006]. Here we will use the channel fidelity [Reimpell and Werner, 2005a] between the error correction operationREC and the desired operation L S : f = 1 n 2 s r,e,c |Tr L † S R r E e C c | 2 . (2.6) where 0≤ f ≤ 1 and from [Nielsen and Chuang, 2000, Thm.8.2], f =1 if and only if there are constants δ rec such that, R r E e C c = δ rec L S , r,e,c |δ rec | 2 =1. (2.7) This suggests the indirect measure of fidelity (using the Frobenius norm, X 2 F = Tr X † X), d = r,e,c R r E e C c − δ rec L S 2 F = c RE(I E ⊗ C c )− ∆ c ⊗ L S 2 F (2.8) where ∆ c ≡ [δ rec ], dim ∆ c = m R × m E , (2.9) 27 and E is the n C × n C m E rectangular “error matrix,” E =[E 1 ··· E m E ], (2.10) R is the m R n S × n C matrix obtained by stacking the m R matrices R r as in (2.5), and I E is the m E × m E identity. Hence, we have c Tr ∆ † c ∆ c = r,e,c |δ rec | 2 =1, and R † R = r R † r R r = I C . We show in Appendix 5 that there exists a recovery and encoding pair,R,C, which achieves perfect error correction (equivalently d=0,f =1), iff for c, c =1,...,m C (I E ⊗ C † c )E † E(I E ⊗ C c )=∆ † c ∆ c ⊗ I S (2.11) This is a generalization to non-unitary CP 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 single n C × n S matrix element C, C † C = I S , whose n S columns are the codewords.As f and d are explicitly dependent on the channel elements, they are convenient for optimization. Consider then the following optimization problems. Fidelity Maximization maximize f(R,C) subject to R † R = I C ,C † C = I S (2.12) Distance Minimization minimize d(R,C, ∆ 1 ,..., ∆ m C ) subject to R † R = I C ,C † C = I S , c ∆ c 2 F =1 (2.13) 28 Here C is the n C ×n S matrix obtained in (2.3) . The first approach was used in [Reimpell and Werner, 2005a, Fletcher et al., 2007, Fletcher, 2006, Reimpell et al., 2006, Kosut and Lidar, 2009, Kosut et al., 2008]. Note that f and d are related as follows: f(R,C)= 1− ˆ d(R,C)/2n S 2 ˆ d(R,C)=min d(R,C, ∆ 1 ,..., ∆ m C |∆ c 2 F =1∀c (2.14) This shows that minimizing the distance (2.13) is equivalent to maximizing fidelity (2.12). In order to show this relation, consider the following problem, minimize d = c RE(I E ⊗ C c )− ∆ c ⊗ L S 2 F subject to c ∆ c 2 F =1 (2.15) Form the Lagrangian, L = d + λ( c Tr ∆ † c ∆ c − 1) (2.16) with λ the Lagrange multiplier. Then, ∇ δrec L =0 when (n S + λ)δ rec = Tr R r E e C c L † S . To enforce the constraint c ∆ c 2 F =1 requires that (n S + λ) 2 = r,e,c |Tr R r E e C c L † S | 2 . Hence, ∆ c = ¯ ∆ c / c ¯ ∆ c 2 F ¯ δ rec = Tr(R r E e C c L † S )/n S (2.17) Observe that c ¯ ∆ c 2 F = f. This together with r R † r R r = I C , c C † c C c = I S gives the optimal distance as given implicitly by (2.14). Note also that with no constraint, λ=0, the ¯ ∆ c are the optimal least-squares (unconstrained) solution. 29 2.3 Distance Minimization We consider the encoding operatorC as a unitary operator acting on both the encoding ancillas and the input qubit. Using the constraints in (2.13), we can express the distance measure (2.8) as d(R,C, ∆) = RE(I E ⊗ C)− ∆ ⊗ L S 2 F (2.18) = n S + Tr E(I E ⊗ CC † )E † − 2Re TrRE(∆ † ⊗ CL † s ) where ∆ is the single m R × m E matrix in (2.9) with m R = n CA n RA (note that in this case, since there is only a single ∆ c matrix, we drop the subscript c). 2.3.1 Optimal solutions Since only the last term in (2.18) depends on the recovery matrix R, minimizing d(R,C, ∆) with respect to R is equivalent to maximizing the last term. In Appendix 5, we show that this maximization results in max R † R=I C Re Tr RE(∆ † ⊗ CL † s )= Tr E(Γ⊗ CC † )E † , (2.19) where the m E × m E matrix Γ is defined as, Γ=∆ † ∆ , (2.20) and the associated n C n RA × n C optimal recovery matrix is, R=[v 1 ... v n C ][u 1 ... u n C ] † (2.21) 30 where v i ,u i ,i =1,...,n C are, respectively, the right and left singular vectors in the singular value decomposition of the n C ×n C n RA matrix E(∆ † ⊗CL † S ), with the singular values, as usual, in descending order. Thus, to obtain the optimal recovery, we need first to find Γ which maximizes (2.19) – this is equivalent to minimizing d over R. Following this we need to determine ∆ satisfying (3.6). To find Γ, observe that Γ ≥ 0 by definition (3.6), and the constraint ∆ F =1 from (2.13) is equivalent to Tr Γ = 1. Hence, optimal recovery can be obtained by first solving for Γ from, maximize Tr E(Γ⊗ CC † )E † subject to Γ≥ 0, TrΓ=1 (2.22) We can show that this optimization problem is equivalent to SDP problem. Note that the problem is of the form, maximize Tr F(Γ) subject to Γ≥ 0, TrΓ=1 (2.23) where F(Γ) is linear in Γ. Consider the relaxed problem, maximize Tr Y subject to F(Γ)− Y 2 ≥ 0, Γ≥ 0, TrΓ=1 (2.24) This is an SDP in Γ and Y with Lagrangian, L(Γ,Y,P,Z)= −Tr Y −Tr P(F(Γ)− Y 2 ) −Tr ZΓ+ λ(Tr Γ− 1) (2.25) 31 The dual function is, g(P,λ,Z)= inf Γ,Y L(Γ,Y,P,Z) =      inf Y Tr(PY 2 − Y )− λ, Z = λI− A(P) −∞ otherwise (2.26) with A(P)= ∂ ∂Γ Tr PF(Γ), which is not dependent on Γ because F(Γ) is linear in Γ. Performing the indicated inf Y gives Y =(1/2)P −1 and g =−(λ+(1/4)Tr P −1 ). The dual optimization associated with (2.24) is to maximize g, or equivalently, minimize its negative, i.e., minimize λ + 1 4 Tr P −1 subject to P> 0,λI− A(P)≥ 0 (2.27) This is an SDP in the dual variables P, λ. For this problem strong duality holds [Boyd and Vandenberghe, 2004]. Consequently, at optimality of (2.24) and (2.27) the com- plementary slackness condition is P opt (F(Γ opt )− Y 2 opt )=0. Since P opt > 0,wehave Y opt = F(Γ opt ). This establishes that solving the SDP (2.24) is equivalent to solving the original problem (2.23). The next step is to use (3.6) to obtain ∆ from Γ. The following choice adheres to the given dimensions:    n CA ≤ m E n RA n CA = m E    ⇒      ∆= √ Γ R is tall (n S m E × n C )    n CA >m E n RA =1    ⇒            ∆=    √ Γ 0 n CA −m E ×m E    R is unitary (n C × n C ) (2.28) 32 Clearly the choice of ∆ is not unique. In fact, the result does not change if ∆ is multi- plied by a unitary, i.e., ∆ → U∆ . This is exactly the unitary freedom in choosing the OSR elements [Nielsen and Chuang, 2000]. Interestingly, however, from many numeri- cal calculations we observe that the following holds:        rank(Γ) = n CA if n CA ≤ m E rank(Γ) = m E if n CA >m E . (2.29) Since the m E × m E matrix Γ is Hermitian (= ∆ † ∆ ), and ∆ is m R × m E with m R = n CA n RA , it follows that if (2.29) is true then, n RA =1. (2.30) If, in the optimized error correction, the dimension of the recovery ancillas space is one, then the optimal recovery matrix R is always a unitary – recovery ancillas are redundant in maximizing the fidelity. Note that we started with a generic n RA parameter, and the properties of the optimal solution led us to the above conclusion. Although we do not have a rigorous proof that the recovery ancillas are redundant, a compelling heuristic argument is offered in Section 2.4 along with supporting numerical results. Optimal Encoding For a given R and ∆ , the optimal encoding C can be found by solving (2.13) for C, that is, minimize d(R,C, ∆) = RE(I E ⊗ C)− ∆ ⊗ L S 2 F subject to C † C = I S (2.31) 33 Form the Lagrangian, L = d +Tr P(C † C− I S ) (2.32) with P the Lagrange multiplier. Then,∇ C L =0 when C = ¯ C(I S + P) −1 with ¯ C as defined in (2.31). To enforce the constraint C † C = I S requires that (I S + P) 2 = ¯ C † ¯ C. Hence, C = ¯ C( ¯ C † ¯ C) −1/2 . Note that with no constraint, P =0, and ¯ C is the optimal least-squares (unconstrained) solution. The left-hand column of Table 2.2, labeled Algorithm-1, summarizes the preced- ing method for recovery and encoding optimization. For optimal recovery alone, solve (2.22) for Γ, then determine ∆ via (2.28), and finally R from (2.21). For optimal encod- ing alone, solve (2.31) for C.To find a combined optimal encoding and recovery repeat steps 1 and 2 in Table 2.2 until d stops decreasing. (By virtue of (2.14), fidelity increases in every step). Since in each step the distance measure, d ind , can only decrease, never increase, the converged solution to the combined optimization is only guaranteed to be a local optimal solution to (2.13). 2.3.2 Alternative algorithm for recovery optimization An alternative to the above optimal recovery procedure (Step 1 in Algorithm-1of Table 2.2) is to iterate between solving (2.13) directly by minimizing over ∆ and then using (2.21) to find R. Specifically, for a given R and C, Step 2a in Algorithm-2of Table 2.2 requires solving the following constrained least-squares problem for ∆ : minimize d(R,C, ∆) = RE(I E ⊗ C)− ∆ ⊗ L S 2 F subject to ∆ F =1 (2.33) 34 As shown in section 2.2.2, the solution is, ∆= ¯ ∆ / ¯ ∆ F ¯ ∆ re = Tr(R r E e CL † S )/n S , (2.34) where ¯ ∆ is the unconstrained (least squares) solution to min ∆ d ind . This solution is then used in (2.21) to find R (Step 1b), then back to (2.34) (Step 1a), and so on until d stops decreasing (Step 1c). Table 2.2: Iterative Algorithms for Optimal QEC Algorithm-1 Algorithm-2 Initialize encoding C Repeat 1. Optimal recovery (a) solve (2.22) for Γ (b) Γ→ ∆ via (2.28) (c) ∆ → R via (2.21) 2. Optimal encoding (a) solve (2.31) for C Until d stops deceasing Initialize encoding C and recovery R Repeat 1. Optimal recovery – Repeat a-c (a) solve (2.33) for ∆ (b) ∆ → R via (2.21) (c) Until d stops decreasing 2. Optimal encoding (a) solve (2.31) for C Until d stops deceasing The difference between the two algorithms is in computing the optimal recovery (Steps 1). In Step 1 of Algorithm-1no iterations are required; the optimal recovery is achieved once through. For Step 1 of Algorithm-2, an optimal recovery is the result of some num- ber of iterations. Although at present a proof is not available, in every case we have tried the optimal fidelity in both recovery algorithms converges to the same result. Addition- ally, the total CPU-time in MATLAB to compute the optimal recovery in Algorithm-2 35 (including the iterations) is significantly less than the CPU-time for the recovery step in Algorithm-1using YALMIP [Lofberg, 2004] to call the solver SDPT3 [Toh et al.]. 2.3.3 Robust error correction An important advantage of the method presented here is that unlike the standard error correction model, it accounts for uncertainty in knowledge of the channel. Such uncer- tainty may exist for many reasons. For example, different runs of a tomography exper- iment can yield different error channels {E α } α=1 . Or, a physical model of the error channel might be generated by a Hamiltonian H(θ) dependent upon an uncertain set of parameters θ. In any case, not accounting for the 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 these Hamiltonian parametric uncertainties is to take a sample from the set of Hamiltonians, say,{H(θ α )} α=1 . Tracing out the m E bath states will result in a set of error channels {E α } α=1 where each error channel has OSR elements{E α,k } κ k=1 , where κ is the largest of the number of OSR elements in each sample. In those samples with a smaller number we can set the corresponding OSR elements to zero. Two standard measures of robustness are the average-case and worst-case. For the average-case, suppose that each OSR setE α is known to occur with probability p α . Then define the average error channel by the OSR, E avg ={ √ p α E α,k |α=1,...,,k =1,...,κ} (2.35) The average error channel in this form has κ OSR elements, potentially a very large number. However, this number is readily reduced to no more than m E = n 2 C using a 36 singular value decomposition [Nielsen and Chuang, 2000, Thm.8.3]. Associated with E avg is the average channel fidelity, f avg = α p α f α = 1 n 2 S r,e ,c |Tr R r E avg e C c | 2 (2.36) where E avg e ,e =1,...,κ are the OSR elements ofE avg in (2.35). For average-case robust error correction we replace f in (2.12) with f avg in (2.36), and using the relationship (2.14), replace d in (2.13) with, d avg = RE avg (I E ⊗ C)− ∆ ⊗ L S 2 F E avg =[E avg 1 ··· E avg κ ] (2.37) Worst-case error correction was considered in [Kosut et al., 2008]; we do not consider it here. The examples presented in Sec.2.5 show that this approach yields a high degree of robustness to uncertainty in the optimal codes. 2.4 Optimized distribution of ancillas In our formalism, the dimension of the Recovery ancillas’ space, i.e., the required num- ber of recovery ancilla qubits, is determined by the rank of the m E × m E matrix Γ. 37 2.4.1 Rank minimization of Γ In this section, we study the rank of Γ through a heuristic argument by noting the sim- ilarity between our problem and the so called “Rank Minimization Problem” (RMP) [Vandenberghe and Boyd, 1996]: minimize rank (X) subject to X∈X (2.38) The matrix X is the optimization variable andX is a convex set denoting the constraints. Although several special cases of the RMP have well-known solutions, in general the RMP is known to be computationally intractable. However, there are a number of heuristic approaches to solving this problem. Restate (2.22) as follows, minimize Tr (Γ) subject to Γ≥ 0, Tr E(Γ⊗ CC † )E † ≥ const. (2.39) where the constant is the maximum which arose in (2.22). A well known heuristic for RMP when X is positive semidefinite [Fazel, 2002, Fazel et al., 2002, Mesbahi and Papavassilopoulos, 1997] is to replace the rank objective with Tr [X] and solve, minimize Tr [X] subject to X∈X,X ≥ 0 (2.40) By comparing (2.39) with (2.40), we can view our problem in (2.22) as an RMP that minimizes the rank of Γ. Thus, the rank of the optimal Γ is the smallest possible consis- tent with not changing the rank of our objective matrix, E(Γ⊗ CC † )E † . Noting that 38 rank(CC † )= n S and rank(E)= n C and with a straightforward linear algebra analysis we find that this property holds if        rank(Γ)≥ n CA if n CA ≤ m E rank(Γ) = m E if n CA >m E . (2.41) That is, in the first case, if rank(Γ) <n CA , rank( E(Γ⊗ CC † )E † ) decreases by decreasing the rank of Γ. But if rank(Γ) ≥ n CA , rank( E(Γ⊗ CC † )E † )= n C , and it does not depend on rank(Γ). In the second case, Γ should be full rank. Therefore the rank of the optimal Γ is        rank(Γ opt )= n CA if n CA ≤ m E rank(Γ opt )= m E if n CA >m E , (2.42) which agrees with (2.29). Note that the same argument also applies in the average case (2.37) with E replaced by E avg . 2.4.2 Numerical result for randomly generated error maps Here, we examine the result above for randomly generated error maps. Namely, we find the rank of the optimal Γ for each random map by applying the distance minimization method. The error map is modeled as shown in Fig. 2.2 as a unitary U E acting on the joint codespace-bath Hilbert space. The unitary U E arises from a randomly selected m E n C ×m E n C time-independent Hamiltonian H E , i.e., U E = e −itH E (we work in units 39 where =1). The unitary evolution operator generated by this Hamiltonian at time t=1 is U E =exp(−iH E )=          E 1 ... E 2 ... . . . E m E ...          (2.43) That is, we pick the first n C columns of the matrix U E . Here, E 1 ...E m are the n C ×n C OSR elements of the error operation, and from (2.10), E =[E 1 ··· E m E ]. Figure 2.3 presents the channel fidelity vs. the number of iterations in Algorithm 1 for 100 random error maps. In this experiment, the system is a single qubit and one qubit is used as an encoding ancilla, i.e., n S =2, n CA =2. Each error map has 4 OSR elements, i.e., m E =4, and is generated using a 16× 16 random Hamiltonian matrix according to (2.43). Therefore, the matrix Γ in (2.22) is 4× 4. Figure 2.4 shows the histogram of the rank of Γ vs. the number of iterations. This histogram indicates that after 20 iterations in the optimization algorithm, the rank of Γ is always two, which is equal to n CA . In fact, those Γ that are not rank 2 after 10 iterations are associated to the error maps with lower rate of fidelity convergence. Figure 2.5, which shows the singular values of the same Γ matrices, is included for comparison of the magnitude of the singular values. In all cases, the nonzero singular values are of the order of 10 −1 . The numerical precision of all the results is 10 −8 .We repeated the experiment for more than 1000 random maps with different dimensions (only 100 are shown), and the result holds for all of them. Namely, after sufficiently many iterations in Algorithm 1, the rank of the optimal Γ is the same as the dimension of the encoding ancillas space, i.e., rank (Γ opt )= n CA . 40 2 4 6 8 10 12 14 16 18 20 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 number of iterations f Figure 2.3: Channel fidelity f for random error maps on two-qubit codes. 2.5 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. In these examples, the error channel E consists of single-qubit errors occurring independently on all qubits with probability p. We examine two cases of bit-flip and bit-phase-flip errors. 2.5.1 3-qubit bit-flip errors In this example, we consider the independently occurring bit-flip error as the noise chan- nel, where the bit-flip operator is X = 2 4 01 10 3 5 . We used q C =2 encoding ancilla qubits. There are 2 3 =8 OSR error elements for 3-qubit encoding: {E i } 8 i=1 = A i 1 ⊗ A i 2 ⊗ A i 3 ,i 1 ,i 2 ,i 3 ∈{1, 2} A 1 = (1− p) I (no error) A 2 = √ pX (bit− flip error) (2.44) 41 1 2 3 4 0 10 20 30 40 50 60 70 80 90 100 rank of Γ population no iteration 10 iterations 20 iterations Figure 2.4: Rank of the optimal Γ for random error maps on two-qubit codes. Figure 2.6 shows f vs. bit-flip probability p in the range p ≤ 0.9 for the standard 3-qubit code, optimal recovery at each p, average-case recovery over the p range, and no recovery. For the average case, we computed an optimized encoding and recovery for the single channel obtained by averaging over the error channels corresponding to p =0, 0.1, ..., 0.9 as defined in (2.37). We then applied this encoding and recovery to each of these 10 channels, thus producing the 10 fidelity values shown. Note that the optimal recovery can be achieved equivalently by either the constrained least squares method or the convex optimization method. Interestingly, the standard 3-qubit code not only provides optimal recovery for the range p≤ 0.5, it is optimal for both recovery and encoding in this range. Forp> 0.5 the standard code is clearly no longer optimal. Only in this range does the optimal recovery outperform the standard code, a phenomenon similar to what was reported for amplitude-damping errors in [Reimpell and Werner, 2005a]. Analysis of our optimal encoding recovery results reveals the following simple picture. The optimal code is the standard 3-qubit code for the entire p range, i.e., | ¯ 0 = 42 20 40 60 80 100 1 2 3 4 0 0.2 0.4 0.6 0.8 1 experiment number singular Value index magnitude of singular values of Γ Figure 2.5: Singular values of the optimal Γ for random error maps on two-qubit codes. For all cases tested only two of the singular values are significantly different from zero, meaning that the rank of the Γ matrices is 2. |000 and | ¯ 1 = |111. The optimal recovery is the standard recovery [Nielsen and Chuang, 2000] in the range 0≤ p≤ 0.5. In the range 0.5≤ p≤ 1 the optimal recovery is a bit-flip on all qubits followed by the standard recovery. Figure 2.7 shows channel fidelity f in two ranges: p< 0.5 and 0.5 <p ≤ 0.9. Unlike the previous case, here we compute the optimization twice, once for each range. I.e., for the average case, we computed an optimized encoding and recovery for the single channel obtained by averaging over the error channels corresponding to p = 0, 0.1, ..., 0.4. We then applied this encoding and recovery to each of these 5 channels, thus producing the 5 fidelity values shown in the range 0≤ p≤ 0.5. We then repeated this procedure for p=0.5, 0.6, ..., 0.9. encoding and recovery for the worst case in each of the two ranges. Forp< 0.5, the standard, optimal, average-case, all coincide. For p> 0.5, the optimal and average case codes coincide and divert again from the standard. 43 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 optimal avg −case standard no recovery f p Figure 2.6: Channel fidelity f vs. bit-flip probability p for 3-qubit encoding. The optimal encoding and recovery are the same as in Figure 2.6, i.e., the standard 3-qubit code, with standard recovery in the range 0≤ p≤ 0.5, and bit-flips preceeding standard recovery in the range 0.5 ≤ p ≤ 0.9. We conclude from the examples in Figures 2.6 and 2.7 that optimal encoding and recovery has no advantage over standard encoding and recovery for low bit-flip probabilities (p< 0.5), and thus increasing the codespace would be required to improve fidelity. For large errors (p> 0.5), optimization is more effective in that it identifies an optimal recovery. In both cases the achieved optimal fidelity is independent of the number of recovery ancillas used, hence in all examples shown in Figures 2.6 and 2.7 there are no recovery ancillas. It is striking that the average case fidelity matches the optimal in Figure 2.7, but not in Figure 2.6. This is entirely due to the range of p values over which the average is performed. The lesson is that the more information is available about the noise channel, the more robust the encoding and recovery will be: in Figure 2.7 we know that the probability is in the range [0, 0.5] or [0.5, 0.9], while in Figure 2.6 we only know that it 44 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 optimal avg −case standard no recovery f p Figure 2.7: Channel fidelity f vs. bit-flip probability p for 3-qubit encoding in two ranges: p< 0.5 and 0.5<p≤ 0.9. is in the range [0, 0.9]. Absent such information, robustness may still be attainable by experimenting with tuning the encoding and recovery over a range of channels. 2.5.2 Bit-Phase flip error In this example, the noise channel consists of bit-phase flip errors Y = 2 4 0 −i i 0 3 5 occur- ring independently with probability p. We do not allow for more than three to occur simultaneously (i.e., we consider weight-3 errors). We examine two cases: 1. Consider- ing a fixed number of encoding ancillas, we compare the fidelity using different numbers of recovery ancillas. 2. We fix the total number of available ancilla qubits, and compare the fidelity for various distributions of encoding and recovery ancillas. 5-qubit bit-phase flip error In this example, the bit-phase flip errors occur independently on the input qubit and 4 ancillas. There are 26 error OSR elements: 1 for no error, 5 for a single error, 10 for 45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 optimal, no recovery ancilla optimal, 1 recovery ancilla optimal, 2 recovery ancillas standard error correction f p Figure 2.8: Channel fidelity f vs. bit-phase flip probability p for 3 qubit code and 0, 1, or 2 recovery ancillas, with optimal encoding and recovery. double errors, and 10 for triple errors. Thus the matrix Γ in (2.22) is 26× 26 and the rank of Γ opt is equal to n CA =16, meaning that the optimal distribution of ancillas is having all four in the encoding block and none in the recovery block. Figure 2.8 shows f vs. bit-phase flip error probability p for the optimal encod- ing/recovery in the case of zero, one and two recovery ancillas. The result shows that all cases yield the same fidelity. Therefore, the fidelity of the system is independent of the number of recovery ancillas. bit-phase flip errors with a fixed number of ancillas In this example, we consider five ancilla qubits that can be used either in the encoding block or in the recovery block. We compare the fidelity for the following distributions: three encoding ancillas and two recovery ancillas, fore encoding ancillas and one recov- ery ancilla, and five encoding ancillas with no recovery ancilla. Figure 2.9 shows that the channel fidelity increases significantly by using the ancillas in the encoding instead 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 3 encoding ancillas 4 encoding ancillas 5 encoding ancillas f p Figure 2.9: Channel fidelity f vs. bit-phase flip probability p with a fixed total of 6 ancillas, and optimal encoding and recovery. of the recovery. Thus the most efficient use of ancillas is achieved when they are all used for encoding. 2.6 Conclusion We have presented an optimization approach to quantum error correction that yields codes which achieve robust performance, when tuned to a specific noise channel. An important aspect of developing optimal codes which are tuned to a class of errors, or are robust over a range of errors, is that the optimized performance levels may be sufficient for the intended purposes. Hence, no further increases in codespace dimension and/or concatenation may be necessary. We also showed that the fidelity of such a system is independent of the number of the recovery ancillas. This is entirely due to the structure of the error correction optimization problem, for which we found that a unitary recovery operator maximizes the fidelity of the system. However, the fidelity increases significantly by increasing the dimension of 47 the encoding ancillas space. Therefore, in the optimal quantum error correction scheme, one should use all the available ancilla qubits in the encoding block. Although not further developed here, the resulting codes, unlike standard codes, have support over all basis states. Some of the recovery structure is revealed via the indirect approach. This in turn leads to a method for approximating optimal recovery involving only a singular value decomposition, making it potentially useful in evaluating very large blocks of encoding to see if further performance improvement is possible. We stress that there is an important difference between the standard error correction schemes [Shor, 1995, Gottesman, 1996, Steane, 1996, Laflamme et al., 1996, Knill and Laflamme, 1997, Nielsen and Chuang, 2000] and the approach presented here. While in the standard case only the class of errors should be known, in our method the exact form of the noise map is required for optimization. In general, the noise map can be identified using quantum process tomography [Poyatos et al., 1997]. In most cases this extra knowledge is equivalent to identifying the probability of the error, which can also be found using our method. In order to identify the probability in a particular error model, one should calibrate the fidelity of the system using a fixed pair of recovery and encoding operators. Once the relation between the fidelity associated to this pair and the error probability is known, a measurement of the fidelity yields the probability. 48 Chapter 3 Optimized Entanglement Assisted Error Correction 3.1 Introduction In this chapter, the error correction procedure benefits from a pre-noise entanglement shared between the encoding and recovery ancilla qubits. This initial shared entangle- ment can in principle transfer information about the initial state to the recovery block. We show that this additional information can enhance the performance of the recovery ancillas and increase the error correction fidelity in most cases. This fidelity increase can in fact lead to perfect error correction for an important class of error channels. The error correction procedure for this class of channels is also interpreted through a teleportation argument. The entangled recovery ancillas are fundamentally different than the regular recov- ery ancillas considered in previous optimization problems Reimpell and Werner [2005a], Kosut et al. [2008], Taghavi et al. [2010]. In the previous chapter, we showed that reg- ular recovery ancillas are redundant in increasing the error correction fidelity [Taghavi et al., 2010]. The fidelity increase here shows the importance of relevant information stored in the ancillas in quantum error correction. The optimization procedure here, similar to previous works in this area [Reimpell and Werner, 2005a, Kosut et al., 2008, Taghavi et al., 2010], assumes the scenario where 49 Figure 3.1: In entanglement assisted error correction, the encoding and recovery ancillas are entangled before the procedure begins. one has knowledge of the noise channel. Namely, we assume that a channel identifi- cation procedure, e.g., quantum process tomography [Poyatos et al., 1997], has been performed prior to the error correction. 3.2 Problem Formulation A quantum channel can be considered as a process defined by an initial state ρ s and a final state ˆ ρ s , with the dynamics described by an operatorE. In the presence of deco- herence the dynamics of the systemE is not unitary, and it can be described using the Operator Sum Representation as explained in the previous chapter. Here the error cor- rection procedures, encoding and recovery, as well as the error operations are described using Kraus operators. The whole procedure on the system can be represented as ρ 0 C → ρ C E → ρ R R → ρ T , where C, E, and R are the encoding, error and recovery operators respectively. ρ 0 through ρ T are the state of the entire system including the ancillas at each step, for example ρ 0 = ρ S ⊗ ρ e . As shown in Figure 3.1, all encoding ancillas are entangled to 50 recovery ancillas prior to the operation. Regular unentangled encoding and recovery ancillas could also be considered in the procedure. However, regular recovery ancillas are redundant in improving the error correction fidelity, and regular encoding ancillas, if added, do not provide any extra insights into our argument. Hence, we omit them for simplicity. Using the OSR for each of these operations the overall evolution can be described as ρ T = c,e,r (R r E e C c )ρ 0 (R r E e C c ) † , where R r , E e , and C c are the Kraus operators of the recovery, error and encoding chan- nels. The number of these operators for each map depends on the manner of implemen- tation and the basis representation [Nielsen and Chuang, 2000]. In fact any OSR can be equivalently expressed as a unitary with ancilla states. In this formalism, the encoding operatorC is considered as a unitary operator because it represents the actual unitary U C acting on the (tensor) product of system state ρ 0 and the ancillas’ state. The encoded state is therefore produced by ρ C = U C (ρ 0 ⊗|BB|)U † C , where|B is the state of the ancillas, which are assumed to be prepared in a maximally entangled state. Since the encoding does not act on the second half of the entangled pairs, the recov- ery ancillas, the encoding operatorC should be represented asC n C ⊗ I ( n A 2 ) . Here n A is the dimension of the ancillas’ space and n C = n S ( n A 2 ) where n S is the dimension of the initial input state. The purpose of optimization here, similar to the previous chapter, is to design the encoding C and recovery R for a given error channel E such that the overall map ρ S → ρ T is as close as possible to our desired unitary operator. Again we consider the channel fidelity [Reimpell and Werner, 2005b] as our performance measure, and we 51 maximize the fidelity measured between the error correction operation REC and our desired unitary operation L S : f = 1 n 2 S r,e Tr L † R r E e C 2 , (3.1) where L is our desired unitary operator. As shown in (2.14) maximizing this fidelity is equivalent to minimizing the following distance: d = r,e R r E e C− δ re L 2 , (3.2) where X 2 =TrX † X is the Frobenius norm, and δ re are constants satisfying the condition r,e |δ re | 2 =1. Again we can rewrite this distance as d =RE(I⊗ C)− ∆ ⊗ L 2 where ∆ ≡ [δ re ], and E is the rectangular error matrix built using the error Kraus operators, E =[E 1 ...E m E ].R is the matrix achieved by stacking the Kraus operators R r as R =       R 1 . . . R m R       , and I is the identity matrix. Therefore, we have ∆ 2 = Tr ∆ † ∆= r,e |δ re | 2 =1, and R † R = r R † r R r = I C . Considering all the constraints above, our optimization problem can be summarized as: minimize d(R,C, ∆) subject to R † R = I, C = C ⊗ I n A 2 ,C † C = I, ∆ 2 =1 (3.3) 52 There are three parameters in this expression (R,C, ∆ ) that are to be optimized. In order to identify the optimal values of these parameters we use a similar iterative algorithm. Starting with an arbitrary initial encoding operator, at each step we find the optimized recovery operator for the encoding operator obtained in the previous step and vice versa. ∆ is an intermediary parameter that must also be recalculated at each step. This iteration continues until the distance (3.2) stops decreasing. The details of the operations at each step are provided in the next two sections. The design of the optimized recovery operator at each step is not affected by the extra constraints added by introducing entanglement in the error correction procedure. This is because the recovery operator optimized for a general given encoding operator is also optimized for encoding operators of the form C = C ⊗ I. However, these extra constraints play an important role in identifying the optimized encoding operator which has to be in the desired format of C = C ⊗ I. In addition, the encoding ancillas here have to be initialized in a maximally entan- gled state, for example as an EPR pair. However, our distance measure defined in (3.2) is independent of the initial state of the qubits. In order to overcome this problem we add an extra step, an entangling operator U, to the procedure. By adding this entangling operator, all ancillas can be considered to have an initial state of|0. 3.2.1 Optimized Encoding Operator By adding the entangling operator, the distance measure (3.2) can be written as d = r,e R r E e CU− δ re L 2 where L is our desired unitary operator, and U is the entangling operator. Using the definition of Frobenius norm, we can express this distance as 53 d = r,e Tr{(U † C † E † e R † r − δ ∗ re L † )(R r E e CU− δ re L)} =Tr{(U † C † CU)− 2 r,e δ ∗ re L † R r E e CU} + K where the term K does not depend on C. Hence, the optimized encoding can be achieved by solving: minimize d=Tr{(U † (C † ⊗ I)(C ⊗ I)U)− 2 r,e δ ∗ re L † R r E e (C ⊗ I)U} subject to C † C = I To minimize this distance, we form the Lagrangian, L=Tr{(U † (C † ⊗ I)(C ⊗ I)U)− 2 r,e δ ∗ re L † R r E e (C ⊗ I)U}+TrP(C † C − I) where P is the Lagrange multiplier matrix. Considering that Tr(C † C ⊗ I)= dim I Tr(C † C ), ∂L ∂C =2dim I C +2PC − 2Tr 1 { re δ re U † (R r E e ) † L}=0 where the trace is over the space that C acts on. The solution to this equation is: C = B(B † B) − 1 2 (3.4) where B =Tr 1 { re δ re U † (R r E e ) † L}. This encoding operator is optimized for the given recovery operator R. 54 3.2.2 Optimized Recovery operator The derivation of the optimized recovery operator is essentially similar to what was done previously in [Taghavi et al., 2010] except that we should consider the entangling oper- ator. Here we briefly provide the required steps for optimizing the recovery operator. Using the constraints in (3.3), we can express the distance in (3.2) as d =RE(I E ⊗ CU)− ∆ ⊗ L S 2 = n S +Tr E(I E ⊗ CUU † C † )E † − 2ReTrRE(∆ † ⊗ CUL † S ). (3.5) Minimizing this expression with respect to R is equivalent to maximizing the last term. As shown in [Taghavi et al., 2010] this maximization results in max R † R=I C ReTr RE(∆ † ⊗ CUL † S )=Tr E(Γ⊗ CUU † C † )E † where the matrix Γ is defined as, Γ=∆ † ∆ . (3.6) The constraint ∆ 2 =1 from (3.3) is equivalent to TrΓ = 1, and Γ ≥ 0 by definition. Therefore, the optimization problem for Γ is maximize Tr E(Γ⊗ CUU † C † )E † subject to Γ≥ 0, TrΓ = 1 (3.7) The optimized Γ can be obtained by solving an equivalent SDP (semidefinite pro- gramming) problem [Taghavi et al., 2010].The matrix ∆ can be obtained from Γ by 55 using the definition 3.6. Once the optimized ∆ is known, the optimized recovery matrix can be achieved: R=[v 1 ... v n C ][u 1 ... u n C ] † (3.8) where v i ,u i are, respectively, the right and left singular vectors in the singular value decomposition of the matrix E(∆ † ⊗ CUL † S ), with the singular values in descending order. The algorithm below summarizes the preceding method for encoding and recovery optimization in the presence of entanglement between the encoding and recovery blocks. Initialize C Repeat a) Optimal recovery solve (3.7) for Γ obtain ∆ via (3.6) obtain R via (3.8) b) Optimal encoding obtain C via (3.4) Until distance (3.2) stops decreasing Since in each iteration of this algorithm the distance d only decreases the converged solution to this optimization is guaranteed to be a local optimal solution to (3.3). 56 3.3 Random Unitary Channels An error channelE is called a random unitary channel if we can decompose it into the probabilistic application of one of a finite set of unitary operations: E(ρ)= n i=1 p i V i ρV † i , (3.9) where V i ,i=1,...,n are unitary operators and p i ,i=1,...,n is a probability distribu- tion. Random unitary channels describe the noise processes that can be corrected using classical information extracted from the environment [Gregoratti and Werner, 2003]. It is clear that any random unitary channel should be unital, meaning thatE(I)=I. How- ever, the inverse relation holds only for channels on qubits, and is not true for larger dimensions [Tregub, 1986, Kummerer and Maassen, 1987]. There are two operations in (2.1) that are not random unitary. These operations are the introduction of the ancillas and the partial trace over the environmentB. Therefore, an error channel may not be written as random unitary (3.9) in general. The necessary and sufficient conditions for a channel to be a random unitary are discussed in [Aude- naert and Scheel, 2008]. Buscemi finds an upper bound on the number of unitaries needed for a random unitary in (3.9) [Buscemi, 2006]. Here we show that if the number of unitary operators required in (3.9) is less than or equal to two, our optimized error correction method can perfectly correct the error using only one pair of ebits as ancilla qubits. In fact, for this case, the optimized encoding and recovery operators are known and have an interpretation based on teleportation. To see this, consider a channel that can be decomposed into two unitaries, V 1 and V 2 , as follows: 57 E(ρ)=(1− p)V 1 ρV † 1 + pV 2 ρV † 2 (3.10) Suppose Alice wants to send Bob a message, which is stored in one qubit of data, through this noisy channel. Here we explain how one pair of entangled qubits enables Alice and Bob to make this communication error free. This pair of qubits was already prepared by an entanglement source and sent to Alice and Bob, each taking one of the qubits. Alice can apply an encoding operator on her qubits, the data qubit and one of the ebits, before sending them to Bob through the noisy channel. Bob applies the recovery operator on all qubits once he receives the message. The optimized encoding operator that Alice can use in this scenario is achieved by first applying V † 1 on her qubits to make the error channel equivalent to E(ρ)=(1− p)IρI + pV 2 V † 1 ρV 1 V † 2 (3.11) Alice continues the encoding operation by applying: C = 1 2 (|v 1 v 1 B 1 | +|v 1 v 2 B 2 | +|v 2 v 1 B 3 | +|v 2 v 2 B 4 |) (3.12) where v 1 and v 2 are the eigenvectors of the unitary V 2 V † 1 , and B 1 to B 4 are some orthonormal basis of the maximally entangled space, for example Bell states ( 1 √ 2 (|00± |11) and 1 √ 2 (|10±|01)). This encoding of data enables us to pass two bits of classical information safely through the noisy channel. These two bit of classical information together with the second half of the ebit enables us to recover the original state. To see this more precisely suppose that the initial state of the data qubit is |φ i = a|0 + b|1 58 Figure 3.2: Quantum error correction using teleportation The state of the whole system after applying the entangling unitary is |ψ i = 1 √ 2 {a(|000 +|011)+ b(|100 +|111)} By applying the encoding (3.12) and error map on the first two qubits, we can find the final state of the entire system as follows: |ψ tot = 1 2 √ 2 (|v 1 v 1 (a|0+b|1)+|v 1 v 2 (a|0−b|1)+|v 2 v 1 (a|1+b|0)+|v 2 v 2 (a|1−b|0)) This is the state of the system once Bob receives it. At this point as shown in Figure 3.2 he can recover the original state|φ i by measuring the state of the first two qubits. Based on the result of the measurement he applies the appropriate Pauli operator to the last qubit, second half of the ebit, to recover the original state. For example if he measures the first two qubits as|v 2 v 1 , he should apply σ X to the last qubit to recover the original state. 59 3.3.1 Example 1 As an example of the random unitary channels with two unitary operators we consider the bit flip error. The error, σ X =    01 10    , which is occurring independently on each qubit can be represented as E(ρ)=(1− p)IρI + pσ x ρσ x . As shown in Figure 3.3, this error can be corrected using an ebit as the error correct- ing resource. The ebit is again shared between the encoding and recovery parts. The encoding acts on the initial data qubit and the first entangled qubit. Considering that the error is already presented in the form (3.11) we do not need to apply the initial unitary operator here. Therefore, the optimized encoding operator based on (3.12) is C =|++B 1 | +|+−B 2 | +|−+B 3 | +|−−B 4 | , where|+ = 1 √ 2 (|0 +|1) ,|− = 1 √ 2 (|0−|1) and B 1 to B 4 form an orthonormal basis of the space of maximally entangled states. The first entangled qubit together with the initial data qubit are corrupted by the bit flip error while the other entangled bit stays intact. The recovery operation acts on all qubits and can reproduce the initial state by measuring the first two qubits as has been discussed above. 3.3.2 Example 2 In this example both bit flip and phase flip errors occur with equal probabilities: E(ρ)=(1− p)IρI + p 2 σ x ρσ x + p 2 σ z ρσ z . 60 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f Bit flip , n s =2, n CA =2, 1 ebit no ebit no recovery 1 ebit p Figure 3.3: Fidelity of error correction using entanglement for bit flip error. The error occurs independently on different qubits. This error channel clearly can not be represented in the form of (3.10). Hence our teleportation argument does not apply here. While the entanglement cannot make the error correction perfect in this case, it can still pass useful information about the initial state to the recovery block, and therefore increases the fidelity of the error correction. Figure 3.4 presents the fidelity of error correction for this channel for different values of p. The difference between the green and red plot is only in the presence of entanglement. Remember that without entanglement the extra qubit used in the recovery would act as a regular recovery ancilla, and therefore it could not increase the fidelity. The minimum fidelity in both cases occurs at p = 2 3 , where the channel is symmetric about I, σ X , and σ Z :E(ρ)= 1 3 (IρI + σ x ρσ x + σ z ρσ z ). 3.3.3 Depolarizing Channel In this example we consider a particular simple model which is extensively used to describe noise effacing quantum systems, a depolarization channel [Nielsen and 61 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f weight 2 Adepolarizing: only σ x and σ z no ebit no R 1 ebit p Figure 3.4: Fidelity of the error correction for bit-phase flip error with one entangled pair of qubits as ancillas. Chuang, 2000]. This channel is an important class of quantum maps occurring when the coherence in the process vanishes naturally or via engineered quantum operations [Briegel et al., 1998]. It employs unbiased noise generating bit flip and phase flip errors and is represented by: E(ρ)=(1− p)IρI + p 3 σ x ρσ x + p 3 σ y ρσ y + p 3 σ z ρσ z , (3.13) where σ x , σ y and σ z are the Pauli operators. The depolarizing channel shrinks the radius of the Bloch sphere by a factor of 1− p, while preserving its shape. In this particular case of error channel the ebit does not increase the fidelity. This is due to the highly symmetric property of the channel that makes all output states unitarily equivalent. The optimized fidelity of the channel with and without using the ebit is pre- sented in Figure 3.5. Note the break point which occurs at p = 3 4 , when the coefficients in (3.13) become equal. The fidelities of both cases are the same forp< 3 4 , but the ebit can slightly increase the fidelity forp> 3 4 . This is due to the asymmetric property of the 62 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f weight 2 depolarizing no ebit 1 ebit no R p Figure 3.5: Optimized fidelity for depolarizing channel. channel in this range of probability. We can see this by multiplying (3.13) by one of the Pauli operators, for example by σ x , E(ρ)= p 3 IρI +(1− p)σ x ρσ x + p 3 σ z ρσ z + p 3 σ y ρσ y . This equivalent channel clearly is not symmetric with respect to the Pauli operators. 3.4 Conclusion The optimization method presented here returns the optimized fidelity for a given noise channel in the presence of entanglement between the encoding and recovery block. We showed that this entanglement can substantially increase the error correction fidelity in most cases. This fidelity increase can lead to perfect error correction for error channels with specified characteristics. In these cases, the error correction procedure can be inter- preted using quantum teleportation. For the cases in which perfect error correction is not possible, our error correcting method can automatically return the optimized fidelity. 63 Considering that regular recovery ancillas are redundant in the error correction pro- cedure for given noise channels, this result shows how information about the encoded data transferred to the recovery block through entanglement with the encoding can increase the fidelity. 64 Chapter 4 Error Protection for Linear Maps 4.1 Introduction A key assumption I considered in the previous chapters is that the evolution of the quantum systems can be described using completely positive (CP) maps [Nielsen and Chuang, 2000, Kraus, 1983]. A linear mapE is called CP if it is positive (E≥ 0) and E⊗ I n ≥ 0∀n ∈ Z + , where I n is the n-dimensional identity operator. CP maps can always be represented in terms of Kraus operators as discussed in section 2.2. The allure of CP maps is that they are positive for any initial state ρ(0). However, the assumption of CP maps introduced in previous chapters is unnecessary, and in fact, if the initial total system state is entangled, the dynamics of the system can not be described by CP maps. This dynamics given a general initial state can always be represented as a linear, Hermitian map [Shabani and Lidar, 2009a]. A remarkable theory was developed in recent years to address the problem of deco- herence in quantum information processing tasks. Decoherence-free subspaces [Zanardi and Rasetti, 1997, Lidar et al., 1998, 2001] and subsystems [Knill et al., 2000, Kempe et al., 2001] are among the methods which have been proposed. However, these theories are based on the assumption that the system’s evolution is completely positive. Consid- ering that this assumption is not accurate in general, one is naturally led to ask about the impacts of linear maps in the formalism of decoherence free subspaces and subsystems. Here we study these concepts in the framework of linear maps. 65 4.2 Linear maps Given an arbitrary initial total system state, the dynamics of the system can be charac- terized using linear maps. A mapE is linear if and only if it can be represented as E(ρ S )= α E α ρ S E † α . (4.1) where{E α ,E α } are the “left and right Kraus operators”, and ρ S is the system density matrix. Note thatE is trace preserving if α E † α E α =I. (4.2) Canonical form representation The density matrix may also be treated as a vector − → ρ . In this case the linear superopera- tor is represented by a matrix L. By reshuffling L we get another matrix which is called “dynamical matrix”: Dmµ nν = Lmn µν Consider a singular value decomposition of D D = k i=1 λ i X i X † i where k is the rank of the matrix D. By reshaping the singular vectors to make them N× N matrices, and also defining F i = √ λ i X i and F i = √ λ i X i , the mapE becomes: ρ S = i F i ρ S F † i 66 This representation is called canonical form, and the operators{F i ,F i } are the set of canonical operation elements. Definition 1. Let the matrices u and v satisfy uv † = I. An operator sum representation of a linear mapE, i E i ρE i , is called a “non-redundant representation” if E i = j u ij F j and E i = j v ij F j where{F j ,F j } are the set of canonical operation elements and their number is equal to the rank of the dynamical matrix of the mapE. 4.3 Noiseless Subsystems 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 subspace H out may itself have a tensor product structure, i.e., additional factors similar toH NS may be contained in it, 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. Also, suppose{E i ,E i } is a non-redundant representation of the noise map which is acting on the system. We allow for the most general situation of a system that is not necessarily initially NS. Define projectors as follows: P NS−in = I NS ⊗I in 0 , (4.3) P d =    I NS ⊗I in 0 00    , P d ⊥ =    00 0I NS ⊗I in    . (4.4) 67 The system density matrix takes the corresponding block form ρ S =    ρ NS−in ρ ρ † ρ out    , (4.5) and note that P NS−in ρ S P † NS−in = I NS ⊗I in 0    ρ NS−in ρ ρ † ρ out       I NS ⊗I in 0    = ρ NS−in . (4.6) Definition 2. Let the system Hilbert space H S decompose as H S = H NS ⊗H in ⊕ H out , and partition the system state ρ S accordingly into blocks, as in (4.5). Assume ρ NS−in (0) =P NS−in ρ S (0)P † NS−in =0. Then the factorH NS is called a decoherence-free (or noiseless) subsystem if the following condition holds: Tr in {ρ NS−in (t)} =U NS Tr in {ρ NS−in (0)}U † NS , (4.7) whereU NS is a unitary matrix acting onH NS . Definition 3. Perfect initialization (DF subsystems): ρ =0 and ρ out =0 in (4.5). Definition 4. Imperfect initialization (DF subsystems): ρ and/or ρ out in (4.5) are non- vanishing. According to Definition 2, a quantum state encoded into theH NS factor at some time t is unitarily related to the t=0 state. The factorH in is unimportant, and hence is traced 68 over. Clearly, a NS reduces to a DF subspace whenH in is one-dimensional, i.e., when H in = C. Using (4.6), the NS definition (4.7) becomes, in terms of the full state: Tr in { α P NS−in E α ρ S (0)E † α P † NS−in }=Tr in {U⊗I P NS−in ρ S (0)P † NS−in U † ⊗I}. (4.8) Let us represent the error 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 ρ S =    ρ 1 ρ 2 ρ † 2 ρ 3    , (4.9) E α =    P α A α D α B α    ,E α =    P α A α D α B α    , (4.10) with appropriate normalization constraints, considered below. Equation (4.8), in this matrix form, becomes: Tr in { α P α ρ 1 P † α +P α ρ 2 A † α +A α ρ † 2 P † α +A α ρ 3 A † α }=Tr in {U⊗Iρ 1 U † ⊗I}. (4.11) The following theorems present the necessary and sufficient conditions for a subsys- tem to be decoherence free. 69 Theorem 1. 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 linear maps iff the Kraus operators have the matrix representation E α =    U⊗C α 0 0B α    ,E α =    U⊗C α 0 0B α    (4.12) Theorem 2. Assume perfect initialization. Then the Kraus operators have the relaxed form E α =    U⊗C α A α D α B α    ,E α =    U⊗C α A α D α B α    (4.13) where α C † α C α =I in and α D † α D α =0. 4.4 Proofs of the theorems 4.4.1 Sufficiency Imperfect Initialization For notational simplicity let us write the system density matrix as in (4.9), with ρ 1 = r aba b |ab|⊗|a b |, (4.14) 70 where{|a} and{|a } are bases forH NS andH in respectively. For a proof of sufficiency we must show that our NS condition (4.11) is satisfied, where now by assumptionP α = U⊗C α ,P α =U⊗C α ,A α =A α =0, our NS condition (4.11) reduces to: Tr in { α U⊗C α ρ 1 U † ⊗C † α }=Tr in {U⊗Iρ 1 U † ⊗I}. (4.15) Substituting, we find for the LHS of (4.15): Tr in { α U⊗C α ρ 1 U † ⊗C † α } = αi aba b r aba b U|ab|U † i |C α |a b |C † α |i = αaba b r aba b U|ab|U † i b |C † α |i i |C α |a = aba b r aba b U|ab|U † b | ! α C † α C α " |a = aba b r aba b U|ab|U † δ a b = aba r aba a U|ab|U † . (4.16) On the other hand the RHS of (4.15) is: Tr in {U⊗Iρ 1 U † ⊗I} = i aba b r aba b U|ab|U † i |a b |i = i ab r abi i U|ab|U † , which equals the last line in (4.16), as required. 71 Perfect Initialization In this case ρ S (0) =P d ρ S (0)P d =    ρ 1 0 00    , i.e., ρ 2 =0 and ρ 3 =0. Then our NS condition (4.11) reduces again to (4.15), which we already proved. The CP map case Since the CP case is obtained by dropping the prime subscript on all noise operatorsE α , both proofs above also apply for the case of completely positive maps. Note that in this case the condition α D † α D α =0 means that D α =0. 4.4.2 Necessity Imperfect Initialization We now wish to show that (4.11) implies (4.12) for any non-redundant representation, i.e., thatP α = U⊗C α ,P α = U⊗C α , andA α = A α =0. To derive constraints on the various terms in (4.11) we consider various special cases, which yield necessary conditions. First, consider an initial state ρ S (0) such that ρ 2 = 0 but ρ 3 = 0. As the LHS of (4.11) is independent from ρ 3 , and ρ 3 is arbitrary, the last term must vanish: α A α ρ 3 A † α =0 =⇒      ifA α =0 thenA α =0 ifA α =0 thenA α =0 . (4.17) 72 Next, consider an initial state ρ S (0) such that ρ 3 =0 but ρ 2 =0. As the LHS of (4.11) is independent from ρ 2 , and ρ 2 is arbitrary, the middle term must vanish, which together with (4.17) yields α P α ρ 2 A † α +A α ρ † 2 P † α =0 =⇒      ifA α =0 thenP α =0(sinceA α =0) ifA α =0 thenP α =0(sinceA α =0) . (4.18) We see that (4.18) implies that whenever eitherA α =0 orA α =0, the corresponding summand (with index α) in (4.11) vanishes. Therefore, without loss of generality we can restrict our attention to the caseA α =A α =0. Thus what is left of (4.11) is Tr in {U⊗Iρ 1 U † ⊗I}=Tr in { α P α ρ 1 P † α }. (4.19) The remaining task is to show thatP α = U⊗C α , P α = U⊗C α . Assume ρ 1 = |ii|⊗|i i |. Note that the partial matrix elementj |P α |i is an operator on theH NS factor,|ii|. Then (4.19) reduces to Tr in {U|ii|U † ⊗|i i |}=Tr in { α P α |ii|⊗|i i |P † α } =⇒|ii| = α,j P αj i |ii|(P αj i ) † , (4.20) whereP αj i ≡ U † j |P α |i andP αj i ≡ U † j |P α |i . The last equality means that the linear mapP i = {P αj i ,P αj i } is the identity map,P αj i ∝ P αj i ∝ I. Since |i and|j are arbitrary this condition implies that P α = U ⊗ C α and P α = U ⊗ C α . Substituting this result in (4.19), we will have: Tr in {ρ 1 }=Tr in { α (I⊗ C α )ρ 1 (I⊗ C † α )} (4.21) 73 Further suppose ρ 1 = iji j a iji j |ij|⊗|i j |, and use ( 4.21) to obtain iji j a iji j |ij| = iji j k ,α a iji j |ij|k |C α |i j |C † α |k = iji j k ,α a iji j |ij|j |C † α |k k |C α |i = iji j a iji j |ij|j | α C † α C α |i from which it follows that α C † α C α = I. (4.22) Considering the normalization constraint, E † α E α = I, together with the new addi- tional constraint, we can derive: α P † α P α + α D † α D α = I NS ⊗ I in ⇒ I NS ⊗ α C † α C α + α D † α D α = I NS ⊗ I in but α C † α C α = I, thus α D † α D α =0, (4.23) which finalizes all the conditions required in the corollary. Perfect Initialization In this case ρ s (0) =P d ρ S P † d , we have ρ 2 = ρ 3 =0, and (4.11) reduces to: Tr in { α P α ρ 1 P † α }=Tr in {U⊗Iρ 1 U † ⊗I} 74 Thus, the argument leading to the vanishing of the A α and A α does not apply any- more. However the other argument which leads to P α = U⊗ C α and P α = U⊗ C α still holds. By substituting this result in (4.19), and with a similar calculation, the constrain α D † α D α =0 can be found. 4.5 Conclusion Here we studied quantum error protection methods, noiseless subspaces and subsys- tems, in the framework of linear maps, which provides a more general description of open quantum system dynamics. We introduced the “non-redundant representation” of a linear map based on Kraus operators. Using this representation, we characterized the necessary and sufficient conditions under which a subsystem is decoherence-free (or noiseless) with respect to linear maps. These conditions, as expected, are more relaxed than the required conditions already known for completely positive maps. 75 Chapter 5 Conclusion We started with a presentation of quantum robots as quantum systems that move in, and interact with, an external environment of quantum systems. We showed how such robots, armed with a set of basic operators, can perform an arbitrary Hamiltonian on the particles and therefore are universal. The main part of this thesis is devoted to a computational approach to the quantum error correction problem. The problem was previously addressed by focusing on finding procedures for perfect recovery of quantum states passing through noisy channels. We showed that the optimization method presented here yields codes which achieve a higher fidelity than such standard procedures. We also characterized the error correction fidelity with regard to the number of encoding and recovery ancillas. While the fidelity is an increasing function of the num- ber of encoding ancillas, it is indifferent to the number of regular recovery ancillas. 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If V is partitioned as V =[V 1 V 2 ] with V 1 being n RA n C × n C then the objective function in (2.19) becomes, ReTr RW =ReTr S 0 X, X = V † 1 RU (5.1) SinceX≤ 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 † , which is precisely the result in (2.21). This also establishes that the optimal objective function isTr S 0 which, by definition, is equal toTr √ WW † , thus max R † R=I C ReTr RW =Tr √ WW † (5.2) which establishes (2.19). Condition (2.11) follows directly from (2.7) by multiplying both sides by their respective conjugate (with indices c and c ) which also eliminates R because R † R = I C . This immediately establishes that (2.11) is a necessary condition for (2.7). To prove sufficiency, first expand (2.8) to get, d = c Tr (I E ⊗ C † c )E † E(I E ⊗ C c ) +Tr Γ c ⊗ I S − 2ReTr RE(∆ † c ⊗ C c L † S ) Γ c =∆ † c ∆ c (5.3) 81 From (5.2), we get, min R † R=I C d = c [Tr (I E ⊗ C † c )E † E(I E ⊗ C c ) +Tr Γ c ⊗ I S ]− 2Tr √ WW † W = c E(∆ † c ⊗ C c L † S ) (5.4) Using (2.11) we get,Tr √ WW † = c E(I E ⊗ C c C † c )E † . This, together with repeated uses of (2.11) shows that min R † R=I C d=0. Since d is a norm, and is zero, then so is its argument, which by definition establishes (2.7) and thus shows sufficiency of (2.11). 82 Appendix B Unitary freedom in (2.19) In (2.19),Γ=∆ † ∆ remains unchanged if ∆ is multiplied by a unitary. This unitary freedom is exactly the unitary freedom in describing the error map OSR. To see this, recall again from [Nielsen and Chuang, 2000, Thm.8.2] that two error maps with OSR elements E =[E 1 ...E m E ] and F =[F 1 ...F m E ] are equivalent if and only if E i = j W ij F j where the m E × m E matrix W is unitary. Equivalently from (5.3), E = F(W ⊗ I C ). Substituting this for E into the left hand side of (2.19) gives, ReTr RE(∆ † ⊗ CL † S )=ReTr RF((∆ † ⊗ CL † S ) (5.5) with ∆ =∆ W † . Hence, ∆ † ∆ = W∆ † ∆ W † =Γ, which establishes the claim. 83

Abstract (if available)

Abstract Two subjects in the area of quantum computation are considered here. In the first chapter I present a universal model for a quantum Robot. Chapters two, three, and four are dedicated to the problem of quantum error correction/protection.

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Creator Taghavi, Soraya (author)

Core Title Quantum computation and optimized error correction

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2010-05

Publication Date 04/14/2010

Defense Date 03/04/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag convex optimization,OAI-PMH Harvest,quantum error correction

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lidar, Daniel A. (committee chair), Brun, Todd A. (committee member), Haas, Stephan (committee member)

Creator Email soraya_taghavi@yahoo.com,taghavi@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2922

Unique identifier UC1488143

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Document Type Dissertation

Rights Taghavi, Soraya

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu