Quantum error correction and fault-tolerant quantum computation

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Content QUANTUM ERROR CORRECTION AND FAULT-TOLERANT QUANTUM COMPUTATION by Ching-Yi Lai A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2013 Copyright 2013 Ching-Yi Lai To my parents Rong-Chang Lai and Mei-Hui Huang ii Acknowledgments I would like to thank my advisor, Todd Brun, for oering a great learning environment. He is always kind and patient with my questions. His deep knowledge and thoughtful suggestions have helped me greatly in research. I am also grateful to Daniel Lidar and Ben Reichardt for serving in both my candidacy and thesis committees. I am equally thankful to Giuseppe Caire and Aiichiro Nakano for being in my candidacy committee. Their comments and questions helped me to better my thesis and presentation skills. I cannot thank them enough for teaching me in many courses on quantum error correction, quantum algorithm, information theory, and computational physics, respectively. I also want to thank my collaborators Mark Wilde, Alexei Ashikamin, Martin Suchara and Gerardo Paz for valuable discussions. It was a wonderful experience to collaborate with them, which stimulated me to work harder. I am especially grateful to Mark for comments and suggestions on my articles and research career. I also thank everyone in the group, Yicong Zheng, Kung-Chuan Hsu, Jos e Raul Gonzalez Alonso, Scout Kingery, Jan Florjanczyk, Shesha Raghunathan, Bilal Shaw, Martin Varbanov, and Christopher Cantwell. Yicong has always being patient in teaching me physics and explaining other research topics to me. Finally I would like to thank my parents, my brother, and my friends for their support and company. I won't be able to persist in my research without them. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vii List of Figures viii Abstract xi I Introduction 1 Chapter 1: Preliminaries 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Quantum Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Stabilizer Group and Space Decomposition . . . . . . . . . . . . . 7 1.3.2 Cliord Encoder and the Check Matrix . . . . . . . . . . . . . . . 8 1.3.3 Syndrome Representatives and Decoding . . . . . . . . . . . . . . 11 1.4 The Basics of Entanglement-Assisted Quantum Error-Correcting (EAQEC) Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Fault-Tolerant Quantum Computation . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Shor's syndrome extraction . . . . . . . . . . . . . . . . . . . . . . 19 1.5.2 Steane's Syndrome Extraction . . . . . . . . . . . . . . . . . . . . 21 II Entanglement-Assisted Quantum Codes 22 Chapter 2: Duality of Entanglement-Assisted Quantum Codes 23 2.1 The Dual Codes of EAQEC Codes . . . . . . . . . . . . . . . . . . . . . . 24 2.2 The MacWilliams Identity and the Linear Programming Bounds . . . . . 28 2.3 Linear Programming Bounds for EAQEC Codes . . . . . . . . . . . . . . 32 2.3.1 The Linear Programming Bounds for Maximal-Entanglement Quan- tum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 iv 2.3.2 The Linear Programming Bound for Non-Maximal-Entanglement Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Improving the Linear Programming Bounds . . . . . . . . . . . . . . . . . 36 2.5 Bounds on EAQEC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.1 Gilbert-Varshamov Bound for EAQEC Codes . . . . . . . . . . . . 42 2.5.2 Maximal-Entanglement EAQEC Repetition and Accumulator Codes 43 2.5.3 Existence of Other EAQEC Codes . . . . . . . . . . . . . . . . . . 52 2.5.4 The Plotkin bound for EAQEC Codes . . . . . . . . . . . . . . . . 53 2.5.5 Table of Lower and Upper Bounds on the Minimum Distance of Maximal-Entanglement EAQEC Codes . . . . . . . . . . . . . . . 55 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 3: A Construction of Entanglement-Assisted Quantum Error-Correcting Codes: Random Optimization 58 3.1 The Encoding Optimization Procedure for EAQECCs . . . . . . . . . . . 59 3.1.1 Selecting Symplectic Partners and Logical Operators . . . . . . . . 60 3.1.2 Unitary Row Operators . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.1 Results of the Encoding Optimization Procedure . . . . . . . . . . 68 3.2.2 Random Optimization Procedure . . . . . . . . . . . . . . . . . . . 71 3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 4: Random EAQEC Codes 75 4.1 The Weight Enumerator Bound on the Block Error Probability under Maximum A Posteriori Decoding . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Hashing Bounds for Pauli Channels . . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 Hashing Bound for Stabilizer Codes . . . . . . . . . . . . . . . . . 83 4.2.2 Entanglement-Assisted Quantum Error-Correcting Codes . . . . . 88 4.2.2.1 Maximal-Entanglement Codes . . . . . . . . . . . . . . . 88 4.2.2.2 Non-Maximal-Entanglement Codes . . . . . . . . . . . . 89 4.2.2.3 Entanglement-Assisted Codes with Imperfect Ebits . . . 90 4.2.3 EAQEC Codes for Classical Communication . . . . . . . . . . . . 92 Chapter 5: EAQEC Codes with Imperfect Ebits 94 5.1 Determining the Syndrome Representatives for EAQEC Codes . . . . . . 96 5.2 EAQEC Codes that are Equivalent to Standard Stabilizer Codes . . . . . 98 5.3 Quantum Codes with Two Encoders . . . . . . . . . . . . . . . . . . . . . 103 5.4 Channel Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.1 Formula for the Channel Fidelity over the Depolarizing Channel . 106 5.4.2 Channel Fidelity for EAQEC Codes . . . . . . . . . . . . . . . . . 110 5.4.3 Approximation of Channel Fidelity . . . . . . . . . . . . . . . . . . 114 5.5 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.6 EAQEC and Entanglement Distillation . . . . . . . . . . . . . . . . . . . . 117 5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 v III Fault-Tolerant Quantum Computation 127 Chapter 6: Performance and Error Analysis of Knill's Postselection Scheme in a Two-Dimensional Architecture 128 6.1 Basics of the Knill C 4 =C 6 Scheme with Postselection . . . . . . . . . . . . 130 6.2 The Two-Dimensional Qubit Layout of the C 4 Code . . . . . . . . . . . . 135 6.3 Error Analysis of the 2-Dimensional Knill Postselection Scheme . . . . . . 142 6.3.1 Error Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3.2 Pseudo-Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Chapter 7: Encoded Quantum Syndrome Measurement 149 7.1 Encoded Quantum Syndrome Measurement . . . . . . . . . . . . . . . . . 149 7.2 Using Fewer Ancillas in Syndrome Measurement . . . . . . . . . . . . . . 156 7.3 Error-Correcting Codes for Measurement Errors . . . . . . . . . . . . . . . 159 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Bibliography 163 vi List of Tables 2.1 Upper and lower bounds on the minimum distance of any [[n;k;d;nk]] maximal-entanglement EAQEC codes. . . . . . . . . . . . . . . . . . . . 56 3.1 Four types of unitary row operators . . . . . . . . . . . . . . . . . . . . . 64 3.2 Optimization over the [[7; 1; 3]] quantum BCH code . . . . . . . . . . . . . 69 3.3 Optimization over Shor's [[9; 1; 3]] code . . . . . . . . . . . . . . . . . . . . 70 3.4 Optimization over Gottesman's [[8; 3; 3]] code . . . . . . . . . . . . . . . . 71 3.5 Optimization over a [[15; 7; 3]] Quantum BCH code . . . . . . . . . . . . . 71 3.6 Optimization over the [[13; 1; 5]] quantum QR code . . . . . . . . . . . . . 72 5.1 Channel delity of dierent quantum codes in the depolarizing channel . . 110 6.1 The numbers of the quantum operations contained in each higher-level quantum operation of the C 4 code and its following error detection (1- Rec). Each entry represents the number of the elementary gate (U (0) ) corresponding to that row contained in the higher-level gate (U (1) ) corre- sponding to that column. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Comparison of the error thresholds of three concatenated codes. . . . . . . 145 6.3 Comparison of the 4 4 and 5 5 tiles. . . . . . . . . . . . . . . . . . . . 147 vii List of Figures 1.1 Transversal CNOT gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 Construction and verication of the Shor state. . . . . . . . . . . . . . . 20 1.3 The fault-tolerant syndrome measurement circuit of Steane's 7-qubit code for bit- ip error [PVK96][Sho96]. . . . . . . . . . . . . . . . . . . . . . . 20 1.4 The circuit for syndrome measurement of bit- ip errors by Steane's method. Note that the transversal CNOT gates are compressed into a single CNOT gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 (a) The encoder for a [[9; 1; 9; 8]] EA repetition code consists of a periodic cascade of CNOT gates. The encoder for arbitrary [[n; 1;n;n 1]] EA repetition codes extends naturally from this design. We can implement these encoders with a simple quantum shift-register circuit that uses only one memory qubit [Wil09]. (b) Considering the circuit in (a) but changing the information qubit to an ebit and all of the ebits to information qubits gives the encoder for the dual of the EA repetition code, namely, the [[9; 8; 2; 1]] EA accumulator code. This circuit naturally extends to encode the [[n;n 1; 2; 1]] EA accumulator codes. Simple variations of the above circuit encode the even n repetition and accumulator EAQEC codes, and we discuss them in Section 2.5.2. . . . . . . . . . . . . . . . . . . . . . . . 25 4.1 The gure plots both the EA hashing bound 11=2 [H 2 (p) +p log 2 3] from Ref. [BSST99] and the \asymptotic weight enumerator bound" from (4.3) as a function of the depolarizing parameter. The two bounds become close for high depolarizing noise. Interestingly, the thresholds of the maximal- entanglement EA turbo codes from Ref. [WH11] are just shy of the asymp- totic weight enumerator bound (see Figures 6(b) and 7(b) of that paper). 83 4.2 The gures plot the weight enumerator bound in (4.2) as a function of the depolarizing parameter p for various nite-length codes. (a) The weight enumerator bound for maximal-entanglement repetition codes of length 3 to 12. (b) The expected weight enumerator bound for random rate 1=n maximal-entanglement codes of length 3 to 12. (c) The weight enumera- tor bound for maximal-entanglement accumulator codes of length 3 to 12. (d) The expected weight enumerator bound for random rate (n 1)=n maximal-entanglement codes of length 3 to 12. Observe that the per- formance of the maximal-entanglement repetition and accumulator codes with respect to this upper bound is comparable to the expected perfor- mance of random maximal-entanglement codes. . . . . . . . . . . . . . . . 84 viii 5.1 (Color online) Contour plot of q 2 r 0 q 0 r 1 . The shaded region is where q 2 r 0 q 0 r 1 > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Illustration of the weight distributions of the syndrome representatives of the two decoding method. . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Comparison of two [[3; 1; 3; 2]] EAQEC codes in terms of channel delity. The [[3; 1; 3; 2]] AB code performs better in Region B, while the repetition code performs better in Region A. The region forp a < 0:015 andp b < 0:2 is enlarged on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4 (Color online) Contour plot of the dierence between the two delities: F C (D re 3 (T 3 pa T 2 p b )U re 3 )F C (D AB 3 (T 3 pa T 2 p b )U AB 3 ). The shaded region is where F C (D re 3 (T 3 pa T 2 p b )U re 3 )F C (D AB 3 (T 3 pa T 2 p b )U AB 3 )> 0. . . . 112 5.5 (Color online) The approximations of the channel delity of the [[11; 1; 5]] quantum code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 (Color online) The approximations of the channel delity of the [[24; 1; 8]] quantum code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.7 (Color online) Channel delity of dierent quantum codes for p b =p a = 0:99, 0:65, 0:5, 0:35, 0:1, 0:01. . . . . . . . . . . . . . . . . . . . . . . . . 123 5.8 (Color online) Contour plot of the dierence (F single F seq ) between decod- ing methods for the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code. The shaded region is where F single F seq > 0. . . . . . . . . . . . . . . . . . . . . . . 124 5.9 (Color online) Comparison of the [[8; 3; 3]] distillation protocol plus [[5; 2; 3; 3]] EAQEC code, and the [[10; 2; 7; 8]] EA EAQEC code without distillation. The performance of the [[10; 2; 7; 8]] AB EAQEC code is better forp a < 0:45.124 5.10 (Color online) Comparison of the [[5; 1; 3]] distillation protocol plus [[4; 1; 3; 1]] EAQEC code, and the [[6; 1; 5; 5]] EA EAQEC code without distillation. The performance of the [[6; 1; 5; 5]] AB EAQEC code is slightly better for p a < 0:11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.11 (Color online) Comparison of dierent distillation protocols with the same [[4; 1; 3; 1]] EAQEC code. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.12 (Color online) Comparison of dierent combinations of distillation proto- cols and EAQEC codes that encode one information qubit. . . . . . . . . 126 6.1 Knill Syndrome extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2 State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.3 Circuits for fault-tolerant quantum error detection. . . . . . . . . . . . . . 133 6.4 The circuit for implementing the logical S gate. . . . . . . . . . . . . . . . 133 6.5 The circuit for implementing the logical T gate. . . . . . . . . . . . . . . 134 6.6 The circuit for preparing the logical statej+ii. . . . . . . . . . . . . . . . 134 6.7 The decoding circuit for C 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.8 The distillation circuit for the statej +ii. . . . . . . . . . . . . . . . . . . 134 6.9 The twirl operation for the statej +ii. . . . . . . . . . . . . . . . . . . . . 134 6.10 The 1-exRec of the CNOT gate. . . . . . . . . . . . . . . . . . . . . . . . 142 6.11 The 1-exRec of the CNOT gate followed by two perfect ED + . . . . . . . . 143 ix 7.1 P se for encoded syndrome measurement, repetitive syndrome measure- ment, encoded syndrome measurements with [7; 3; 4] simplex code and with [10; 3; 5] code. Note that the plot for the [10; 3; 5] is the upper bound on its P se . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 The circuit for syndrome measurement of four data blocks with three ancilla blocks by a classical [3; 1; 3] repetition code. . . . . . . . . . . . . 159 x Abstract Quantum computers need to be protected by quantum error-correcting codes against decoherence. One of the most interesting and useful classes of quantum codes is the class of quantum stabilizer codes. Entanglement-assisted (EA) quantum codes are a class of stabilizer codes that make use of preshared entanglement between the sender and the receiver. We provide several code constructions for entanglement-assisted quantum codes. The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We nd new constraints on the simplied stabilizer group and the logical group, which help improve the linear programming bounds on entanglement- assisted quantum codes. The results also can be applied to standard stabilizer codes. In the real world, quantum gates are faulty. To implement quantum computation fault-tolerantly, quantum codes with certain properties are needed. We rst analyze Knill's postselection scheme in a two-dimensional architecture. The error performance of this scheme is better than other known concatenated codes. Then we propose several methods to protect syndrome extraction against measurement errors. xi Part I Introduction 1 Chapter 1 Preliminaries 1.1 Background Quantum computer needs to be protected by quantum error-correcting codes against decoherence. The theory of quantum error correction is important for both quantum computation and quantum communication [Sho95, Ste96a, Ste96c, EM96, BDSW96, KL97]. Quantum stabilizer codes are the most extensively studied quantum codes [CRSS97, Got97], and have the advantage that their properties can be analyzed using group algebra. Quantum stabilizer codes are closely related to classical linear codes, and can be obtained by the Calderbank-Shor-Steane (CSS) and Calderbank- Rains-Shor-Steane (CRSS) code constructions from weakly self-dual classical codes [CS96, Ste96b, CRSS97, CRSS98, Got97, NC00]. When entanglement between sender and receiver is available, a new error correction scheme becomes possible: entanglement-assisted quantum error-correction (EAQEC). This coding scheme has the advantage that it allows any classical linear code, not neces- sarily weakly self-dual, to be transformed into an entanglement-assisted quantum error- correcting code (EAQECC) [BDH06b]. In addition, EAQEC codes can increase both the transmission rate and error-correcting ability [LB13a, WH11]. Also, some problems or limitations in quantum LDPC codes and turbo codes can be solved using EAQEC codes [WH11, HBD09]. The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We nd new constraints on the simplied stabilizer group and the 2 logical group, which help improve the linear programming bounds of the entanglement- assisted quantum codes. The results can be applied to standard stabilizer codes. Further improvements are discussed. In real world quantum gates are faulty. To implement quantum computation fault- tolerantly, quantum codes with certain properties are needed [Sho96, DS96, Ste97, Got98, NC00, Pre98, Ste99a, Kni05]. A quantum computation can be implemented with arbi- trary accuracy provided that the error rate of any physical gate is below an error thresh- old. Knill's postselection scheme has the highest threshold among known concatenated codes [Kni05]. However, assumptions such as interactions between any two qubits may not be possible in physical implementations. We design a two-dimensional architecture of Knill's postselection scheme and show that it still has a high threshold. To perform quantum error correction, we have to measuring error syndromes with faulty measurements. Currently Shor's syndrome extraction [Sho96], Steane's syndrome extraction [Ste97], and Knill's syndrome extraction [Kni05] are three methods adopted in fault-tolerant quantum computation. We will generalize Shor and Steane's methods using the idea of classical error-correcting codes in dierent directions. (Knill's method and Steane's method bare some similarity and can be done similarly.) 1.2 Quantum Mechanics The state space of an isolated quantum system is a Hilbert space. In particular, we consider a 2-dimensional Hilbert spaceH with an (ordered) orthonormal basisfj0i;j1ig. A quantum statejvi inH is called a qubit and is denoted by a unit vector jvi =j0i +j1i orjvi = 2 4 3 5 ; 3 where and are complex numbers andjj 2 +jj 2 = 1. The concept of qubitsj0i andj1i is similar to the classical bits 0 and 1, except that a quantum state can be in a superposition state of qubitsj0i;j1i. The Pauli matrices I = 2 4 1 0 0 1 3 5 ;X = 2 4 0 1 1 0 3 5 ;Z = 2 4 1 0 0 1 3 5 ;Y = 2 4 0 i i 0 3 5 =iXZ form a basis of the space of the linear operators on the single-qubit state spaceH. The Pauli matrices X, Y , Z have eigenvalues1 and they anticommute with each other. They satisfy X 2 =Y 2 =Z 2 =I, X y =X,Y y =Y , and Z y =Z, where A y denotes the Hermitian conjugate of an operator A. The operations of X and Z on the basis vectors are Xj0i =j1i; Xj1i =j0i; Zj0i =j0i; Zj1i =j1i: Thus X is often referred to as a \bit- ip" operator, and Z is often referred to as a \phase- ip" operator. Y is the combined \bit-phase ip" operator. The state space ofn qubits is the tensor product of n single qubit state space and is denoted byH H =H n . Ifjv i i is the state of the i-th qubit, the joint state is jv 1 i jv n i orjv 1 ijv n i for simplicity. Let G n =feM 1 M n :M j 2fI;X;Y;Zg;e2f1;igg be the n-fold Pauli group. The weight wt(g) of g =eM 1 M n 2G is the number of M j 's that are not equal to the identity operator I. Every element in the Pauli group has eigenvalues +1 or1. We use the subscriptX j ,Y j , orZ j to denote a Pauli operator 4 on qubit number j. That is, X j = I j1 X I nj , Y j = I j1 Y I nj , and Z j = I j1 Z I nj for j = 1; ;n. We dene X u = Q i:u i =1 X i for some binary n-tuple u = (u 1 u n ) and similarly Z v = Q j:v j =1 Z j for some binary n-tuple v = (v 1 v n ). Any element g =eM 1 M n 2G n can be expressed as g =e 0 X u Z v for some e 0 2f1;ig and two two binary n-tuples u;v2Z n 2 . The overall phase of a quantum state is not important and we would sometimes consider the quotient of the Pauli group by its center G n =G n =f1;ig, which is an Abelian group and can be generated by a set of 2n independent generators. For g 1 = X u 1 Z v 1 , g 2 =X u 2 Z v 2 2 G n , the symplectic inner product in G n is dened by g 1 g 2 =u 1 v 2 +u 2 v 1 mod 2; (1.1) where is the usual inner product for binary n-tuples. Note that is commutative. We dene a map :G n ! G n by (eX u Z v ) =X u Z v . For g;h2G n , (g)(h) = 0 if they commute, and (g)(h) = 1, otherwise. The evolution of a closed quantum system is described by a unitary transformation. After a unitary quantum operator U applies to a statej i, the state becomes Uj i. When the quantum system is described by a density operator = P i p i j i ih i j, where fp i ;j i ig is an ensemble of pure statesj i i with probability p i , the evolution of the density operator under a unitary operator U is = X i p i j i ih i j U !UU y = X i p i Uj i ih i jU y : A quantum operation (evolution)E can be described by the operation-sum representa- tion: E() = X i E i E y i 5 with a set of operatorsfE i g such that X i E y i E i =I: In the following, error processes are modeled by the depolarizing channel T n p indepen- dently operating on n qubits, where T p () = (1 3 4 p) + p 4 (XX +YY +ZZ); with depolarizing ratep for 0p 1 and the density operator of a single qubit. Since the operation elements of the depolarizing channelT p are q 1 3 4 pI; q 1 4 pX; q 1 4 pY; and q 1 4 pZ, the operation elements ofT n p aref p p i E i g, where E i is a Pauli operator in the n-fold Pauli groupG n and p i is the probability that error E i happens. If E i is of weight w, then p i is q w , 1 3 4 p nw 1 4 p w : (1.2) The discussion below can be easily generalized to the case of an arbitrary Pauli channel with operation elementsf p p I I; p p X X; p p Y Y; p p Z Zg, where p I +p X +p Y +p Z = 1. Quantum measurement can be described by a collection of measurement operators. In particular we consider the projective measurement described by a (Hermitian) observable M with a spectral decomposition M = X m mP m where P m is the projector onto the eigenspace of M with eigenvalue m. Note that P m are orthogonal to each other and P m P m =I. 6 1.3 Quantum Stabilizer Codes 1.3.1 Stabilizer Group and Space Decomposition SupposeS is an Abelian subgroup of then-fold Pauli groupG n that does not includeI, and its has a set ofnk independent generatorsfg 1 ;g 2 ; ;g nk g. Lets =s 1 s nk 2 Z nk 2 be a binary (nk)-tuple. Since the stabilizer generators have eigenvalues1, we deneH s S to be a subspace ofH n such that forjvi2H s S , g j jvi = (1) s j jvi forj = 1; ;nk. That is,H s S lies in the (1) s j -eigenspace of g j forj = 1; ;nk and the projector ontoH s S is P s = nk Y j=1 I + (1) s j g j 2 : Note that the P s 's are orthogonal to each other and P s P s =I. Thus the n-qubit state spaceH n can be decomposed as H n = M s=s 1 s nk 2Z nk 2 H s S : (1.3) An [[n;k;d]] quantum stabilizer codeC(S) corresponding to the stabilizer groupS is dened to be the 2 k -dimensional subspace of the n-qubit state spaceH n xed byS, and d is the minimum distance that will be dened later. That is, C(S) =fj i2H n :gj i =j i;8g2Sg =H 0 S : For an error operator E2G n andj i2C(S), we have Ej i2H s S 7 for some binary (nk)-tuple s =s 1 s nk , where s j = 1 if E anti-commutes with g j , and s j = 0, otherwise. The (nk)-tuple s is called the error syndrome of E and can be obtained by measuring the stabilizer generators g 1 ; ;g nk . If the error syndrome is nonzero, we can detect the error. Otherwise, either no error or an undetectable error (logical error) happened. The normalizer group ofS inG n is N (S) =ff2G n :fgf y 2S;8g2Sg: In the case of Pauli operators,N (S) is the subgroup of the elements inG n that commute with anything inS, and they have error syndrome zero. The denition of the minimum distance of a stabilizer code comes from the error correction conditions [CRSS98, Got97, NC00]:fE i g is a collection of correctable error operators inG n forC(S) if and only if E y m E j = 2N (S)n ~ S; 8m;j; where ~ S =feg : g2S;e2fI;iIgg. Therefore, the minimum distance d of the stabilizer code is dened to be the minimum weight of an element inN (S)n ~ S or(N (S))n (S). An [[n;k;d]] quantum stabilizer code can correct errors of weight up tob d1 2 c. For a code with minimum distance d, if the error syndromes of error operators of weight smaller than or equal tob d1 2 c are distinct, we call that code nondegenerate. Otherwise, it is degenerate. 1.3.2 Cliord Encoder and the Check Matrix Consider the initial n-qubit statej i =j0i nk ji, where there are nk ancilla qubits j0i's and an arbitrary k-qubit information stateji. A set of stabilizer generators of this state isfZ 1 ; ;Z nk g. The operatorsZ nk+1 ; ;Z n ; andX nk+1 ; ;X n act to modiy the information stateji, and these operator are called logical operators. This stabilizer code is called a canonical code. 8 We would like to consider a special type of encoding operator belonging to the n-fold Cliord group CL(n): the Cliord encoder. The n-fold Cliord group CL(n) (together with the phase operator) is the normalizer group of the n-fold Pauli group G n in the space of linear operators onH n and it can be generated by the Hadamard gate ^ H = 1 p 2 2 4 1 1 1 1 3 5 , the phase gate ^ S = 2 4 1 0 0 i 3 5 , and the controlled-NOT gate CNOT = 2 6 6 6 6 6 6 6 4 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 3 7 7 7 7 7 7 7 5 . In particular, the Pauli matrices X, Y , Z belong to CL(1). Suppose the sender (Alice) has an n-fold unitary Cliord encoder U E 2 CL(n), which leaves the n-fold Pauli group G n invariant under conjugation. The encoded state U E j0i nk ji is xed by the stabilizer groupS with a set of generatorsfU E Z 1 U y E ; ; U E Z nk U y E g. In particular, we can chooseg i =U E Z i U y E fori = 1; ;nk. The logical operators on U E j i are Z 1 =U E Z nk+1 U y E ; ; Z k =U E Z n U y E ; X 1 =U E X nk+1 U y E ; ; X k =U E X n U y E : Observe that X i anticommute with Z i fori =i; ;k, respectively, and they all commute with the stabilizers. Thus the normalizer group ofS is N (S) =hg 1 ;g 2 ; ;g nk ; Z 1 ; Z 2 ; ; Z k ; X 1 ; X 2 ; ; X k i with n +k independent generators. Note that the operation of g X i onj i2C(S) is equivalent to X i for any g2S, and similarly for Z i . A stabilizer code is a quantum analogue of classical linear block codes and we can analyze it by a binary check matrix. We dene a map :G n !Z 2n 2 by (eX u Z v ) = (u;v) for eX u Z v 2G n and u;v2Z n 2 . A check matrix H corresponding to the stabilizer 9 S =hg 1 ; ;g nk i is dened as a binary (nk) 2n matrix such that the the i-th row vector ofH is(g i ). For convenience, H is denoted by [H X jH Z ]; whereH X andH Z are two (nk)n binary matrices. SinceS is Abelian, the check matrix H must satisfy the commutation condition H 2n H T =H X H T Z +H Z H T X =O (nk)(nk) ; where 2n = 2 4 O nn I nn I nn O nn 3 5 ,O ij is anij zero matrix, andI nn is ann-dimensional identity matrix. The canonical code has the check matrix h O (nk)n I (nk)(nk) O nkk i : (1.4) Given a check matrix H of a stabilizer group, the unitary Cliord encoder of the stabilizer code can be implemented by applying a certain quantum circuit. For example, Wilde gave an algorithm [Wil08] to nd an encoding circuit consisting of ^ H, ^ S, and CNOT gates for a given quantum stabilizer code. This algorithm applies a series of CNOT gates, Hadamard gates, Phase gates, SWAP gates, and row operations to the check matrix H such that H takes the form (1.4). This process is like performing Gaussian elimination on a matrix, but using CNOT gates, Hadamard gates, Phase gates, and SWAP gates, in addition to the elementary row operations of Gaussian elimination. The series of operations used in the algorithm serve as a unitary operation U y E such that U y E g i U E =Z i , and hence the inverse operator U E is a desired encoding operation. Note that the encoding circuit is not unique. 10 1.3.3 Syndrome Representatives and Decoding After the receiver (Bob)the noisy quantum state, he does the following three steps to recover the information stateji: syndrome measurement, correction, and decoding. He rst applies a series of projective measurements with projectors P s = nk Y j=1 I + (1) s j g j 2 on the output state of the noisy channel to nd the error syndrome s =s 1 s 2 s nk 2 Z nk 2 . P 0 is the projector on the code space, that is, P 0 U E j0i nk ji =U E j0i nk ji for any k-qubit stateji. Given a stabilizer groupS, there are 2 nk distinct error syndromes and the n-qubit state spaceH n can be decomposed as a direct sum of 2 nk subspacesH s S as in Eq. (1.3). For each nonzero error syndrome s, we choose a Pauli operator E s (not inS), whose error syndrome is s; for the error syndrome s = 0, we choose E 0 =I. The error operators E s are called syndrome representatives. We implicitly assume that E s is the most likely error operator with error syndrome s. If the measurement result is s, the correction operator C s = E s is applied, followed by the decoding unitary operator U y E . Finally, Bob throws away the ancilla qubits, which is the same as applying a partial trace over the ancilla qubits. We dene a setT containing the syndrome representatives fE s g. ThenjTj = 2 nk and T is a set of correctable error operators. Note that the choice of T determines the decoding process. (S determines the encoding process.) In fact, we have many more correctable error operators than T . For g2S, the action of E s g andE s on the encoded state is the same (E s gU E j ij0i nk =E s U E j ij0i nk ) and can be corrected by the same correction operator C s . The error operator E s g is called 11 a degenerate error of E s . The error operators in TS are correctable and are the only correctable error operators. that satises the error correction conditions. Since TS is the set of correctable errors, we would like to choose T to maximize X E2TS Pr(E): This is equivalent to choosing a syndrome representative E s for each syndrome s such that X E2EsS Pr(E) is maximized. Assume the syndrome representative E s has the minimum weight in the coset E s S. It can be shown that this criterion leads to high channel delity. For a channel with low error rate, P E2EsS Pr(E) is dominated by Pr(E s ). (It is possible that a degenerate error of E s has the same weight as E s .) We would like to choose the set of syndrome representatives T to contain lower-weight error operators, since these errors are more likely to occur in the depolarizing channel model. A simple algorithm to dene a set T is as follows: (a) Let T =; and T 0 =; (b) Find an error operator E2G n n (T[T 0 ) with the lowest weight, and compute its syndrome s. (c) If there is no E s 2T , set E s E and T T[fE s g. (d) If there is an E s 2T , set T 0 T 0 [fEg. (e) IfjTj< 2 nk , go to (b). Else, output T . This algorithm nds a set T of minimum weight, that is, it minimizes the quantity X E2T wt(E): 12 1.4 The Basics of Entanglement-Assisted Quantum Error- Correcting (EAQEC) Codes Brun, Devetak and Hsieh proposed a theory of quantum stabilizer codes when shared entanglement between the sender (Alice) and the receiver (Bob) is available [BDH06b]. Suppose Alice and Bob share c maximally-entangled pairs (ebits) j + i AB = 1 p 2 (j00i +j11i); which is an eigenstate of both XX and ZZ with eigenvalue +1. Using an additional nkc ancilla qubits in the statej0i, Alice encodes herk-qubit stateji with a Cliord encoder U E into ann-qubit state (including thec halves of the ebits on Alice's side) and then sends it to Bob through a noisy channel. It is assumed that Bob's c qubits suer no error during the process. This is called an [[n;k;d;c]] EAQEC code, where d is the minimum distance and will be dened later. We can think that standard stabilizer codes are a special case of c = 0. A set of stabilizer generators of the initial statejij + i c AB j0i nkc is fZ A k+1 Z B k+1 ; ;Z A k+c Z B k+c ;Z A k+c+1 I B ; ;Z A n I B ; X A k+1 X B k+1 ; ;X A k+c X B k+c g; inG n+c ; where the superscript A or B indicates that the operator acts on the qubits of Alice or Bob, respectively. Let g j =U E Z j U y E and h j =U E X j U y E for j = 1; ;n. The encoded state (U A E I B )jij + i c AB j0i nkc has a set of stabilizer generators fg A k+1 Z B k+1 ; ;g A k+c Z B k+c ;g A k+c+1 I B ; ;g A n I B ;h A k+1 X B k+1 ; ;h A k+c X B k+c g: 13 Since we assume that Bob's qubits suer no errors, we only consider the operators on Alice's qubits and we omit the superscript A in the following. The simplied stabilizer subgroupS 0 ofG n is S 0 =hg k+1 ; ;g k+c ;h k+1 ; ;h k+c ;g k+c+1 ; ;g n i: This subgroup is not Abelian now. The generators satisfy the commutation relations as follows: (g i )(g j ) = 0 for i6=j; (1.5) (h i )(h j ) = 0 for i6=j; (1.6) (g i )(h j ) = 0 for i6=j; (1.7) (g i )(h i ) = 1 for all i: (1.8) We say thatg i andh i are symplectic partners fori = 1; ;k +c. The logical subgroup L ofG n of the encoded state is L =hg 1 ; ;g k ;h 1 ; ;h k i: The symplectic subgroupS S ofS 0 is the subgroup generated by thec pairs of symplectic partners ofS 0 : S S =hg k+1 ; ;g k+c ;h k+1 ; ;h k+c i The isotropic subgroupS I ofS 0 is the subgroup generated by the generators g i ofS 0 such that g i g = 0 for all g inS 0 : S I =hg k+c+1 ; ;g n i: Notice thatS 0 =S S S I inG n . The minimum distance d of the EAQEC code is the minimum weight of any element in (S S S I ) ? n(S I ). 14 We dene a simplied check matrixH 0 as a binary (nk +c) 2n matrix such that thenk+c row vectors ofH 0 are(g 0 i ) fori = 1; ;nk and(h 0 j ) forj = 1; ;c. For simplicity, we usually order the generators g 0 i and h 0 j so that (g 0 i ) is the i-th row vector of H 0 for i = 1; ;nk, (h 0 j ) is the (j +nk)-th row vector of H 0 for j = 1; ;c, and the j-th and (j +nk)-th row vectors are a symplectic pair. H 0 must satisfy the commutation relations (1.5{1.8), and in the case c =nk, H 0 2n H 0T = 2 6 4 O nk(nk) I (nk)(nk) I (nk)(nk) O (nk)(nk) 3 7 5: (1.9) For example, the simplied check matrix corresponding to the set of generators of a stabilizer group of the initial statej i EA is 2 6 4 O rn I rr O r(nr) I rr O r(nr) O rn 3 7 5: (1.10) Conversely, an (nk +c) 2n binary matrix ~ H, serving as a simplied check matrix, can dene a simplied stabilizer group and hence an EAQEC code. The number of ebits required to construct an EAQEC code [WB08] is c = 1 2 rank( ~ H ~ H T ): (1.11) Note that Wilde's encoding circuit algorithm [Wil08] can also be applied to a simplied check matrix to nd an encoding unitary operator of the EAQEC code, just as for a standard stabilizer code. Similarly, we dene a simplied logical matrix L 0 corresponding to the logical oper- ators by putting '( Z 0 i ) to be the i-th row vector of L 0 for i = 1; ;k, and '( X 0 j ) to 15 be the (j +k)-th row vector of L 0 for j = 1; ;k: Since the logical operators commute withfg 0 1 ; ;g 0 r ;h 0 1 ; ;h 0 r g, we have H 0 2n L 0T =O (r+c)2k : (1.12) Since the logical operators satisfy the commutation relations (1.5{1.8), we have L 0 2n L 0T = 2 6 4 O kk I kk I kk O kk 3 7 5: For example, the simplied logical matrix corresponding to the initial statej i EA is 2 6 4 O kn O kr I kk O kr I kk O kn 3 7 5: (1.13) An [[n;k;d;c]] EAQEC code must satisfy some upper bounds on the minimum dis- tance. For example, we have the singleton bound for EAQEC codes [BDH06b] n +ck 2(d 1); (1.14) and the Hamming bound for non-degenerate EAQECCs [Bow02] t X j=0 3 j n j 2 nk+c : (1.15) 1.5 Fault-Tolerant Quantum Computation A quantum computation can be implemented with arbitrary accuracy provided that the error rate of any physical gate is below an error threshold. The basic idea of fault- tolerant quantum computation is to compute on the encoded quantum states and perform 16 quantum error correction periodically (or right after each logical operation) to prevent accumulation of errors. Decoding is never required during the computation. Any quantum computation can be implemented by a universal set of elementary gates. For example, the Cliord operators H, S, CNOT , together with a gate outside the Cliord group such as the T gate T = 2 4 1 0 0 e =4 3 5 , form a universal set. Since we are dealing with the encoded state of a certain quantum error-correcting code, the encoded version of these gates are required. Each encoded gate is composed of a series of non-perfect physical gates. Moreover, errors may propagate between qubits through multiple-qubit gates. For example, we know that errors will propagate through the CNOT gate: X 1 C 1 (X 2 ) ! X 1 X 2 : Z 2 C 1 (X 2 ) ! Z 1 Z 2 : A single qubit error could evolve to multiple qubit errors through a series of CNOT gates. Apparently, a quantum circuit with CNOT gates involved dierent qubits would be bad. We hope the error rate of the output of any encoded gate is small enough so that the followed error-correction routines can recover the quantum states. To analyze the error performance, we have to dene the notion of fault-tolerant rst. Denition 1.5.1. A procedure is fault-tolerant if it has the property that if only one component (or more generally, a small number of components) in the procedure fails, the errors produced by this failure are not transformed by the procedure into an uncorrectable error. We would like to design the quantum circuit of an encoded gate so that it is fault- tolerant. 17 Steane's [[7; 1; 3]] code is the rst quantum code that has an error threshold, which is of the magnitude 10 5 . This quantum code comes from the classical [7; 4; 3] Hamming code and has a set of stabilizer generators g 1 =XIIXIXX g 2 =IXIXXIX g 3 =IIXXXXI g 4 =ZIIZIZZ g 5 =IZIZZIZ g 6 =IIZZZZI: The left half of the matrix 2 6 6 6 6 4 (g 1 ) (g 2 ) (g 3 ) 3 7 7 7 7 5 (or the right half of the matrix 2 6 6 6 6 4 (g 4 ) (g 5 ) (g 6 ) 3 7 7 7 7 5 ) is a parity check matrix of the classical [7; 4; 3] Hamming code. The logical operators are X =XXXXXXX; Z =ZZZZZZZ: The logical Hadamard gate is H =HHHHHHH so that H X H y = Z and vice versa. The logical Phase gate is S = (ZS) 7 : 18 1st block . . . . . . 2nd block . . . 8 > > < > > : 8 > > > > < > > > > : Figure 1.1: Transversal CNOT gate. The bitwise implementation of these gates are in a fashion called transversality. It can be observed that a transversal gate is fault-tolerant: an error on a qubit does not propagate to other qubits. The encoded CNOT gate is also in a bitwise fashion as in Fig. 1.1. We have shown that the encoded Cliord operators of the Steane code can be fault- tolerantly implemented. However, the implementation of a fault-tolerant encoded T gate involves the preparation of certain special ancilla states, which is true for general quantum codes. 1.5.1 Shor's syndrome extraction Shor's syndrome extraction can be applied to all stabilizer codes [Sho96, Gai08]. Suppose we are measuring a stabilizer generator g j of weight w. First we prepare a w-qubit cat state jcati = 1 p 2 j0i w +j1i w = 1 p 2 (j000 0i +j111 1i): Applying Hadamard transform on the cat state, we obtained the so-called Shor state jShori =H w jcati = 2 w 1 X b=0 jbi + 2 w 1 X b 0 =0 (1) b 0 1 jb 0 i = X b:b 1=0 jbi; 19 j0i H H j0i H j0i H j0i H j0i m Figure 1.2: Construction and verication of the Shor state. H H Steane code H H H H H H jShori 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 8 > > > > > > > > < > > > > > > > > : Figure 1.3: The fault-tolerant syndrome measurement circuit of Steane's 7-qubit code for bit- ip error [PVK96][Sho96]. which is the superposition of all even-weight states. The Shor state can be prepared by the circuit in Fig. 1.2 (Fig.11 of [Pre98]). Then we use the Shor state to measure the generator g j to obtain a syndrome bit s j =m 1 + +m w mod 2, where m i is the measurement result of i-th qubit of the Shor state. For example, g 2 = X 2 X 4 X 6 X 7 for the Steane code and measurement ofg 2 is shown in Fig. 1.3. The second bit of the error syndrome is s 2 =m 1 +m 2 +m 3 +m 4 mod 2. 20 = j+i E = = m Figure 1.4: The circuit for syndrome measurement of bit- ip errors by Steane's method. Note that the transversal CNOT gates are compressed into a single CNOT gate. There arenk stabilizer generatorsg j 's, each of which needs aw j -qubit Shor state. Shor's method for syndrome measurement can be applied to the case k 1. 1.5.2 Steane's Syndrome Extraction Suppose we are using the CSS code construction with a classical code C such that C ? C. Steane [Ste97] proposed a scheme for syndrome measurement, which uses two N-qubit ancillas, prepared in the encoded statej+i E . j+i E = ^ H N j0i E = ^ H N X u2C ? jui = 2 N 1 X v=0 X u2C ? (1) uv jvi = X v2C jvi; since P u2C ?(1) uv = 0; if v = 2 C. To measure the error syndrome, we rst prepare a j+i E as in Fig. 1.4 to measure the bit- ip errors. (Similarly, we apply the Hadamard transform and repeat the procedure to measure phase- ip errors.) After a decoder logic, we obtain an error vector representing the bit- ip errors. 21 Part II Entanglement-Assisted Quantum Codes 22 Chapter 2 Duality of Entanglement-Assisted Quantum Codes In classical coding theory, a well-established notion is that of a dual code. Suppose thatC is an [n;k] linear code over an arbitrary eld GF(q) with a generator matrix G and a corresponding parity check matrix H such that HG T = 0: The dual code ofC is the [n;k 0 = nk] linear codeC ? with H as a generator matrix and G as a parity check matrix. The dimensions of the codeC and its dual codeC ? satisfy the relation k +k 0 =n: It is well known that the MacWilliams identity gives a relation between the weight enumerator ofC and the weight enumerator of its dual codeC ? [MS77], which can be used to determine the minimum distance of the dual codeC ? , given the weight enumerator ofC. The MacWilliams identity for quantum codes connects the weight enumerator of a classical quaternary self-orthogonal code associated with the quantum code to the weight enumerator of its dual code [SL97, Rai95, Rai99a, AL99]. This leads to the linear programming bounds (upper bound) on the minimum distance of quantum codes. This type of the MacWilliams identity for quantum stabilizer codes can be directly obtained by applying the Poisson summation formula from the theory of orthogonal groups. Though the orthogonal group of a stabilizer group with respect to the symplectic inner product (1.1) does not dene another quantum stabilizer code. However, the notion of a dual code can be dened in the case of entanglement-assisted quantum error-correcting codes (EAQECCs). 23 2.1 The Dual Codes of EAQEC Codes The orthogonal group of a subgroup V of G n with respect to is V ? =fg2 G n :gh = 0;8h2Vg: In other words, (N (S)) is the orthogonal group of (S) in G n with respect to: (N (S)) = ((S)) ? : Observe that the orthogonal group ofS 0 =S S S I in G n isLS I , which denes another EAQEC code with symplectic subgroup L and isotropic subgroupS I . The number of a set of independent generators ofS 0 =S S S I isK = 2c+(nkc) =nk+c, and the number of a set of independent generators of its orthogonal groupLS I is K 0 = 2k + (nkc) =n +kc. These parameters satisfy the following relation: K +K 0 = 2n =N; where N is the number of a set of independent generators of the full Pauli groupG n . This equation is parallel to the classical duality between a code and its dual code, which motivates the denition of the dual code of an EAQEC code as follows. Denition 2.1.1. The dual of an [[n;k;d;c]] EAQEC code, dened by a simplied stabi- lizer groupS 0 =S S S I and a logical groupL, is the [[n;c;d 0 ;k]] EAQEC code withLS I being the simplied stabilizer group andS S being the logical group for some minimum distance d 0 . When c =nk, we call such a code a maximal-entanglement EAQEC code. In this case,S I is the trivial group that contains only the identity, and the simplied stabilizer group isS S . Its dual code is a maximal-entanglement EAQEC code dened by the logical groupL. 24 info ebit ebit ebit ebit ebit ebit ebit ebit ebit info info info info info info info info (a) (b) Figure 2.1: (a) The encoder for a [[9; 1; 9; 8]] EA repetition code consists of a periodic cascade of CNOT gates. The encoder for arbitrary [[n; 1;n;n 1]] EA repetition codes extends naturally from this design. We can implement these encoders with a simple quantum shift-register circuit that uses only one memory qubit [Wil09]. (b) Consid- ering the circuit in (a) but changing the information qubit to an ebit and all of the ebits to information qubits gives the encoder for the dual of the EA repetition code, namely, the [[9; 8; 2; 1]] EA accumulator code. This circuit naturally extends to encode the [[n;n 1; 2; 1]] EA accumulator codes. Simple variations of the above circuit encode the even n repetition and accumulator EAQEC codes, and we discuss them in Sec- tion 2.5.2. Whenc = 0, the code is a standard stabilizer code, with a stabilizer groupS =S I = hg k+1 ; ;g n i, and a logical groupL =hg 1 ; ;g k ;h 1 ; ;h k i. S S is the trivial group in this case. The simplied stabilizer groupL denes an [[n; 0;d 0 ;k]] EAQEC code|that is, a single entangled stabilizer state that encodes no information. As an example, consider the class of [[n; 1;n;n 1]] EA repetition codes for n odd [LB13a]. The two logical operators for the one logical qubit of this code are as follows: X X X X X Z Z Z Z Z (2.1) 25 The simplied stabilizer generators are as follows: Z Z I I I I Z Z I I . . . . . . . . . . . . I I I Z Z X X I I I I X X I I . . . . . . . . . . . . I I I X X (2.2) One can determine the symplectic pairs by performing a symplectic Gram-Schmidt orthogonalization of the above operators [Wil08]. If we interchange the roles of the stabilizer subgroup and the logical operator subgroup, we obtain an [[n;n 1; 2; 1]] EA accumulator code. To make this more precise, consider the encoding circuits in Figure 2.1. The circuit in Figure 2.1(a) is the encoder of a [[9; 1; 9; 8]] EA repetition code. Swapping the infor- mation qubit for an ebit and all of the ebits for information qubits gives the encoder of Figure 2.1(b), which encodes a [[9; 8; 2; 1]] EA accumulator code. To illustrate that the circuit is working as expected, let us consider it acting on the rst ve qubits only (just to simplify the analysis). Inputting the following two operators at the information qubit slot X I I I I Z I I I I gives the following two logical operators: X X X X X Z Z Z Z Z ; 26 and these operators match the form of the logical operators for the repetition code in (2.1). Inputting the following operators at the ebit slots I X I I I I Z I I I I I X I I I I Z I I I I I X I I I I Z I I I I I X I I I I Z ; gives the following operators X X I I I I Z Z Z Z I X X X X Z Z I I I I I X X I I I I Z Z I I I X X I I Z Z I ; which we can transform by row operations to be the same as the operators in (2.2). 27 2.2 The MacWilliams Identity and the Linear Program- ming Bounds The MacWilliams identity for general quantum codes can be obtained by applying the Poisson summation formula from the theory of orthogonal groups [Kna06]. We rst introduce the notation needed to give the Poisson summation formula. A multiplicative character of an Abelian groupG is a homomorphism fromG into the complex numbersC such thatj(g)j = 1 for allg2G. If; 0 are two multiplicative characters of G, we dene ( 0 )(g) = (g) 0 (g) for all g 2 G. The multiplicative charactersfg of G form an Abelian group b G and they form an orthogonal basis of the functions that map G to C. Consider a function f : G ! C. By the Fourier inversion formula, f can be expanded overfg as f(g) = 1=jGj P 2 b G b f()(g); where b f() = P g2G f(g)(g) is the Fourier coecient of f at . Let V be a subgroup of G. Then the Poisson summation formula is as follows: X h2V f(h) = 1 jG=Vj X 2 b G;j V =1 b f(): (2.3) Now consider a subgroupV of G n and its orthogonal groupV ? . The weight enumerator ofV is a polynomialW V (x;y) = P n w=0 B w x nw y w , whereB w is the number of elements ofV with weightw. We are ready to prove the MacWilliams identity between the weight enumerators of V and V ? . Theorem 2.2.1. Suppose W V (x;y) = P n w=0 B w x nw y w and W V ?(x;y) = P n w 0 =0 A w 0x nw 0 y w 0 are the weight enumerators of subgroup V of G n and its orthogo- nal group V ? in G n . Then W V (x;y) = 1 jV ? j W V ?(x + 3y;xy): (2.4) 28 or B w = 1 jV ? j n X w 0 =0 P w (w 0 ;n)A w 0; for w = 0; ;n; (2.5) where P w (w 0 ;n) = P w u=0 (1) u 3 wu w 0 u nw 0 wu is the Krawtchouk polynomial [MS77]. Proof. Let c G n and \ G n =V be the group of multiplicative characters of G n and G n =V , respectively. We can dene a group isomorphism that maps c G n to G n , where for any 2 c G n we map 7! a with (g) = (1) ag for some a 2 G n and any g2 G n . The Fourier coecient of a function f at 2 c G n can be rewritten as b f(a ) = X g2 Gn f(g)(1) ag : We can also dene a group isomorphism that maps \ G n =V toV ? . For any2 c G n so that (g) = 1 for any g2 V , we map 7! a with (g) = (1) ag for some a 2 V ? and any g2 G n . Thus we havej G n =Vj =j \ G n =Vj =jV ? j. The Poisson summation formula (2.3) can be rewritten as X h2V f(h) = 1 jV ? j X g2V ? b f(g): (2.6) Suppose x;y2C are two complex numbers. Dene f 0 : G 1 !fx;yg by f 0 (I) = x and f 0 (X) =f 0 (Y ) =f 0 (Z) =y. Thus b f 0 (I) = X M2 G 1 (1) IM f 0 (M) =x + 3y and b f 0 (X) = X M2 G 1 (1) XM f 0 (M) =xy = b f 0 (Y ) = b f 0 (Z): 29 Let g = M g 1 M g n for any g2 G n , where M g j 2 G 1 for j = 1; ;n. Note that (1) hg = Q n j=1 (1) M h j M g j for g;h2 G n . Dene f : G n !C by f(g) = n Y j=1 f 0 (M g j ) =x nwt(g) y wt(g) : (2.7) Then b f(h) = X g2 Gn f(g)(1) hg = X g2 Gn n Y j=1 f 0 (M g j )(1) M h j M g j = n Y j=1 X M2 G 1 f 0 (M)(1) M h j M = n Y j=1 b f 0 (M h j ) =(x + 3y) nwt(h) (xy) wt(h) : (2.8) By substituting (2.7) and (2.8) in (2.6), we obtain equation (2.4). Expanding the RHS of (2.4), we nd that the coecients can be summarized by the Krawtchouk polynomials. Equation (2.5) is obtained by equating the coecients on both sides. Applying Theorem 2.2.1 to the simplied stabilizer groupS S S I and the isotropic subgroupS I , respectively, we obtain the MacWilliams Identity for EAQEC codes. Theorem 2.2.2. The MacWilliams identities for EAQEC codes are as follows: W LS I (x;y) = 1 jS S S I j W S S S I (x + 3y;xy); (2.9) W S I (x;y) = 1 jLS S S I j W LS S S I (x + 3y;xy): (2.10) Suppose W S I (x;y) = P n w=0 C w x nw y w is the weight enumerator ofS I . Given the weight enumeratorW LS I (x;y) = P n w=0 B w x nw y w ofLS I and the minimum distance 30 d of the EAQEC code, we have C w =B w for w2f0; ;d 1g. Equivalently, d is the smallest positive integer w such that B w C w > 0: From the MacWilliams identity, we can obtain the weight enumerator W S S S I (x;y) = P n w=0 A w x nw y w ofS S S I . Then d 0 is the smallest positive integer w such that A w C w > 0: In the case of maximal-entanglement EAQEC codes,S I is the trivial group and there is no degeneracy. If we exchange the roles ofS S andL, we obtain an [[n;nk;d 0 ;k]] EAQEC code, which is the dual of the original [[n;k;d;nk]] EAQEC code. The minimum distance d 0 of this [[n;nk;d 0 ;k]] EAQEC code is the minimum weight of a nontrivial element inL ? =S S . Thus d 0 can be determined from the MacWilliams identity and the weight enumerator of the [[n;k;d;nk]] EAQEC code, as in the following example. Example 2.2.1. The dual of the [[n; 1;n;n 1]] repetition code is the [[n;n 1; 2; 1]] accumulator code whenevern is odd. The coecients ofW L (x;y) = P n w=0 B w x nw y w for the odd-n [[n; 1;n;n1]] repetition code areB (n) , (B 0 ; ;B n ) = (1; 0; 0; ; 0; 3). Let 31 A (n) = (A 0 ; ;A n ). Using the MacWilliams identity, we obtain the weight enumerators W S S (x;y) = P n w=0 A w x nw y w of these dual EAQEC codes: A (3) =(1; 0; 9; 6); B (3) =(1; 0; 0; 3); A (5) =(1; 0; 30; 60; 105; 60); B (5) =(1; 0; 0; 0; 0; 3); A (7) =(1; 0; 63; 210; 735; 1260; 1281; 546); B (7) =(1; 0; 0; 0; 0; 0; 0; 3); A (9) =(1; 0; 108; 504; 2646; 7560; 15372; 19656; 14769; 4920); B (9) =(1; 0; 0; 0; 0; 0; 0; 0; 0; 3); 2.3 Linear Programming Bounds for EAQEC Codes The signicance of the MacWilliams identities is that linear programming techniques can be applied to nd upper bounds on the minimum distance of EAQEC codes. We rst provide the linear programming bound for maximal-entanglement quantum codes, which is similar to the case of classical codes, since there is no degeneracy in this case. Then we proceed to the general case. 2.3.1 The Linear Programming Bounds for Maximal-Entanglement Quantum Codes The minimum distance of an EAQEC code (or a standard stabilizer code) is the minimum weight of any element in (LS I )nS I . Suppose the weight enumerators ofS S S I and LS I are W S S S I (x;y) = P n w=0 A w x nw y w and W LS I (x;y) = P n w=0 B w x nw y w , 32 respectively. For c = 0 or c = nr, the minimum distance can be determined from the coecients of the weight enumerators W S S S I (x;y) and W L S S I (x;y), which are constrained by the MacWilliams identities. This gives the constraints we need in the integer programming problem. For an [[n;k;d;nk]] EAQEC code, it must be that B w = 0 forw = 1;:::;d 1. If we cannot nd any solutions to an integer program with the following constraints: A 0 =B 0 = 1; A w 0;B w 0; for w = 1; ;n; A w jS S j;B w jLj; for w = 1;:::;n; n X w=0 A w =jS S j; n X w=0 B w =jLj; B w = 1 jS S j n X w 0 =0 P w (w 0 ;n)A w 0; for w = 0;:::;n; B w = 0; for w = 1;:::;d 1; for a certain d, this result implies that there is no [[n;k;d;nk]] EAQEC code. If d is the smallest such d, then d 1 is an upper bound on the minimum distance of an [[n;k;d;nk]] EAQEC code. This bound is called the linear programming bound for EAQEC codes with maximal entanglement. If we replace the constraints B w = 0; for w = 1;:::;d 1 with A w = 0; for w = 1;:::;d 1; we get the linear programming bound on the minimum distance of the dual code [[n;n k;d;k]]. 33 Example 2.3.1. Consider the [[8; 3; 5; 5]] EAQEC code from the random optimization algorithm in Ref. [LB13a]. The linear programming bound shows that there is no [[8; 3;d; 5]] EAQEC code with d> 5, and thus the [[8; 3; 5; 5]] code is optimal. 1 Example 2.3.2. Consider the [[15; 7; 6; 8]] EAQEC code from the random optimization algorithm in Ref. [LB13a]. The linear programming bound shows that no [[15; 7;d; 8]] EAQEC code with d > 7 exists; however, it does not rule out the existence of a [[15; 7; 7; 8]] code. 2.3.2 The Linear Programming Bound for Non-Maximal- Entanglement Quantum Codes Now we consider the case 0<c<nk, and bothS I andS S are not trivial. Since the min- imum distance of an EAQEC code is the minimum weight of any element in (LS I )nS I , constraints on the weight enumerator ofS I are required. If we view the group (LS S ) as a logical group of a standard stabilizer code with a stabilizer groupS I , this is just the dual relation of the standard stabilizer codes. Let the weight enumerators ofS I and LS S S I beW S I (x;y) = P n w=0 C w x nw y w andW LS S S I (x;y) = P n w=0 D w x nw y w , respectively. Then W S I (x;y) and W LS S S I (x;y) are related by the MacWilliams iden- tity (2.10) in Corollary 2.2.2. The minimum distance of the EAQEC code is the minimum nonzero integer w such that B w C w > 0. With constraints on B w 's and C w 's, we can 1 We used the optimization software LINGO from LINDO Systems to solve the integer programming problem for these examples. 34 nd the linear programming bound on the minimum distance of the EAQEC code. To sum up, we have the following constraints: A 0 =B 0 =C 0 =D 0 = 1; A w 0;B w 0;C w 0;D w 0; for w = 1;:::;n; A w jS S S I j;B w jLS I j; for w = 1;:::;n; C w jS I j;D w jLS S S I j; for w = 1;:::;n; D w A w ;D w B w ;D w C w ; for w = 1;:::;n; A w C w ;B w C w ; for w = 1;:::;n; n X w=0 A w =jS S S I j; n X w=0 B w =jLS I j; n X w=0 C w =jS I j; n X w=0 D w =jLS S S I j; B w = 1 jS S S I j n X w 0 =0 P w (w 0 ;n)A w 0; for w = 0;:::;n; D w = 1 jS I j n X w 0 =0 P w (w 0 ;n)C w 0; for w = 0;:::;n; B w =C w ; for w = 1;:::;d 1; If we cannot nd any solutions to the integer programming problem with the above constraints for a certain d, this result implies that there is no [[n;k;d;c]] EAQEC code. If d is the smallest such d, then d 1 is an upper bound on the minimum distance of an [[n;k;d;c]] EAQEC code. This is the linear programming bound for EAQEC codes with 0<c<nk. If we replace the constraints B w =C w ; for w = 1;:::;d 1; 35 with A w =C w ; for w = 1;:::;d 1; this gives the constraints of the linear programming bound on the minimum distance of the [[n;c;d;k]] dual code. Example 2.3.3. The linear programming bound of the [[7; 1; 4]] code is d 6, which is the same as the singleton bound. Hence the [[7; 1; 5; 4]] EAQEC code from the encoding optimization algorithm in Example 4 of [LB13a] does not achieve the linear programming bound. It follows that the [[7; 1; 5; 5]] EAQEC code does not achieve the upper bound either. Example 2.3.4. The linear programming bound of the [[8; 3; 3]] EAQEC code is d 4, which improves the singleton bound: d 5. Hence the [[8; 3; 4; 3]] EAQEC code from the encoding optimization algorithm in Example 8 of [LB13a] is optimal. On the other hand, the linear programming bound of the [[8; 3; 4]] EAQEC code isd 5, which is the same as the singleton bound. Hence the [[8; 3; 4; 4]] EAQEC code does not achieve the upper bound. Example 2.3.5. The linear programming bounds of [[9; 1;c]] EAQEC codes forc = 3; 4; 5 are d 8; however, the singleton bounds are d 6, d 7, and d 7, respectively. From these three examples, we can determine that the linear programming bound might be better or worse than the Singleton bound when 0<c<nk. 2.4 Improving the Linear Programming Bounds We can apply the constraints on the coecientsfB w g andfA j g to an integer program to upper bound the minimum distance of quantum codes of given parameters n;k;c. Rains introduced the idea of shadow enumerators to improve the linear programming bound for quantum codes [Rai99b]. 36 The shadow Sh(V ) of a group V in G n is the set fE2 G n :Eg = wt(g) mod 2; 8g2Vg: It is easy to verify the following lemma. Lemma 2.4.1. For g;h2 G n , wt(gh) wt(g) + wt(h) +gh mod 2: If V contains no element of odd weight, then Sh(V ) = V ? . By Theorem 2.2.1, we have W Sh(V ) (x;y) = 1 jVj W V (x + 3y;xy) = 1 jVj W V (x + 3y;yx); where the last equality follows because V contains no element of odd weight. IfV contains some elements of odd weight, it is easy to characterizeSh(V ) in the case thatV is (weakly) self-orthogonal with respect to. LetV be generated byg 1 ;g 2 ; ;g r . IfV contains an element of odd weight, say g 1 without loss of generality, we can replace any otherg i that has odd weight with g i g 1 fori6= 1. Thus we can assume that g 1 is the only generator of odd weight. Let V e =hg 2 ; ;g r i. The shadow of V is V ? e nV ? . Let W Ve = P n w=0 E w x nw y w . Let W V = P n w=0 A w x nw y w . Since the elements of V e are of even weight, we have E w = 0 for odd w and E w =A w for even w. Thus 2W Ve (x;y) =W V (x;y) +W V (x;y): (2.11) 37 Then we have W Sh(V ) (x;y) =W V ? e (x;y)W V ?(x;y) = 1 jV e j W Ve (x + 3y;xy) 1 jVj W V (x + 3y;xy) = 1 jVj W V (x + 3y;yx): To sum up, we have W Sh(V ) (x;y) = 1 jVj W V (x + 3y;yx): (2.12) When V is a stabilizer group ofG n , (V ) is self-orthogonal. Eq. (2.12) provides addi- tional constraints on the coecients of W V (x;y), and the linear programming bounds for standard quantum codes can be improved [Rai99b]. Consider the simplied stabilizer groupS 0 =S S S I of an [[n;k;d;c]] EAQEC code. Assume c > 0. We can applied Eq. (2.12) to the isotropic subgroupS I . However, it's complicated if V is not self-orthogonal and equation (2.11) cannot be applied here. Let S e be the subgroup generated by the generators ofS 0 of even weight. Lemma 2.4.2. LetS S =hg k+1 , ; g k+c , h k+1 , ; h k+c i, andS I =hg k+c+1 , ; g n i. There are three types of a set of generators: I. All the generators of S 0 are of even weight. II. g k+c+1 has odd weight and all the other generators ofS I andS S are of even weight. III. g k+1 and h k+1 has odd weights and all the other generators ofS S andS I are of even weight. Proof. SupposeS I contains some elements of odd weight, say g k+c+1 without loss of generality. If g j or h j is of odd weight for j6= k +c + 1, we replace it by g j g k+c+1 or h j g k+c+1 , which is of even weight by equation (1.5), (1.7), and Lemma 2.4.1. This is type II. 38 Next, supposeS I contains no elements of odd weight andS s has some elements of odd weight. Consider the generators ofS s . Assume g k+1 has odd weight without loss of generality. If g i has odd weight for i = k + 2; ;k +c, we replace it with g i g k+1 , which is of even weight by (1.5) and Lemma 2.4.1. Notice that h k+1 has to be replaced byh k+1 h i to maintain the relation in (1.7) at the same time. Thus we have g k+1 of odd weight and g k+2 ; ;g k+c of even weight. If h i has odd weight for i =k + 2; ;k +c, we replace it with h i g k+1 , which is of even weight by (1.6) and Lemma 2.4.1. Similarly, h k+1 has to be replaced by h k+1 g i to maintain the relation in (1.6) in the meantime. Therefore, we can assume h k+2 ; ;h k+c are of even weight. It remains to consider whether h k+1 is of odd weight or not. If h k+1 of odd weight, this is type III. If h k+1 has even weight, we replace g k+1 by g k+1 h k+1 , which is of even weight by (1.8) and Lemma 2.4.1. This case reduce to type I. Let N even i (c;r) and N odd i (c;r) be the numbers of even and odd elements in a sub- groupS ofG n with c pairs of symplectic partners and r isotropic generators of type i. We show how to calculate these two numbers. We rst consider the case r = 0. Theorem 2.4.1. If the generators are of type I, N even I (c; 0) = 1 2 (4 c + 2 c ); N odd I (c; 0) = 1 2 (4 c 2 c ): If the generators are of type III, N even III (c; 0) = 1 2 (4 c 2 c ); N odd III (c; 0) = 1 2 (4 c + 2 c ): Proof. These formulas can be proved by recursion. We rst consider the case that the generators are of type (a). It is obvious that N even I (0; 0) = 1 and N odd I (0; 0) = 0. (In 39 this case,S is the trivial subgroup.) Forc = 1, we have two generatorsg 1 andh 1 of even weight such that g 1 h 1 = 1. Thus wt(g 1 h 1 ) wt(g 1 ) + wt(h 1 ) +g 1 h 1 1 mod 2. We have N even I (1; 0) = 3 and N odd I (1; 0) = 1. LetS c denote a group generated byc pairs of symplectic partners. Now we add a new pair of symplectic partnerg 0 andh 0 of even weight such thatg 0 andh 0 are orthogonal to any thing in S c and g 0 h 0 = 1. An element E in the new group S c+1 , is of even weight if one of the following cases hold: E2 S c and wt(E) 0 mod 2 ; E = g 0 E 0 for some E 0 2S c and wt(E) 0 mod 2 ; E =h 0 E 0 for some E 0 2S c and wt(E) 0 mod 2 ; or E =g 0 h 0 E 0 for some E 0 2S c and wt(E) 1 mod 2. Thus N even I (c + 1; 0) = 3N even I (c; 0) +N odd I (c; 0): Similarly, N odd I (c + 1; 0) = 3N odd I (c; 0) +N even I (c; 0): Solving this system of two recursions, we have N even I (c; 0) = 1 2 (4 c + 2 c ); N odd I (c; 0) = 1 2 (4 c 2 c ): In the same way, we can nd the formulas for the generators of type (c). Note that N even i (c; 0)N odd (c; 0) i = 2 c and N even i (c; 0) +N odd i (c; 0) = 4 c for i =I, III and c6= 0. 40 Theorem 2.4.2. N even I (c;r) = 2 r1 (4 c + 2 c ); N odd I (c;r) = 2 r1 (4 c 2 c ): N even III (c;r) = 2 r1 (4 c 2 c ); N odd III (c;r) = 2 r1 (4 c + 2 c ): For r> 0, N even II (c;r) = 2 2c+r1 ; N odd II (c;r) = 2 2c+r1 : Theorem 2.4.2 provides additional constraints on the coecients of W S S S I and W LS I . In addition, we have W Sh(S I ) (x;y) = 1 jS I j W S I (x + 3y;yx) from Eq. (2.12). Therefore, the linear programming bounds for EAQECCs can be improved. Consider the integer program in the previous section with the variablesA j 's andB w 's, which are the coecients of the weight enumerators of a simplied stabilizer group and its orthogonal group, and the constraints in Theorem 2.2.2. With the new constraints from Theorem 2.4.2, we are able to improve the linear programming bounds. Example 2.4.1. We eliminate the existence of [[4; 2; 3; 2]], [[5; 3; 3; 2]], [[5; 2; 4; 3]], [[6; 3; 4; 3]], [[9; 2; 7; 7]], [[10; 2; 8; 8]], [[10; 3; 7; 7]], [[11; 3; 8; 8]], [[14; 2; 11; 12]], [[14; 3; 10; 11]], [[15; 2; 12; 13]], [[15; 3; 11; 12]], [[15; 8; 7; 7]], [[15; 9; 6; 6]] EAQEC codes. 41 The constraints from Theorem 2.4.2 can be applied to the normalizer group of the stabilizer group of a standard stabilizer code. although I have not found any example that the existing upper bounds are improved. 2.5 Bounds on EAQEC Codes This section establishes a table of upper and lower bounds on the best minimum distances of maximal-entanglement EAQEC codes forn 15. We begin by discussing the existence of arbitrary EAQEC codes, followed by some specic EAQEC code constructions. The existence of an [[n;k;d]] stabilizer code implies the existence of an [[n;k;d 0 d;c]] EAQEC code, since we can replace ancilla qubits with ebits and then optimize the encoding operator [LB13a]. Therefore, the lower bound on the minimum distance of regular stabilizer codes [CRSS98] can be applied here. Similarly, the existence of an [[n;k;d;c]] EAQEC code wherec<nk implies the existence of an [[n;k;d 0 d;c 0 >c]] EAQEC code. This establishes the existence of many [[n;k;c]] EAQEC codes. 2.5.1 Gilbert-Varshamov Bound for EAQEC Codes Consider the stabilizer groupS of an [[n;k;d;c]] EAQEC code, which is a subgroup of the Pauli groupG n+c . We consider only error operators in the groupG n because the entanglement-assisted paradigm assumes that the ebits on Bob's side of the chan- nel are not subject to errors. The EAQEC code is dened in an (n +c)-qubit space, but only the rst n qubits suer from errors. Following the argument of the quantum Gilbert-Varshamov bound [EM96], we can get the Gilbert-Varshamov bound for EAQEC codes. However, we will show that there are maximal-entanglement EAQEC codes with minimum distance higher than this bound for n 15. Theorem 2.5.1. Given n;d;c, let k = & log 2 2 n+c P d1 j=0 3 j n j !' : 42 The Gilbert-Varshamov bound for EAQEC codes states that if 0knc, then there exists an [[n;k;d;c]] EAQEC code. Equivalently, d1 X j=0 3 j n j 2 k 2 n+c : 2.5.2 Maximal-Entanglement EAQEC Repetition and Accumulator Codes In this section, we construct EA repetition codes with parameters [[n; 1;n;n1]] for oddn and [[n; 1;n1;n1]] for evenn. The duals of these codes are the EA accumulator codes with parameters [[n;n 1; 2; 1]] for odd n. [[n;n 1; 1; 1]] for even n. Theorems 2.5.4 and 2.5.5 show that both of these code constructions are optimal, in the sense that [[n; 1;n;n1]] and [[n;n1; 2; 1]] EAQEC codes do not exist for evenn. Thus, the results here complete our understanding of the dual classes of EA repetition and accumulator codes for arbitrary n. Theorem 2.5.2. There are [[n; 1;n;n1]] EAQEC codes for oddn and [[n; 1;n1;n1]] EAQEC codes for evenn. These codes are optimal, and are not equivalent to any regular stabilizer code for n 5. Proof. Suppose ^ H n is an (n 1)n parity-check matrix of a classical [n; 1;n] repetition code: ^ H n = 2 6 6 6 6 6 6 6 4 1 1 0 0 0 1 1 . . . 0 . . . . . . . . . . . . . . . 0 0 1 1 3 7 7 7 7 7 7 7 5 : The [[n; 1; 1]] n-qubit bit- ip code has a check matrix h O (n1)n ^ H n i : 43 We want to introduce (n 1) simplied generators to the generating set of the stabilizer group such that the minimum distance of the code is increased ton. Consider a simplied check matrix H 0 = 2 4 O (n1)n ^ H n ^ H n O (n1)n 3 5 : By (1.11), the number of symplectic pairs in H 0 is 1 2 rank(H 0 H 0T ) = rank( ^ H n ^ H T n ) =n 1; forn odd. It can be veried thatH 0 is a simplied check matrix with minimum distance n. Therefore, there exists a set of symplectic partners of the generators of the stabilizer group of then-qubit bit ip code such that the minimum distance of the code is n. It is easy to verify that (1.14) is saturated by the parameters [[n; 1;n;n 1]]. These [[n; 1;n;n1]] codes are not equivalent to any regular stabilizer code, for there are no regular [[2n 1; 1;n]] quantum codes for n> 3. This is because they violate the quantum Hamming bound, which says that an [[n;k;d = 2t + 1]] code satises 2 nk t X i=0 n i 3 i : Let n = 2t + 1. The [[2n 1; 1;n]] = [[4t + 1; 1; 2t + 1]] code would have P t i=0 4t+1 i 3 i error syndromes if it exists. The last term 4t+1 t 3 t is of order O (12t + 3) t , which is larger than the total number of possible syndromes 2 4t for suciently large t. We have checked that it holds when t> 1 or n> 3. In the case of evenn, the above construction gives a series of [[n; 0;n;n2]] EAQEC codes with no information qubits. A series of [[n; 1;n 1;n 1]] EAQEC codes for n even is constructed in [LB13b]. These EAQEC codes are optimal, since it is proved that there is no [[n; 1;n;n 1]] EAQEC codes for n even in [LB13b]. These EAQEC codes are not equivalent to any regular stabilizer codes for n> 4 by the same argument as in the case of n odd. 44 Theorem 2.5.3. There are [[n; 1;n 1;n 1]] EA repetition codes for n even. The duals of these codes are the [[n;n 1; 1; 1]] EA accumulator codes. Proof. Suppose H (n1) is an (n 2) (n 1) parity-check matrix of a classical [n 1; 1;n 1] binary repetition code: H (n1) = 2 6 6 6 6 6 6 6 4 1 1 0 0 0 0 1 1 0 0 . . . . . . . . . . . . 0 0 1 1 3 7 7 7 7 7 7 7 5 : We dene two (n 1)n matrices H 1 = 2 6 6 6 6 6 6 6 4 0 . . . H (n1) 0 1 1 1 0 3 7 7 7 7 7 7 7 5 ; and H 2 = 2 6 6 6 6 6 6 6 4 0 H (n1) . . . 0 1 1 1 1 3 7 7 7 7 7 7 7 5 : Consider a simplied check matrix of the form H 0 = 2 4 O H 1 H 2 O 3 5 : 45 Consider the matrix H 1 H T 2 . We have that H 1 H T 2 i;j = 8 > > > > < > > > > : 1; if i =j for j = 1; ;n 1; or i =j 2 for j = 3; ;n 2 0; else. For example, when n = 6, H 1 H T 2 = 2 6 6 6 6 6 6 6 6 6 6 4 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 5 : Thus the number of symplectic pairs in H 0 is as follows [WB08]: 1 2 rank(H 0 H 0T ) = rank(H 1 H T 2 ) =n 1: The simplied logical matrix is L 0 = 2 4 00 00 11 1 11 10 00 0 3 5 ; which implies the minimum distance is n 1. Therefore, H 0 and L 0 dene an [[n; 1;n 1;n 1]] EAQEC code. One obtains the dual [[n;n 1; 1; 1]] codes simply by swapping the roles of the logical matrix and the simplied check matrix. This completes the family of EA repetition and accumulator codes for any n. The encoding circuit of Figure 2.1 encodes these even-n repetition codes with the exception 46 that the last qubit is removed, the last CNOT in the rst string does not act, and the last CNOT in the second string does not act. Example 2.5.1. The coecients ofW L (x;y) = P n w=0 B w x nw y w for the even-n [[n; 1;n 1;n1]] EA repetition code areB (n) = (1; 0; ; 0; 1; 2). Using the MacWilliams identity, we obtain the weight enumerators W S S (x;y) = P n w=0 A w x nw y w of these dual even-n EAQEC codes: A (4) =(1; 1; 15; 27; 20); B (4) =(1; 0; 0; 1; 2); A (6) =(1; 1; 40; 130; 305; 365; 182); B (6) =(1; 0; 0; 0; 0; 1; 2); A (8) =(1; 1; 77; 357; 1435; 3395; 5103; 4375; 1640); B (8) =(1; 0; 0; 0; 0; 0; 0; 1; 2); A (10) =(1; 1; 126; 756; 4326; 15246; 38304; 65604; 73809; 49209; 14762); B (10) =(1; 0; 0; 0; 0; 0; 0; 0; 0; 1; 2); A (12) =(1; 1; 187; 1375; 10230; 47850; 168630; 432894; 811965; 1082565; 974303; 531443; 132860); B (12) =(1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 2): The even-n [[n; 1;n 1;n 1]] EA repetition codes do not saturate the quantum singleton bound or the linear programming bounds. Were an even-n [[n; 1;n;n 1]] code to exist, it would have a weight enumerator W LS I (x;y) = P n w=0 B w x nw y w with B 0 = 1;B n = 3; andB w = 0 forw6= 0;n. The weight enumerator of its dual would also have the coecients A w = 1 4 (3 w + 3(1) w ) n w ; 47 which are positive integers forw = 0; ;n. It would only be able to correct up to n1 2 channel qubit errors, which is the same number of errors that our even-n repetition codes can correct. We prove below that even-n [[n; 1;n;n1]] EAQEC codes do not exist, and thus our even-n repetition codes from Theorem 2.5.3 are optimal. Theorem 2.5.4. There is no [[n; 1;n;n 1]] EAQEC code for n even. Proof. We prove it by contradiction. Suppose there is an [[n; 1;n;n 1]] EAQEC code for n even with a 2 2n logical matrix 2 4 u 1 u 2 v 1 v 2 3 5 ; whereu 1 ,u 2 ,v 1 , andv 2 are binary row vectors of lengthn. These vectors should satisfy the following condition in order for the above matrix to be a valid logical matrix: u 1 v 2 +u 2 v 1 = 1 mod 2: Let gw() be the \general weight" function dened by gw (ujv) X i:u i =1 or v i =1 1; where u i denotes the i th bit of the binary n-tuple u. The above binary vectors should satisfy the further constraints gw u 1 jv 1 = gw u 2 jv 2 = gw u 1 +u 2 jv 1 +v 2 =n; in order for the code to have distance n as claimed. We now use the above constraints to obtain a contradiction. We rst partition the rst row of the matrix into subsets A,B, andC ofX,Y , andZ operators, respectively. There should not be any identity operators in the rst row in order for the code to have 48 distance n. Up to permutations on the qubits (under which the distance is invariant), the logical matrix has the following form: 2 4 1 1 0 u 2 A u 2 B u 2 C 0 1 1 v 2 A v 2 B v 2 C 3 5 ; where 1 is a vector of all ones, 0 is a vector of all zeros, and we have split up the vector u 2 jv 2 into dierent components corresponding to the subsets A, B, and C. Consider the vectoru 2 A . Suppose that a component u 2 A i = 0. Then v 2 A i should equal 1 so that the code's distance is not less than n. Now suppose that u 2 A i = 1. Then v 2 A i should also equal 1 so that the code's distance is not less thann. Otherwise, we could add (1j0) to ( u 2 A i j v 2 A i ) and obtain (0j0) as a component of another logical operator, and such a result would imply that the code's distance is less than n. These steps imply that the logical matrix should have the following form: 2 4 1 1 0 u 2 A u 2 B u 2 C 0 1 1 1 v 2 B v 2 C 3 5 : Similar reasoning withv 2 C andu 2 C implies thatu 2 C should equal 1, and the logical matrix should then have the following form: 2 4 1 1 0 u 2 A u 2 B 1 0 1 1 1 v 2 B v 2 C 3 5 : Finally, consider the vector u 2 B . Suppose that a component u 2 B i = 1. Then v 2 B i should equal 0 so that the code's distance is not less than n. Otherwise, we could add (1j1) to ( u 2 B i j v 2 B i ) and obtain (0j0) as a component of another logical operator, and such a result would imply that the code's distance is less than n. Now suppose that u 2 B i = 0. Then v 2 B i should equal 1, by reasoning similar to the above. Thus, the 49 logical matrix should have the following form in order for the code's distance to be equal to n: 2 4 1 1 0 u 2 A u 2 B 1 0 1 1 1 u 2 B v 2 C 3 5 ; (2.13) where u 2 B is the binary complement of u 2 B . Now, the symplectic product of the above two vectors is (jAj +jBj +jCj) mod 2 =n mod 2 = 0; which contradicts the assumption that the original matrix is a valid logical matrix. Theorem 2.5.5. There is no [[n;n 1; 2; 1]] EAQEC code for n even. Proof. We prove the theorem by contradiction, in a fashion similar to the previous theo- rem. Suppose there is an [[n;n 1; 2; 1]] EAQEC code forn even, and suppose its 2 2n simplied check matrix [H X jH Z ] has the form 2 4 u 1 u 2 v 1 v 2 3 5 ; whereu 1 ,u 2 ,v 1 , andv 2 are binary vectors of lengthn. These vectors should satisfy the following condition in order for the above matrix to be a simplied check matrix of a maximal-entanglement EAQEC code with one ebit: u 1 v 2 +u 2 v 1 = 1 mod 2: We now partition the rst row of the simplied check matrix into subsets A, B, and C ofX,Y , andZ operators, respectively. Up to permutations on the qubits (under which the distance is invariant), the simplied check matrix has the following form: 2 4 1 1 0 u 2 A u 2 B u 2 C 0 1 1 v 2 A v 2 B v 2 C 3 5 ; 50 where we have split up the vector u 2 jv 2 into dierent components corresponding to the subsets A, B, and C. The code has minimum distance two by assumption, and is non-degenerate because it is a maximal-entanglement EAQEC code. Therefore, no column of the above matrix should be equal to the all-zeros vector. Were it not so, then the code would not be able to detect every single-qubitX orZ error and would not have distance two as claimed. These constraints restrict the simplied check matrix to have the following form: 2 4 1 1 0 u 2 A u 2 B 1 0 1 1 1 v 2 B v 2 C 3 5 : Also, no column of the entrywise sum of the matrices to the left and right of the vertical bar should be equal to the all-zeros vector. Were it not so, then the code would not be able to detect every single-qubit Y error and would not have distance two as claimed. These constraints further restrict the simplied check matrix to be as follows: 2 4 1 1 0 u 2 A u 2 B 1 0 1 1 1 u 2 B v 2 C 3 5 : (2.14) Now, the symplectic product of the above two vectors is (jAj +jBj +jCj) mod 2 =n mod 2 = 0; which contradicts the assumption that the original matrix is a simplied check matrix for a maximal-entanglement EAQEC code with one ebit. Interestingly, observe that the non-existent logical matrix in (2.13) has the same form as the non-existent simplied check matrix in (2.14). Were either type of code to exist, we would expect them to be duals of each other, but they both fail to exist because they cannot satisfy the dual constraints imposed on them. 51 2.5.3 Existence of Other EAQEC Codes The following theorem is similar to Theorem 6 in Ref. [CRSS98]. It shows how to obtain new EAQEC codes from existing ones. These results are helpful in our search for lower bounds on the minimum distance of maximal-entanglement EAQEC codes. Theorem 2.5.6. Suppose an [[n;k;d;c]] code exists. Then (i). An [[n + 1;k;d;c + 1]] code exists. (ii). An [[n;k 1;d 0 d;c + 1]] code exists. Proof. (i). Suppose H = [H X jH Z ] is a simplied check matrix of an [[n;k;d;c]] code. Then the simplied check matrix H 0 = 2 6 6 6 6 6 6 6 6 6 6 4 00 0 0 00 0 1 00 0 1 00 0 0 0 0 H X . . . H Z . . . 0 0 3 7 7 7 7 7 7 7 7 7 7 5 denes an [[n + 1;k;d;c + 1]] code. We have the stabilizer group (S I)[ fX n+1 ;Z n+1 g, where (S I) =fE I :E2Sg. (ii). It is obtained by moving a symplectic pair from the logical group to the stabilizer group. The following two Lemmas come from the fact that an maximal-entanglement EAQEC code is equivalent to a classical additive quaternary code. Lemma 2.5.1. The upper bound on the minimum distance of the classical [n;k] additive quaternary codes is an upper bound on the minimum distance of [[n;k;nk]] EAQEC codes. 52 Lemma 2.5.2. The upper bound on the minimum distance of the [n;k] additive quater- nary codes is an upper bound on the minimum distance of the [n+1;k+1] additive quater- nary codes, and hence an upper bound on the minimum distance of the [[n+1;k+1;nk]] EAQEC codes. 2.5.4 The Plotkin bound for EAQEC Codes The Plotkin bound for EAQEC codes is similar to the Plotkin bound for classical codes [MS77]. It is again helpful in our eorts to bound the minimum distance of maximal- entanglement EAQEC codes. Theorem 2.5.7. The Plotkin bound for any [[n;k;d;c]] EAQEC code is d 3n2 2k2 2 2k 1 : Proof. The proof is based on the proof of the classical Plotkin bound in Ref. [MS77]. Let M = 2 n+kc be the number of operators in LS I . We bound the quantity P u;v2LS I nS I wt(uv) in two dierent ways. First, we lower bound it. There are M choices for u, and for each choice of u, there are M 2 nkc choices for v such that uv = 2S I . Furthermore, for a code of minimum distance d, wt(uv) d for any uv = 2S I . So the following lower bound holds M(M 2 nkc )d X u;v2LS I : uv= 2S I wt(uv) X u;v2LS I wt(uv): The equality holds when c =nk because S I is trivial and wt(uv) = 0 if u =v. Now we obtain an upper bound on the quantity. We form anMn matrix whose rows are the elements in the logical groupLS I . Letm j 1 ;m j 2 ;m j 3 , andm j 4 be the number ofI,X,Y , andZ operators in column j of this matrix, respectively. So the equality P 4 l=1 m j l =M holds for all j2f1; ;ng. Each choice of a particular Pauli operator and some other 53 Pauli operator in the same column contributes exactly 2 to the sum P u;v2L wt(uv). Thus, the rst equality below holds for this reason, and the second holds by applying P 4 l=1 m j l =M: X u;v2LS I wt(uv) = n X j=1 4 X l=1 m j l (Mm j l ) = n X j=1 M 2 4 X l=1 (m j l ) 2 ! n X j=1 M 2 M 2 4 = 3n 4 M 2 : The rst inequality follows by applying P 4 l=1 m j l =4 =M=4 and convexity of the squaring function: (M=4) 2 = 4 X l=1 m j l =4 ! 2 4 X l=1 m j l 2 =4: Combining the lower and upper bounds gives us the EA Plotkin bound. Since the proof is independent of the number of ebits c, the EA Plotkin bound applies to arbitrary EAQEC codes. However, note that c does not appear in the bound, and consequently, this bound best describes the characteristics of maximal-entanglement EAQEC codes. However, for largek, the bound is approximately 3 4 n. Hence, this bound is useful only for small values of k. Remark 2.5.1. The EA Plotkin bound has been improved in the case that an EAQEC code is \linear" [GL13]. For EAQEC codes corresponding to classical linear quaternary codes, the linear EA Plotkin bound is d 3 2 2k 8(2 2k 1) (n +c +k): 54 2.5.5 Table of Lower and Upper Bounds on the Minimum Distance of Maximal-Entanglement EAQEC Codes Combining all the results in this and the previous section, we establish Table 2.1, which details lower and upper bounds on the minimum distance of maximal-entanglement EAQEC codes with length n 15. This table is improved in Ref. [LB13c]. Most of the lower bounds in Table 2.1 are slightly higher than the Gilbert-Varshamov bounds for n 15. Codes that achieve the upper bounds are detailed in Ref. [LBW13, LB13c]. The matching upper and lower bounds for k = 1 are from the family of EA repetition codes. The upper bounds for n 15 andk 2 are from the linear programming bound, which is generally tighter than the Singleton bound or the Hamming bound for non-degenerate EAQEC codes. The Plotkin bound and the linear programming bound match for k 2 and n 15. For k = 3 and n = 4; 5; 6; 9; 10; 11; 13; 14; 15, they also match. For k > 3, the Plotkin bound is not as tight as the linear programming bound, the Singleton bound, or the Hamming bound. We are able to build a table of the upper and lower bounds for any maximal- entanglement EAQEC codes with length up to 15 channel qubits in Table 2.1. The upper bounds for n 15 and k 2 are from the linear programming bound, which is generally tighter than the singleton bound [BDH06b] and the Hamming bound for non-degenerate EAQEC codes [Bow02]. Some of the linear programming bounds can be improved by individually discussing the existence of the EAQEC code. For example, it can be proved that [[n; 1;n;n 1]] and [[n;n 1; 2; 1]] EAQEC codes do not exist for evenn [LB13b]. These upper bounds are achieved by the construction of entanglement- assisted repetition code with parameters [[n; 1;n;n1]] forn odd and [[n; 1;n1;n1]] for n even in [LB13b]. More details are can be found in [LB13c]. 55 nnk 1 2 3 4 5 6 7 3 3 2 4 3 2 1 5 5 3 2 2 6 5 4 3 2 1 7 7 5 4 3 2 2 8 7 6 5 4 3 2 1 9 9 6 5-6 5 4 3 2 10 9 7 6 6 5 4 3 11 11 8 7 6 6 5 4 12 11 9 7-8 6-7 6 5-6 4 13 13 10 9 7-8 6-7 6 5 14 13 10 9 7-8 6-8 6-7 6 15 15 11 9-10 8-10 8-9 7-8 6-7 nnk 8 9 10 11 12 13 14 9 2 10 2 1 11 3 2 2 12 4 3 2 1 13 4 4 3 2 2 14 5 4 4 3 2 1 15 6 5 4 4 3 2 2 Table 2.1: Upper and lower bounds on the minimum distance of any [[n;k;d;nk]] maximal-entanglement EAQEC codes. 2.6 Discussion All of the gaps between the lower bound and upper bound d 5 in Table 2.1 are now closed. For n 8, the table is complete. The gaps for n = 14; 15 are still large for some k. Most lower bounds in Table 2.1 are from the optimization algorithm [LB13a]. To make the bounds in Table 2.1 tighter, we need to consider other code constructions to raise the lower bounds. We also plan to explore the existence of other [[n;k;d;nk]] codes to decrease the upper bound. Similar tables for EAQEC codes with 0<c<nk can be constructed by the same techniques. 56 It is possible to further improve the linear programming bounds by introducing new constraints. Consider a [[9; 3; 6; 6]] code if it exists. We may assume that the rst logical operator is g 1 =IIIXXXXXX: We dene the weight enumerator of the logical groupL with respect to g 1 by R L = 3 X a=0 6 X b=0 6b X c=0 A a b;c x 3a 0 x a 1 y 6bc 0 y b 1 y c 2 ; where a is the number of single-qubit Pauli operators on the rst 3 qubits that are not the identity,b is the number of single-qubit Pauli operators on the last 6 qubits that are X, and c is the number of single-qubit Pauli operators on the last 6 qubits that are Z or Y . As in section 2.2, it can be shown that R L ? = 1 jLj R L (x 0 + 3x 1 ;x 0 x 1 ;y 0 +y 1 + 2y 2 ;y 0 +y 1 2y 2 ;y 0 y 1 ): From this equation, we can put more constraints on a solution to the integer program with the variablesA j 's andB w 's and the constraints in Theorem 2.2.2 and Theorem 2.4.2. However, the resulting integer program may have many nontrivial linear equations. I am still trying to gure out an ecient way to implement this integer program problem. 57 Chapter 3 A Construction of Entanglement-Assisted Quantum Error-Correcting Codes: Random Optimization Brun, Devetak and Hsieh showed that an [n;k;d] classical linear quaternary code can be transformed to an [[n; 2kn +c;d;c]] EAQEC code that encodes 2kn +c information qubits inton qubits with the help ofc ebits for somec [BDH06b, BDH06a]. This EAQEC code can correct at leastb d1 2 c qubit errors and has the same minimum distance d as the classical code or higher. If entanglement is used, it boosts the rate of the code. However, it has not been explored how entanglement can instead help increase the min- imum distance. In addition, given parameters n;k;c, it is not clear how to construct an [[n;k;d;c]] EAQEC code directly. We say that an [[n;k;d;c]] EAQEC code is optimal if it saturates any upper bound on the minimum distance d for givenn;k; andc, and that an [[n;k;d;c]] EAQEC code is not equivalent to any standard quantum stabilizer code if there is no standard [[n +c;k;d]] quantum code. We will construct several optimal EAQEC codes that are not equivalent to any standard quantum stabilizer codes. Bowen constructed an EAQEC code from a three-qubit bit- ip code with the help of two pairs of maximally-entangled states (ebits) [Bow02]. He converted the two ancilla qubits to ebits and then applied a unitary transformation (another encoding operator) such that the EAQEC code is equivalent to the ve-qubit code [BDSW96, LMPZ96]. 58 Bowen's code, which can correct an arbitrary one-qubit error, serves as an example that entanglement increases the error-correcting ability of quantum codes. New EAQEC codes are constructed by adding ebits to a given regular stabilizer code. The minimum distance of these EAQEC codes can be optimized over distinct unitary row operators that determine the set of logical operators. We summarize the process in an encoding optimization procedure. If we add fewer than the maximum number of ebits, we have the freedom to choose the set of generators of the stabilizer group, and the freedom to replace dierent ancilla qubits with ebits. This leads to higher computational complexity. When n +k becomes large, the encoding procedure is intractable, and we adopt a random optimization procedure instead. 3.1 The Encoding Optimization Procedure for EAQECCs An [[n; 2k +cn;d;c]] EAQEC code can be constructed from an [n;k;d] classical linear quaternary code by the construction of [BDH06b], andc is determined by (1.11). It seems that only the number of information qubits is increased by introducing ebits. However, with the help of entanglement it is possible to dene more distinct error syndromes for a given codeword size, and hence the set of correctable error operators might be larger. We would like to construct EAQEC codes with a higher minimum distance instead of a higher rate. One way to construct an EAQEC code is to start with a standard QECC and move c of the qubits from Alice's side to Bob's side. So long as c d=2, the resulting code can be encoded by a unitary operator on Alice's side, given c ebits of initial shared entanglement between Alice and Bob [LB12]. While such codes can be interesting and useful, they are not the subject of interest for this chapter; because such codes retain an ability to correct errors on Bob's qubits, they are in a sense not making full use of the fact that Bob's halves of the ebits are noise-free. They therefore are less likely to have the maximum error correcting power on Alice's qubits for the given parameters n,k and 59 c. We are interested in EAQEC codes that can do better than any regular stabilizer code, in this sense. To make this idea precise, we say that an [[n;k;d;c]] EAQEC code is not equivalent to any standard stabilizer code if there is no standard [[n+c;k;d]] quantum code. If there exists a standard [[n+c;k;d]] quantum code, then we may not be achieving the maximum boost to our error correcting power from the c ebits of shared entanglement. We expect added entanglement in general to increase the error-correcting ability of a quantum error- correcting code, such that the EAQEC code is not equivalent to any regular stabilizer code, and indeed this turns out to be possible by our encoding optimization procedure. (Note that this is not always possible|the smallest examples of the [[3; 1; 3; 2]] and [[4; 1; 3; 1]] codes are both equivalent to the standard [[5; 1; 3]] QECC, and this is the best that can be done.) We now consider how added entanglement aects an [[n;k;d]] quantum stabilizer code C(S) dened by a stabilizer groupS =hg 0 1 ;g 0 2 ; ;g 0 r i The basic idea is to replace a set T ofc ancilla qubits by ebits. This introduces the symplectic partnersh 0 j s ofc generators g 0 j s to the generating set of the stabilizer groupS. An EAQEC code is obtained. As we will examine in detail below, the encoding unitary operator for a standard QECC is not uniquely dened. The EAQEC code dened by S 0 =hg 0 1 ; ;;g 0 r ;h 0 1 ; ;;h 0 c i may gain higher error-correcting ability by modifying the encoding operator. We rst discuss the case c = r, where the generator h 0 i is the symplectic partner of g 0 i for all i = 1; ;r. We will treat the case c < r later, by optimizing the choice of c linearly independent generators from the grouphh 0 1 ; ;;h 0 r i. 3.1.1 Selecting Symplectic Partners and Logical Operators Since the symplectic partners of g 0 1 ; ;g 0 r are not unique, we now explain how to select these partners such that the minimum distance of the EAQEC code is higher than the code without entanglement. Suppose W is a unitary Cliord operator that commutes 60 with Z 1 ; ;Z r such that after the operation of W , the simplied check matrix of the initial state (1.10) becomes 2 6 4 O rn I rr O r(nr) I rr A C B 3 7 5; (3.1) where A and B are two r (nr) binary matrices, and C is an rr binary matrix. The simplied check matrix satises the commutation relations (1.5{1.8) if C T +AB T +C +BA T =O rr : (3.2) In addition, it can be checked that the simplied logical matrix is of the form 2 6 4 O kn A T I kk O k(nk) I kk B T O kk 3 7 5 after Gaussian elimination such that (1.12) and (1.13) hold. Since (U E W )Z i (U E W ) y =U E Z i U y E =g 0 i for i = 1; ;r, U E W is also an encoding operator of the quantum stabilizer code C(S). However, the symplectic partners of the g 0 i 's, U E (WX i W y )U y E ; may dier from U E X i U y E for i = 1; ; r, and the logical operators U E (WX i W y )U y E ; U E (WZ j W y )U y E ; for i;j = r + 1; ; n are dierent. Choosing a set of matrices A, B, C such that C T +AB T +C +BA T =O rr determines a unitary operatorW by the encoding circuit algorithm, which in turn determines a set of symplectic partners of g 0 1 ; ;g 0 r and a set of logical operators. Thus we call W the selection operator for EAQEC codes. The minimum distance of the EAQEC code can be optimized by examining each distinct encoding operator U E W . Note that the simplied logical matrix is not aected by the matrix C. Therefore, there are 2 2rk distinct sets of logical operators. 61 Lemma 3.1.1. Given matrices A and B, then a matrix C that satises (3.2) is of the form C =BA T +M; where M is a symmetric matrix. Proof. SupposeC is matrix that satises Eq. (3.2). We can assume thatC =BA T +M for some matrix M. From Eq. (3.2), we have O rr =AB T +BA T +C 0 + (C 0 ) T =M +M T ; which implies that M is symmetric. We construct an EAQEC code that achieves the quantum singleton bound by apply- ing this procedure to a regular stabilizer code in the following example. Example 3.1.1. A check matrix of the regular [[5; 1; 1]] 5-qubit bit ip code (the repetition code) is 2 6 6 6 6 6 6 6 4 00000 11000 00000 01100 00000 00110 00000 00011 3 7 7 7 7 7 7 7 5 : Applying the encoding circuit algorithm to this check matrix, we obtain an encoding operator U E . In particular, if C =O rr in (3.2), then AB T +BA T =O rr : When k = 1; AB T +BA T = O rr holds if and only if A = B or at least one of A and B is the zero vector. Let W be the selection operator determined by the encoding 62 circuit algorithm with A = h 0 0 0 0 i T and B = h 1 0 1 0 i T . Then the encoding operator U E W generates a [[5; 1; 5; 4]] EAQEC code with a simplied check matrix 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 00000 11000 00000 01100 00000 00110 00000 00011 01111 00000 11000 00000 00011 00000 11110 00000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and a simplied logical matrix 2 6 4 11111 00000 00000 11111 3 7 5: With the help of 4 ebits, the minimum distance is increased from 1 to 5. The quantum singleton bound (1.14) is saturated by the parameters [[5; 1; 5; 4]]. Because the minimum distance of a regular [[9; 1]] quantum stabilizer code is at most 3 from the upper bound in [CRSS98], this [[5; 1; 5; 4]] code is not equivalent to any regular 9-qubit code. 3.1.2 Unitary Row Operators Since we have the freedom to choose among dierent sets of generators of a stabilizer group, and also the freedom to choose which ancilla qubits are replaced by ebits when c<r, we will show that the minimum distance can be further optimized over these two freedoms whenc<r. We rst discuss the eect of \unitary row operators" that preserve the overall commutation relations (1.5{1.8). 63 Consider a unitary operator U = 1 p 2 (I +iQ), where Q is a Pauli operator with eigenvalues1. It is easy to verify that UgU y = 8 < : g; if [Q;g] = 0; iQg; iffQ;gg = 0. We dene V 1;2 = V 3 V 2 V 1 , where V 1 = 1 p 2 (I +ig 0 1 h 0 2 ); V 2 = 1 p 2 (Iih 0 2 ); and V 3 = 1 p 2 (Iig 0 1 ). Then V 1;2 g 0 j V y 1;2 = 8 < : g 0 1 g 0 2 ; if j = 2; g 0 j ; if j6= 2. Therefore, V 1;2 is a unitary operator that performs multiplication of g 0 1 to g 0 2 , which corresponds to adding the rst row to the second in the simplied check matrix. On the other hand, V 1;2 h 0 j V y 1;2 = 8 < : h 0 2 h 0 1 ; if j = 1; h 0 j ; if j6= 1. Hence a row operation performed onfg 0 1 ; ;g 0 r g induces a row operation performed on fh 0 1 ; ;h 0 r g in order to preserve the commutation relations (1.5{1.8). We call V 1;2 a unitary row operator. Later we will need unitary row operators that change h 0 j to h 0 j g 0 i , h 0 j to h 0 j Z 0 i , and h 0 j to h 0 j X 0 i seperately. These four types of unitary row operators are summarized in Table 3.1. Type 1: Vh 0 j V y = h 0 l h 0 m ; if j =l; h 0 j ; if j6=l. Vg 0 j V y = g 0 m g 0 l ; if j =m; g 0 j ; if j6=m. Type 2: Vh 0 j V y = h 0 l g 0 m ; if j =l; h 0 j ; if j6=l. Vh 0 j V y = h 0 m g 0 l ; if j =m; h 0 j ; if j6=m. Type 3: Vh 0 j V y = h 0 l Z 0 m ; if j =l; h 0 j ; if j6=l. V X 0 j V y = g 0 l X 0 m ; if j =m; X 0 j ; if j6=m. Type 4: Vh 0 j V y = h 0 l X 0 m ; if j =l; h 0 j ; if j6=l. V Z 0 j V y = g 0 l Z 0 m ; if j =m; Z 0 j ; if j6=m. Table 3.1: Four types of unitary row operators 64 When a dierent set of generators of the stabilizer group is chosen instead of fg 0 1 ; ;g 0 r g, this is equivalent to performing a unitary transformation V , which com- prises a sequence of unitary row operators of type 1 onfg 0 1 ; ;g 0 r g. The eect of V on the simplied check matrix H 0 corresponding tofg 0 1 ; ;g 0 r ;h 0 1 ; ;h 0 r g is to multiply H 0 from the left by a (2n 2k) (2n 2k) matrix of the form M V = 2 4 M Z O (nk)(nk) O (nk)(nk) M X 3 5 : If M X = R m R m1 R 1 , where the R 0 i s are elementary row operations, then M Z = R T m R T m1 R T 1 : It can be checked that MH 0 satises (1.9). If a set T =ft 1 ; ;t c g of c<r ancilla qubits are replaced by ebits, it is possible that after the operation of V , the groupS 0 I =hg j :j = 2Ti changes, and so does the setN (S 0 )S 0 I . In addition, the span of a subset offh 0 1 ; ;h 0 r g can change after the operation of V , though the span of the full set remains unchanged. This means that if we add less than the maximum amount of entanglement to a code, we must optimize over all such unitary row operations. Since the groupS 0 I and the setN (S 0 )S 0 I remain the same under type 1 unitary row operators on theh 0 j forj = 2T , it suces to assume that the operation V consists of type 1 unitary row operators that operate only on the h 0 j for j2T . Let M V be a cr matrix such that the i-th row of M V is the t i -th row of M Z for i = 1; ;c. It is obvious that dierent M V 's can have the same eect on the row space ofH 0 . For example, ifc = 2;fg 0 1 g 0 2 ;g 0 2 ; ;g 0 r ;h 0 1 ;h 0 1 h 0 2 g andfg 0 1 ;g 0 2 ; ;g 0 r ;h 0 1 ;h 0 2 g are two dierent sets of generators, but they generate the same group, and hence their corresponding EAQEC codes have the same minimum distance. Therefore, without loss of generality a distinct unitary row operationV can be assumed to be be represented by a matrix M V in reduced row echelon form. 65 Theorem 3.1.1. The operation of V is equivalent to applying a series of type 1 unitary row operators on the h 0 j for j2T . There are N(r;c), rc X lc=0 lc X l c1 =0 l c1 X l c2 =0 l 2 X l 1 =0 2 c(rc) P c i=1 l i distinct unitary row operations. Proof. The total number of distinct unitary row operations N(r;c) is determined as follows. If we begin with matrices of the form 2 6 6 6 6 6 6 6 4 1 0 0 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 0 1 3 7 7 7 7 7 7 7 5 ; where can be 0 or 1, there are 2 c(rc) distinct unitary row operations. Now we consider matrices in which the leading ones are shifted to the right. Letl j denote the shift amount of the leading 1 of j-th row from its initial position for j = 1; ;c. It can be observed that l j l i if j <i. For a setfl 1 ;l 2 ; ;l c g, the number of is c(rc) P c i=1 l i , and hence there are 2 c(rc) P c i=1 l i distinct unitary row operations. Therefore, summing over all possible sets offi 1 ; ;i c g shows that there is a total of N(r;c) = rc X lc=0 lc X l c1 =0 l c1 X l c2 l 2 X l 1 =0 2 c(rc) P c i=1 l i distinct unitary row operations up to Gaussian elimination. The function N(r;c) has a symmetry given in the following lemma, which can be proved by induction. Lemma 3.1.2. N(r;c) =N(r;rc) for any r and 0cr: 66 On the other hand, the selection operator W in the previous subsection can be decomposed as a series of unitary row operators of type 2, type 3, and type 4. The matrix A determines a series of type 4 unitary row operators, the matrix B determines a series of type 3 unitary row operators, and the symmetric matrixM, satisfyingC =BA T +M, determines a series of type 2 unitary row operators. Unitary row operators of type 2 do not aect the setN (S 0 )S 0 I or the error-correcting ability, and so the symmetric matrix M can be dropped. It is the same as choosing a dierent basis for the same code space. If a set T =ft 1 ; ;t c g of c<r ancilla qubits are replaced by ebits, one can show that N (S 0 ) =hg j :j = 2T; Z 1 ; ; Z k ; Z 1 ; ; Z k i remains unchanged by the operation of type 3 and type 4 unitary row operators on the h 0 j for j = 2 T . It suces to assume that the operationW consists of type 3 and type 4 unitary row operators that act only on the h 0 j for j2T . To sum up, we have the following theorem. Theorem 3.1.2. The operation ofW is equivalent to applying a series of type 4 unitary row operators, followed by a series of type 3 unitary row operators, on the h 0 j for j2T . There are 2 2ck distinct selection operators with C =BA T : Combining the eects of the unitary row operation V with the selection operator W in the previous section, we can optimize an encoding operation of the form U =VU E W over 2 2ck N(r;c) possibilities. We call this the encoding optimization procedure for EAQEC codes. Note that we can nd another unitary row operator W 0 corresponding to W such that W 0 U E and U E W are equivalent encoding operators. While W operates on the raw stabilizer generators and logical operators, W 0 operates on the encoded stabilizer generators and logical operators. Hence, we can also solve the optimization problem for an operator of the form U = VW 0 U E (which is what we actually do in practice, combining VW 0 into a single optimization). 67 3.2 Results 3.2.1 Results of the Encoding Optimization Procedure We applied the encoding optimization procedure to a [[7; 1; 3]] quantum BCH code [GB99, AKS07] and Shor's [[9; 1; 3]] code [Sho95], and the results are shown in Table 3.2 and Table 3.3, whered opt is the minimum distance of the optimized EAQEC codes, and d std is the highest minimum distance of an [[n +c;k]] regular stabilizer code. Example 3.2.1. The check matrix of a regular [[7; 1; 3]] quantum BCH code adopted in the encoding optimization procedure is 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0000000 1001011 0000000 0101110 0000000 0010111 1001011 0000000 1100101 0000000 1011100 0000000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : As shown in Table 3.2, the parameters [[7; 1; 7; 6]], [[7; 1; 5; 3]] and [[7; 1; 5; 2]] achieve the quantum singleton bound for EAQEC codes (1.14) and are not equivalent to any standard quantum stabilizer code. We would like to compare these two EAQEC codes to a competing EAQEC code with n = 7 and d = 5 by the construction of [BDH06b]. According to Grassl's table [Gra], a classical linear code over GF (4) (or GF (2)) that meets our requirement is a [7; 2; 5] linear quaternary code, which can be used to construct a [[7; 2; 5; 5]] EAQEC code. This means that the [[7; 1; 5; 2]] and [[7; 1; 5; 3]] EAQEC codes cannot be obtained by the construction of [BDH06b], and thus are new. 68 In addition, all the [[7; 1; 5; 2]] EAQEC codes we found are degenerate codes, for some simplied stabilizer generators are of weight 4 from the check matrix. For example, the simplied check matrix and simplied logical matrix of a [[7; 1; 5; 2]] EAQEC code are 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0000000 1001011 0000000 1100101 0000000 0010111 1001011 0000000 1100101 0000000 0010111 0000000 1000011 0100011 1101000 0010010 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; 2 4 1001011 0100011 1101000 1001011 3 5 ; with T =f1; 4g. On the other hand, all the [[7; 1; 7; 6]] EAQEC codes are nondegen- erate codes, while [[7; 1; 5; 3]], [[7; 1; 5; 4]], and [[7; 1; 5; 5]] EAQEC codes can be either degenerate or nondegenerate. Table 3.2: Optimization over the [[7; 1; 3]] quantum BCH code c 1 2 3 4 5 6 d opt 3 5 5 5 5 7 d std 3 3 4 5 5 5 69 Example 3.2.2. The check matrix of Shor's [[9; 1; 3]] code is 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000000000 110000000 000000000 011000000 000000000 000110000 000000000 000011000 000000000 000000110 000000000 000000011 111111000 000000000 000111111 000000000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : As can be seen in Table 3.3, the parameters [[9; 1; 9; 8]], [[9; 1; 7; 5]] and [[9; 1; 7; 4]] achieve the quantum singleton bound for EAQEC codes (1.14) and are not equivalent to any regular stabilizer code. A competing EAQEC code with n = 9 and d = 7 by the construction of [BDH06b] is a [[9; 1; 7; 6]] EAQEC code, obtained from a [9; 2; 7] linear quaternary code in Grassl's table. Therefore, the [[9; 1; 7; 5]] and [[9; 1; 7; 4]] EAQEC codes are new. All the [[9; 1; 5; 2]], [[9; 1; 5; 3]], [[9; 1; 7; 4]], [[9; 1; 7; 5]] and [[9; 1; 7; 6]] codes are degenerate codes, and all the [[9; 1; 9; 8]] codes are nondegenerate codes, while the [[9; 1; 7; 7]] codes can be either degenerate or nondegenerate. Table 3.3: Optimization over Shor's [[9; 1; 3]] code c 2 3 4 5 6 7 8 d opt 5 5 7 7 7 7 9 d std 5 5 5 6 6 6 7 70 3.2.2 Random Optimization Procedure It is easy to check that 2 c(n+kc) 2 2ck N(r;c) r c 2 c(n+kc) : A complete encoding optimization procedure for a [[n;k;d]] regular stabilizer code becomes impossible when n +k becomes large. Hence one can consider random search algorithms for the encoding optimization procedure. For each iteration of optimization, we randomly generate two matrices A andB, and randomly choose a unitary row oper- ation V . Then we optimize the minimum distance until a target minimum distance is obtained or a preset of maximum number of iterations is reached. Some examples of random optimization follow: Example 3.2.3. We applied the random optimization algorithm to Gottesman's [[8; 3; 3]] code [Got96] and the results are shown in Table 3.4. By the construction of [BDH06b], the [8; 3; 5] classical linear quaternary codes in Grassl's Table can be transformed to an [[8; 2; 5; 4]] EAQEC code. Hence the [[8; 3; 5; 5]] and [[8; 3; 4; 3]] EAQEC codes are new, and are not equivalent to any regular stabilizer code. In addition, these two EAQEC codes saturate the linear programming bounds and are optimal. Table 3.4: Optimization over Gottesman's [[8; 3; 3]] code c 2 3 4 5 d opt 3 4 4 5 d std 3 3 4 4 Example 3.2.4. We applied random optimization to a [[15; 7; 3]] quantum BCH code and the results are shown in Table 3.5. Note that could not fully optimize parameters in Table 3.5: Optimization over a [[15; 7; 3]] Quantum BCH code c 3 4 5 6 7 8 d opt 3 4 4 5 5 6 d std 4 4-5 4-5 5-6 5-6 5-6 71 this case, since the complexity is very high. However, compared with the [[15; 3; 5; 4]] EAQEC code, obtained by the construction of [BDH06b] from a [15; 7; 5] classical BCH code, the [[15; 7; 5; 7]] and the [[15; 7; 5; 6]] EAQEC codes have 4 more information qubits at the cost of 3 and 2 more ebits, respectively. The [[15; 7; 6; 8]] EAQEC code has 4 more information qubits and a higher minimum distance at the cost of 4 more ebits. In addition, the [[15; 7; 6; 8]] EAQEC code is not equivalent to any known regular stabilizer code. On the other hand, the classical linear quaternary [15; 9; 5] code and [15; 8; 6] code in Grassl's table can be used to construct a [[15; 9; 5; 6]] EAQEC code and a [[15; 8; 6; 7]] EAQEC code by the construction of [BDH06b]. These codes are better than the [[15; 7; 6; 8]] EAQEC code we obtained. This may be because our codes were not fully optimized, but in any case BCH codes may not give the best possible EAQEC codes, even using the encoding optimization procedure. Example 3.2.5. We applied the random optimization algorithm to the [[13; 1; 5]] quantum quadratic residue (QR) code [CRSS97, LL11], and the results are shown in Table 3.6. By the construction of [BDH06b], the [13; 3; 9], [13; 4; 8], and [13; 5; 7] classical linear quaternary codes in Grassl's table can be transformed to [[13; 3; 9; 10]], [[13; 0; 8; 5], and [[13; 1; 7; 4]] EAQEC codes, respectively. The [[13; 1; 11; 11]], [[13; 1; 11; 10]], [[13; 1; 9; 9]], and [[13; 1; 9; 8]] EAQEC codes are new, and are not equivalent to any regular stabilizer code. Table 3.6: Optimization over the [[13; 1; 5]] quantum QR code c 4 5 6 7 8 9 10 11 12 d opt 7 7 7 7 9 9 11 11 13 d std 7 7 7 7 7 7-8 7-9 8-9 9 72 3.3 Discussion We demonstrated the encoding optimization procedure for EAQEC codes obtained by adding ebits to standard quantum stabilizer codes. The four types of unitary row oper- ators play an important role in this encoding optimization procedure, and also help to clarify the properties of EAQEC codes and their relationship to standard codes. Some applications of the encoding optimization procedure were found to have promising results: we constructed [[7; 1; 5; 2]] and [[7; 1; 5; 3]] EAQEC codes from quantum BCH codes; [[8; 3; 5; 5]] and [[8; 3; 4; 3]] EAQEC codes from Gottesman's 8-qubit code; and [[9; 1; 7; 4]] and [[9; 1; 7; 5]] EAQEC codes from Shor's 9-qubit code; together with a fam- ily of EA repetition codes, all of which are optimal. Several of the EAQEC codes found by this encoding optimization procedure are degenerate codes. This procedure serves as an EAQEC code construction method for given parameters n;k;c. Some of our EAQEC codes use large numbers of ebits. However, it is still worthwhile to study EAQEC codes that use large entanglement. The one-shot father protocol is a random EA quantum code, and it achieves the EA hashing bound [BSST99, Bow02, DHW08, DHW04]. Maximal-entanglement EA turbo codes come close the EA hashing bound within a few dB [WH11]. Asymptotically, maximal-entanglement codes achieve the EA capacity [DHW04, DHW08]. The encoding optimization procedure has very high complexity. However, it might be useful to further investigate it for specic families of codes that have special algebraic structures, such as quantum BCH codes and quantum Reed-Muller codes. This is future work. While the encoding optimization procedure in this chapter applies to a standard quantum stabilizer code, it is possible to construct a similar encoding optimization algo- rithm for adding ebits to other EAQEC codes that use less than the maximum amount of entanglement. By adding a small amount of entanglement we may reduce the search 73 space and make optimization more computationally tractable. It also might be pos- sible to generate small or moderately sized EAQECCs randomly, by choosing random selections of simplied generators, and to search in this way for codes with desirable properties. Much work remains to be done in nding the best possible EAQEC codes for dierent applications. 74 Chapter 4 Random EAQEC Codes 4.1 The Weight Enumerator Bound on the Block Error Probability under Maximum A Posteriori Decoding Since maximal-entanglement codes bear many similarities to classical codes, the block error probability when transmitting coded quantum information through the depolarizing channel can be upper bounded with the weight enumerator of a particular maximal- entanglement EAQEC code (similarly to the case for classical codes [RU08, McE02]). This \weight enumerator bound" gives an idea of the performance of maximum-likelihood decoding of an arbitrary maximal-entanglement EAQEC code. We can also determine the expected performance when decoding a random EAQEC code with a maximum likelihood decoding rule. Below, we determine these bounds and plot them for the maximal-entanglement repetition and accumulator EAQEC codes. The result is that these codes perform comparably to a random EA code with respect to this upper bound. Theorem 4.1.1. Suppose that a sender transmits an [[n;k;nk]] maximal- entanglement EAQEC code over a depolarizing channel with parameter p, and further- more, that the receiver decodes this code according to a maximum a posteriori (MAP) decoding rule. Then we have the following upper bound on the block error probability P B : P B B ( ) 1; (4.1) 75 where B (z),W L (1;z) is the weight enumerator of the maximal-entanglement EAQEC code and is the \Bhattacharyya parameter" for the depolarizing channel: 2 r p 3 (1p) + 2 3 p: Proof. LetU be a Cliord encoder for the [[n;k;nk]] maximal-entanglement EAQEC code. The encoded statej i AB isj i AB = (U A I B ) ji (j + i AB ) (nk) : Then Alice transmits her qubits (entangled with Bob's qubits) through n independent uses of a depolarizing channelE where E () = (1p) + p 3 (XX +YY +ZZ); and is the density operator of a single qubit. We assume that p < 3=4 because the channel is completely depolarizing whenp = 3=4. Suppose that an error operator ~ E2G n occurs after the depolarizing channel, and that s x ; s z are the binary vector representa- tions of the error syndrome. Both s x and s z are of length (nk), and Bob observes them by rst decoding the qubits with a decoding unitary U y and then performing Bell measurements on the ebits. This implies that (U y ~ E) A I B j i AB = ~ L 0 ji X s x Z s z A I B j + i AB nk ; and ~ E = U( ~ L 0 X s x Z s z )U y for some logical error ~ L 0 2L 0 , whereL 0 is the set of unencoded logical operators. Poulin et al. devised a maximum a posteriori decoder for standard stabilizer codes [PTO09], 1 and we can modify their decoder to be a maximum a posteriori decoder L MAP (s x ; s z ) for maximal-entanglement EAQEC codes, where L MAP (s x ; s z ) arg max L2L 0 PrfLjs x ; s z g: 1 Poulin et al. described their decoder as a \maximum-likelihood" decoder [PTO09], but a careful study of it reveals that their decoder should more properly be called a maximum a posteriori decoder. 76 This decoder selects the most likely error operator acting on the logical qubits, given the syndrome information s x and s z . We can calculate the above conditional distribution by applying the Bayes rule to the joint distribution PrfL; s x ; s z g: PrfLjs x ; s z g = PrfL; s x ; s z g P L 0 PrfL 0 ; s x ; s z g ; where Pr L;s x ; s z = PrfEgj E=U(L X s x Z s z )U y = (1p) nwt(E) p 3 wt(E) E=U(L X s x Z s z )U y = (1p) n p 3 (1p) wt(E) E=U(L X s x Z s z )U y : The distribution P L 0 PrfL 0 ; s x ; s z g is xed over all choices of L. Since p < 3=4() p= (3 (1p)) < 1, the best choice of L for the maximum a posteriori decoder L MAP (s x ; s z ) is the one that selects a recovery operator L 1 = L such that E = U(L X s x Z s z )U y has the minimum weight. This minimum weight decoder is similar to a classical minimum distance decoder. LetL be the set of encoded logical operators andS 0 be the set of simplied stabilizer generators. Given E 0 2L 0 , let Q(E 0 )fs x ; s z : Prf ~ L 0 E 0 ; s x ; s z g Prf ~ L 0 ; s x ; s z gg: 77 We can now bound the probabilityP B ( ~ L 0 ) of a block error given that the error operator ~ L 0 occurs under this decoding scheme: P B ( ~ L 0 ) = PrfMAP decoder failsj ~ L 0 occursg = Pr n L MAP (s x ; s z )6= ~ L 0 o = Pr n ~ L 0 L MAP (s x ; s z )6=I o = Pr n ~ L 0 L MAP (s x ; s z )2L 0 nI o = X E 0 2L 0 nI Pr n ~ L 0 L MAP (s x ; s z ) =E 0 o X E 0 2L 0 nI X s x ;s z 2Q(E 0 ) Pr n ~ L 0 ; s x ; s z o : Since r Prf ~ L 0 E 0 ;s x ;s z g Prf ~ L 0 ;s x ;s z g 1 for s x ; s z 2Q(E 0 ), we can multiply each term in sum by this factor and then P B ( ~ L 0 ) X E 0 2L 0 nI X s x ;s z 2Q(E 0 ) r Pr n ~ L 0 ; s x ; s z o Pr n ~ L 0 E 0 ; s x ; s z o X E 0 2L 0 nI X s x ;s z 2Z nk 2 r Pr n ~ L 0 ; s x ; s z o Pr n ~ L 0 E 0 ; s x ; s z o = X E2LnI X M2S 0 r Pr n ~ LM o Pr n ~ LEM o X E2LnI X M2G n r Pr n ~ LM o Pr n ~ LEM o ; 78 where ~ L =U( ~ L 0 I)U y , E =UE 0 U y , and M =U(I X s x Z s z )U y 2S 0 . Observe that X M2G n r Pr n ~ LM o Pr n ~ LEM o = X M2G n r PrfMg Pr n ~ LE ~ LM o = X M2G n n Y i=1 r Prf(M) i g Pr n ( ~ L) i (E) i ( ~ L) i (M) i o = n Y i=1 X (M) i 2G r Prf(M) i g Pr n ( ~ L) i (E) i ( ~ L) i (M) i o : It holds that ( ~ L) i (E) i ( ~ L) i 6=I if (E) i 6=I and so X (M) i 2G r Prf(M) i g Pr n ( ~ L) i (E) i ( ~ L) i (M) i o = 2 r p 3 (1p) + 2 3 p = : Otherwise, X (M) i 2G r Prf(M) i g Pr n ( ~ L) i (E) i ( ~ L) i (M) i o = 1: Consequently, P B ( ~ L 0 ) X E2LnI wt( ~ LE ~ L) = X E2LnI wt(E) =B( ) 1: Therefore, the probability P B of a block error is bounded by B( ) 1 when taking the expectation over all ~ L 0 . The above theorem is similar to Theorem 7.5 in Ref. [McE02], which determines an upper bound on the block error probability when transmitting a classical linear code over a binary symmetric channel. Theorem 4.1.2. Suppose that the sender transmits a random [[n;k;nk]] maximal- entanglement EAQEC code over a depolarizing channel with parameterp and furthermore 79 that the receiver decodes this code according to a maximum a posteriori decoding rule. Let U be the Cliord encoder for this code. Then we have the following upper bound on the expected block error probability P B : P B =E U fP B g 2 2k 1 2 2n 1 ((1 + 3 ) n 1); (4.2) where is the Bhattacharyya parameter dened in the previous theorem and the expecta- tion is with respect to the choice of random code. In particular, if the rate k=n satises the following upper bound: k n < 1 1 2 log 2 (1 + 3 ); then the error probability decreases exponentially to zero in the asymptotic limit. Proof. We rst establish a method for choosing a random maximal-entanglement EAQEC code. A natural method for doing so is rst to x a basis of Pauli opera- tors X 1 , Z 1 , X 2 , Z 2 , . . . , X n , Z n , where the rst nk anticommuting pairs correspond to the stabilizer operators for the nk ebits and the next k anticommuting pairs cor- respond to the logical operators for the k information qubits. We then select a Cliord unitary uniformly at random from the Cliord group (see Section VI-A-2 of Ref. [DLT02] for a relatively straightforward algorithm for doing so) and apply it to the above xed basis. This procedure produces 2n encoded operatorsX 1 ,Z 1 ,X 2 ,Z 2 , . . . ,X n ,Z n that specify a random maximal-entanglement EAQEC code. We now need to determine the expected weight enumerator E U fB(z)g = P n w=0 E U fB w gz w for such a random maximal-entanglement code. This will allow us to apply Theorem 4.1.1 to get an upper bound on the expected block error probabil- ity. Each coecientE U fB w g corresponds to the expected number of Pauli operators of weight w that belong to the logical operator group of a random EA code. Equivalently, it corresponds to the expected number of Pauli operators of weight w that commute with the entanglement subgroup of a random code. First, let us considerE U fB 0 g. The identity operator is the only Pauli operator with weight zero. It commutes with all 80 operators with unit probability. Thus,E U fB 0 g = 1. Now, let us considerE U fB w g with w 1. We rst determine the probability that a Pauli operator g with non-zero weight commutes with the 2 (nk) encoded operators X 1 , Z 1 , X 2 , Z 2 , . . . , X nk , Z nk for a random EA code. To simplify the calculation, observe that applying a uniformly random Cliord unitary to the operatorsX 1 ,Z 1 ,X 2 ,Z 2 , . . . ,X nk ,Z nk and then determining the probability that a xed operator g commutes with all of them is actually the same as keeping the basis xed and applying a random Cliord to the operator g itself. This holds because CfC y ggCfC y = 0 () fC y gCC y gCf = 0: Then a uniform distribution on the Cliord unitaries takes this operatorg to an arbitrary Pauli operator g 0 , and the distribution induced is just the uniform distribution on all of the 2 2n 1 n-qubit Pauli operators not equal to the identity (this reasoning is the same as that in Section VI-A-1 of Ref. [DLT02]). At this point, the argument becomes purely combinatorial, and the only operators that commute with the above xed basis are the ones with identity acting on the rst nk qubits. Thus, there are 2 2k 1 Pauli operators besides the identity that commute with the xed basis, and we conclude that the probability that a xed Pauli operator g with non-zero weight commutes with the random set X 1 , Z 1 , X 2 , Z 2 , . . . , X nk , Z nk is 2 2k 1 2 2n 1 : Now we can calculate the expected number of operators that are in the logical subgroup. The number of Pauli operators with weight w is n w 3 w . Consequently, we have E U fB w g = 2 2k 1 2 2n 1 n w 3 w ; 81 which implies E U fB(z)gE U fB 0 g = n X w=1 E U fB w gz w = 2 2k 1 2 2n 1 n X w=1 n w 3 w z w = 2 2k 1 2 2n 1 ((1 + 3z) n 1): Therefore, by exploiting the result in Theorem 4.1.1, an upper bound on the expected block error probability for general EAQEC codes with maximal entanglement is E U fP B gB( )B 0 ; = 2 2k 1 2 2n 1 ((1 + 3 ) n 1): We can drive the expected error probability to be arbitrarily low in the large n and k limit by ensuring that k n < 1 1 2 log 2 (1 + 3 ): (4.3) This bound is not as tight as the EA hashing bound (the optimal limit), and Figure 4.1 displays how these two bounds dier. We can plot the error probability bound in (4.1) as a function of p for specic codes such as the repetition codes or the accumulator codes and then compare the results with the average error probability bound for a random code. Figure 4.2 provides such plots and compares their performance with a random EA code, with respect to these bounds. 4.2 Hashing Bounds for Pauli Channels The hashing bound of a quantum channel is an achievable rate for reliable quantum communication [BDSW96], and as such, it constitutes a lower bound on the quantum capacity of a Pauli channel [Llo97, Sho02, Dev05]. For a Pauli channel, this bound has a 82 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depolarizing parameter Rate Entanglement−assisted hashing bound Weight enumerator bound Figure 4.1: The gure plots both the EA hashing bound 1 1=2 [H 2 (p) +p log 2 3] from Ref. [BSST99] and the \asymptotic weight enumerator bound" from (4.3) as a function of the depolarizing parameter. The two bounds become close for high depolarizing noise. Interestingly, the thresholds of the maximal-entanglement EA turbo codes from Ref. [WH11] are just shy of the asymptotic weight enumerator bound (see Figures 6(b) and 7(b) of that paper). simple form and the proof that it is achievable is particularly simple as well. In this sec- tion, we summarize several variations of the hashing protocol for reliable communication in the asymptotic limit of many channel uses. In particular, one of the hashing bounds demonstrates that a maximal-entanglement EAQEC achieves the entanglement-assisted quantum capacity of a Pauli channel. We consider random stabilizer codes and use some techniques from the previous section. For more details about quantum Shannon theory, we refer interested readers to Ref. [Wil13] and references therein. 4.2.1 Hashing Bound for Stabilizer Codes We rst review the proof of the hashing bound for stabilizer codes [Smi06] in order to have it available for helping to obtain the proofs of the hashing bounds for EAQEC codes. 83 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depolarizing rate Block Error Probability Maximal−Entanglement Repetition Codes 10 −3 10 −2 10 −1 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depolarizing rate Block Error Probability Random Maximal−Entanglement Code with Rate (n−1)/n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depolarizing rate Block Error Probability Random Maximal−Entanglement Code with Rate 1/n 10 −3 10 −2 10 −1 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depolarizing rate Block Error Probability Maximal−Entanglement Accumulator Codes Increasing n Increasing n Increasing n Increasing n (a) (b) (c) (d) Figure 4.2: The gures plot the weight enumerator bound in (4.2) as a function of the depolarizing parameter p for various nite-length codes. (a) The weight enumerator bound for maximal-entanglement repetition codes of length 3 to 12. (b) The expected weight enumerator bound for random rate 1=n maximal-entanglement codes of length 3 to 12. (c) The weight enumerator bound for maximal-entanglement accumulator codes of length 3 to 12. (d) The expected weight enumerator bound for random rate (n 1)=n maximal-entanglement codes of length 3 to 12. Observe that the performance of the maximal-entanglement repetition and accumulator codes with respect to this upper bound is comparable to the expected performance of random maximal-entanglement codes. 84 Theorem 4.2.1 (Hashing Bound). There exists a quantum stabilizer code that achieves the hashing limit R = 1H (p) for a Pauli channelE of the following form: E() =p I +p X XX +p Y YY +p Z ZZ; where is the density operator of a single qubit, p = (p I ;p X ;p Y ;p Z ) and H (p) = P i2fI;X;Y;Zg p i log 2 (p i ) is the entropy of this probability vector. Proof. We need to correct only the typical errors. Dene the typical error set as follows: T p n a n : 1 n log 2 (PrfE a ng)H (p) ; where a n is some sequence consisting of the lettersfI;X;Y;Zg and PrfE a ng is the probability that an independent and identically distributed (IID) Pauli channel issues some tensor-product errorE a nE a 1 E an , whereE a j 2fI;X;Y;Zg. This typical set consists of the likely errors in the sense that X a n 2T p n PrfE a ng 1; (4.4) for all > 0 and suciently large n. The quantum error correction conditions for a stabilizer code dened by a stabilizer groupS in this case are thatfE a n :a n 2T p n g is a correctable set of errors if E y a nE b n = 2N (S)n ~ S; 85 for all error pairs E a n and E b n such that a n ;b n 2 T p n , where ~ S =feg : g2S;e2 fI;iIgg. Also, we consider the expectation of the error probability under a random choice of a stabilizer code and proceed to bound it as follows: E S fp e g =E S ( X a n PrfE a ngI (E a n is uncorrectable underS) ) E S 8 > < > : X a n 2T p n PrfE a ngI (E a n is uncorrectable underS) 9 > = > ; + = X a n 2T p n PrfE a ngE S fI (E a n is uncorrectable underS)g + = X a n 2T p n PrfE a ng Pr S fE a n is uncorrectable underSg +: The rst equality follows by denition: I is an indicator function equal to one if E a n is uncorrectable underS and equal to zero otherwise. The rst inequality follows from (4.4): we correct only the typical errors because the atypical error set has negligible probability mass. The second equality follows by exchanging the expectation and the 86 sum. The third equality follows because the expectation of an indicator function is the probability that the event it selects occurs. Continuing, we have = X a n 2T p n PrfE a ng Pr S n 9E b n :b n 2T p n ; b n 6=a n ; E y a nE b n2N (S)n ~ S o X a n 2T A n PrfE a ng Pr S n 9E b n :b n 2T p n ; b n 6=a n ; E y a nE b n2N (S) o = X a n 2T p n PrfE a ng Pr S 8 > < > : [ b n 2T p n ; b n 6=a n E y a nE b n2N (S) 9 > = > ; X a n ;b n 2T p n ; b n 6=a n PrfE a ng Pr S n E y a nE b n2N (S) o X a n ;b n 2T p n ; b n 6=a n PrfE a ng 2 (nk) 2 2n[H(p)+] 2 n[H(p)+] 2 (nk) = 2 n[1H(p)k=n] : The rst equality follows from the error correction conditions for a quantum stabilizer code. The rst inequality follows by ignoring any potential degeneracy in the code|we consider an error uncorrectable if it lies in the normalizer N (S) and the probability can only be larger because N (S)n ~ SN (S). The second equality follows by realizing that the probabilities for the existence criterion and the union of events are equivalent. The second inequality follows by applying the union bound. The third inequality follows from the fact that the probability for a xed operator E y a nE b n not equal to the identity commuting with the stabilizer operators of a random stabilizer can be upper bounded as follows: Pr S n E y a nE b n2N (S) o = 2 n+k 1 2 2n 1 2 (nk) : The reasoning here is similar to the reasoning in Theorem 4.1.2. The random choice of a stabilizer code is equivalent to xing operatorsZ 1 , . . . ,Z nk and performing a uniformly 87 random Cliord unitary. The probability that a xed operator commutes with Z 1 , . . . , Z nk is then just the number of non-identity operators in the normalizer (2 n+k 1) divided by the total number of non-identity operators (2 2n 1). After applying the above bound, we then exploit the following typicality bounds: 8a n 2T p n : PrfE a ng 2 n[H(p)+] ; T p n 2 n[H(p)+] : We conclude that as long as the rate k=n = 1H (p) 2, the expectation of the error probability becomes arbitrarily small, so that there exists at least one choice of a stabilizer code with the same bound on the error probability. 4.2.2 Entanglement-Assisted Quantum Error-Correcting Codes 4.2.2.1 Maximal-Entanglement Codes Now consider the case of an EAQEC code. At rst, we choose the code to be a maximal- entanglement EAQEC code, so that there are only information qubits or shares of ebits sent into the encoder. The quantum error correction conditions in such a case become thatfE a ng is a correctable set of errors if E y a nE b n = 2N (S S ); for all error pairs E a n and E b n in the error set, whereS S is the symplectic subgroup of the stabilizer code. It follows for a random EAQEC code of this form that Pr S n E y a nE b n2N (S E ) o = 2 2k 1 2 2n 1 2 2(nk) ; 88 because there are 2 2k 1 nonidentity operators that commute with the 2 (nk) operators that generateS S . By modifying the last few steps of the above proof as follows X a n ;b n 2T p n ; b n 6=a n PrfE a ng 2 2(nk) 2 2n[H(p)+] 2 n[H(p)+] 2 2(nk) = 2 2n[1H(p)=2k=n=2] ; we obtain the hashing bound for EAQEC codes: Theorem 4.2.2 (EA Hashing Bound). There exists a maximal-entanglement EAQEC code that achieves the EA hashing limit R = 1H (p)=2 for a Pauli channel with parameters p. 4.2.2.2 Non-Maximal-Entanglement Codes We could also consider codes that do not use the maximal amount of ebits possible. In this case, there are k information qubits, nkc ancilla qubits, and c ebits. The quantum error correction conditions in this case become thatfE a ng is a correctable set of errors if E y a nE b n = 2N (S S ;S I )n ~ S I ; for all error pairsE a n andE b n in the error set, whereS S is the symplectic subgroup and S I is the isotropic subgroup of the EAQEC code, and ~ S I =feg :g2S I ;e2fI;iIgg. Focusing only on non-denegerate errors, the error-correcting conditions become E y a nE b n = 2N (S S ;S I ): Then the relevant probability is Pr S n E y a nE b n2N (S S ;S I ) o = 2 n+kc 1 2 2n 1 2 (nk+c) = 2 n(1k=n+c=n) ; 89 which follows from similar counting arguments. This then leads to the following theorem for general EAQEC codes: Theorem 4.2.3 (EA Hashing Region). There exists an EAQEC code whose achievable rate pair (Q =k=n; E =c=n) obeys the following EA hashing bound for a Pauli channel with parameters p: Q 1H (p) +E: By varying c from 0 to the maximal amount nk, we can interpolate between stabilizer codes and maximal-entanglement EAQEC codes and achieve all rate pairs in the following hashing region: Q 1H (p) +E; Q 1H (p)=2: 4.2.2.3 Entanglement-Assisted Codes with Imperfect Ebits In the case that the ebits of the receiver are not perfect, we can use another stabilizer code to protect the ebits employed in the EAQEC code for transmitting information qubits [LB12, WH11]. Suppose that Alice uses an [[n;k;c]] EAQEC code with a (simplied) stabilizer groupS 1 through a Pauli channel with parameter p 1 to communicate with Bob and Bob's qubits suer a Pauli channel with parameter p 2 . Furthermore, suppose Bob uses an [[m;c]] stabilizer code with a stabilizer groupS 2 to protect hisc qubits. Now we have two typical error sets T p 1 n a n : 1 n log 2 (PrfE a ng)H (p 1 ) ; and T p 2 m u m : 1 m log 2 (PrfE u mg)H (p 2 ) : 90 Suppose Bob uses two decoders in sequence to correct the errors|the rst corrects the errors on the ebits and the second corrects the errors on the information qubits. Similar to the proof of Theorem 4.2.1, we have E S 1 ;S 2 fp e g =E S 1 ;S 2 ( X a n ;u m PrfE a n;E u mgI (E a n is uncorrectable underS 1 or E u m is uncorrectable underS 2 ) ) E S 1 8 > < > : X a n 2T p n X u m 2T p 2 m PrfE a ng PrfE u mgI (E a n is uncorrectable underS 1 ) 9 > = > ; +E S 2 8 > < > : X a n 2T p n X u m 2T p 2 m PrfE a ng PrfE u mgI (E u m is uncorrectable underS 2 ) 9 > = > ; + E S 1 8 > < > : X a n 2T p n PrfE a ngI (E a n is uncorrectable underS 1 ) 9 > = > ; +E S 2 8 > < > : X u m 2T p 2 m PrfE u mgI (E u m is uncorrectable underS 2 ) 9 > = > ; +; which goes to zero if the hashing bounds in Theorem 4.2.1 and Theorem 4.2.3 hold. Hence we have the following hashing bound for combination codes when the ebits are imperfect: Theorem 4.2.4 (Hashing Bounds for Combination Codes). Let = m n . There exists an [[n;k;c]] EAQEC code combined with an [[m;c]] stabilizer code with achievable rate pair (Q =k=n; E =c=n) obeys the following hashing bounds for two Pauli channels with parameters p 1 and p 2 , respectively: 1 E 1H(p 2 ); Q 1H(p 1 ) +E: 91 On the other hand, Bob can treat the combination code as an [[n +m;k]] stabilizer code with a stabilizer groupS. Using a similar argument as in the proof of Theorem 4.2.1, we nd that Q 1 +H(p 1 )H(p 2 ), which agrees with Theorem 4.2.4 if the entanglement consumption rate E can be as large as (1H(p 2 )). This result might be considered surprising because the simulations in Ref. [LB12] suggest that a single decoder has better performance than decoding the two codes in sequence|however, it appears that this is a nite blocklength eect that gets washed away in the asymptotic limit. 4.2.3 EAQEC Codes for Classical Communication Now suppose the goal is to send classical data by exploiting maximal-entanglement EAQEC codes. In this case, the stabilizer structure is similar to that for a maximal- entanglement EAQEC code for sending quantum data, but this time we do not care if Z errors aect the information qubits because they are classical. The error correction conditions then become thatfE a ng is a correctable set of errors if E y a nE b n = 2N (S S ;L X ); for all error pairsE a n andE b n in the error set, whereS S is the symplectic subgroup and L X is the logical X subgroup of the EAQEC code. Then the relevant probability is Pr S n E y a nE b n2N (S S ;L X ) o = 2 k 1 2 2n 1 2 (2nk) = 2 n(2k=n) ; which follows from similar counting arguments. By modifying the last few steps of the proof of Theorem 4.2.1, we obtain the following upper bound: X a n ;b n 2T p n ; b n 6=a n PrfE a ng 2 n(2k=n) 2 2n[H(p)+] 2 n[H(p)+] 2 n(2k=n) = 2 n[2H(p)k=n] ; 92 giving the EA hashing bound for classical communication: Theorem 4.2.5 (EA Hashing Bound for Classical Communication). There exists an EAQEC code for classical communication that achieves the EA classical hashing limit R = 2H (p) for a Pauli channel with parameters p. 93 Chapter 5 EAQEC Codes with Imperfect Ebits In EAQEC codes [BDH06b], it is assumed that the sender (Alice) and the receiver (Bob) share some pairs of qubits in maximally-entangled states before communication, and the qubits on Bob's side are subject to no error. The quantum codes are designed to cope with the noisy channelN A that Alice uses to communicate with Bob. However, noise (such as storage errors) can occur on Bob's ebits in practical situations, which is believed to degrade the performance of the quantum codes. Assume the errors occurring on Bob's qubits are described by a noise processN B . Wilde and Hsieh addressed this question with a channel-state coding protocol in quan- tum Shannon theory and determined the channel capacity when entanglement is not perfect [WH10]. They also performed simulations of entanglement-assisted quantum turbo codes with the depolarizing channel when Bob's ebits also suer errors [WH11]. Wilde and Fattal simulated the performance of an entanglement-assisted Steane code for fault tolerance [WF10]. In this chapter, we discuss two coding schemes to handle the problem when the ebits of Bob are not perfect. Shaw et al. described a six-qubit EAQEC code with one ebit that is equivalent to Steane's seven-qubit code, and can correct a single error on either Alice's or Bob's qubits [SWO + 08]. The entanglement-assisted Steane code, con- structed by Wilde and Fattal, is also equivalent to Steane's seven-qubit code [WF10]. Similarly, Bowen's entanglement-assisted code [Bow02] is equivalent to the ve-qubit code [BDSW96, LMPZ96] and can correct an error on one of Bob's qubits. These three 94 examples motivate the following idea: there are EAQEC codes that are equivalent to standard stabilizer codes, and hence can correct errors on both Alice's and Bob's sides. We show how to obtain an EAQEC code from a (nondegenerate) stabilizer code. Several EAQEC codes from this scheme are found to be optimal. These EAQEC codes will have better performance than their equivalent stabilizer codes when the storage error rate is less than the channel error rate. In the second scheme, Alice uses an EAQEC code to encode her information qubits and Bob uses a standard stabilizer code to protect his halves of the ebits. The com- bination of an EAQEC code and a stabilizer code is called a combination code, and it can be treated either as a single stabilizer code, or by using two sequential decoders. EAQEC codes that are not equivalent to standard stabilizer codes generally have higher error-correcting ability on Alice's qubits and are suitable for this scheme. Minimum distance of a stabilizer code is used as a measure of how good a code is without considering the details of the noisy channel model. However, minimum distance might not always be the best measure, for a quantum code may be able to correct many error operators of weight higher than that indicated by the minimum distance. For instance, in a large code block it is most important that a code be able to correct the set of typical errors [Smi06]. In particular, there is no general denition of minimum distance for the variant coding schemes in this chapter. A perhaps more suitable merit function is the channel delity [Sch96, RW06], which compares the similarity of the mod- ied quantum state with the original quantum state. However, the calculation of the channel delity depends on the channel and has an exponentially increasing complex- ity. We derive a formula for the channel delity of a quantum stabilizer code over the depolarizing channel, which facilitates its computation. The channel delity also can be well approximated by a lower bound when the depolarizing rate is small. Furthermore, Monte Carlo methods can often eciently approximate the channel delity [MU49]. 95 Another natural question arises in EAQEC codes. The perfect entanglement shared between sender and receiver will in practice be generated from a process of entangle- ment distillation [BBP + 96, BDSW96] or a breeding protocol [LD07]. It is known that entanglement distillation with one-way classical communication is equivalent to a quan- tum error-correcting code [BDSW96]. Since we can also communicate using an EAQEC code that is robust to imperfect ebits, we discuss whether it is always necessary to do entanglement distillation before communication, and how much. 5.1 Determining the Syndrome Representatives for EAQEC Codes In the usual paradigm of EAQEC codes, it is assumed that Bob's qubits suer no error. However, this assumption might not be true in practice. Suppose that Alice uses a noisy channelN A to communicate with Bob and Bob's ebits suer from a storage error channelN B . Assume bothN A andN B are depolarizing channels. Let p a and p b be the depolarizing rate ofN A andN B , respectively. Dene q w , 1 3 4 p a nw 1 4 p a w ; (5.1) for w = 0; ;n, and r w 0, 1 3 4 p b cw 0 1 4 p b w 0 : (5.2) forw 0 = 0; ;c. An error operatorE A E B ofN A N B occurs with probabilityq wa r w b , where w a = wt(E A ) and w b = wt(E B ). To correct some errors on Bob's qubits, we have to design a quantum code such that these errors are either syndrome representatives, or degenerate errors of other correctable errors. It is more complicated to determine the syndrome representatives in the case of EAQEC codes, since the error probabilities are dierent on Alice's and Bob's qubits. 96 For an EAQEC code to correct some errors on Bob's qubits, we must sacrice some of its ability to correct channel errors. For example, consider Bowen's [[3; 1; 3; 2]] code with the following stabilizer generators: X Z Z X I Z Z X I X Z Y Y Z I Y Y Z I Z: (5.3) Alice's qubits and Bob's qubits are on the left and the right of the vertical line, respec- tively. The error operators X 4 and Y 1 Y 2 have the same error syndrome. X 4 is an error operator on Bob's side with probability q 0 r 1 = 1 3 4 p a 3 1 3 4 p b 1 4 p b : If Bob's qubits are error free, the code can correct the weight-2 error Y 1 Y 2 on Alice's side, which occurs with probability q 2 r 0 = 1 3 4 p a 1 4 p a 2 1 3 4 p b 2 : We can instead choose to correctX 4 ifq 0 r 1 >q 2 r 0 . This is a tradeo between correcting channel errors or storage errors. We plot q 2 r 0 q 0 r 1 as a function of p a and p b in Fig. 5.1. It can be seen that Y 1 Y 2 is a more likely error than X 4 when p b is small or p a is large. To sum up, the set of syndrome representatives T should be chosen to maximize X E2T Pr(E): We will show that this criterion leads to high channel delity in Section 5.4. We propose two coding schemes in the next two sections. 97 Figure 5.1: (Color online) Contour plot of q 2 r 0 q 0 r 1 . The shaded region is where q 2 r 0 q 0 r 1 > 0. 5.2 EAQEC Codes that are Equivalent to Standard Stabi- lizer Codes The stabilizer generators of the ve qubit code [NC00] are: X Z Z X I I X Z Z X X I X Z Z Z X I X Z After some row operations, these are equivalent to the stabilizer generators of Bowen's [[3; 1; 3; 2]] AB EAQEC code in (5.3). Thus Bowen's quantum code is equivalent to the [[5; 1; 3]] standard stabilizer code, and has the same error-correcting ability on Bob's halves of the ebits as on Alice's qubits. We use the subscript AB to indicate an EAQEC code with this property. If we move the vertical line and put it between the fourth and fth qubits in (5.3), we obtain the stabilizer generators of the [[4; 1; 3; 1]] AB EAQEC code 98 in [BDH06b]. These observations inspire us to nd EAQEC codes that are equivalent to standard stabilizer codes and the answer is straightforward as follows. Given a set of stabilizer generatorsfg 1 ;g 2 ; ;g nk g of an [[n;k;d]] stabilizer code, after Gaussian elimination (and reordering qubits as necessary), they can be written asg 0 1 Z 1 ,h 0 1 X 1 , g 0 2 Z 2 ,h 0 2 X 2 , ,g 0 c Z c ,h 0 c X c ,g 0 c+1 I, ,g 0 nkc I for somecb nk 2 c such that the simplied generators g 0 i andh 0 i anti-commute with each other, and they commute with other simplied generatorsg 0 j ;h 0 l fori = 1; ;c andj;l6=i; whileg 0 c+1 ; ;g 0 nkc commute with all the simplied generators. Consequently,fg 0 1 ; ;g 0 nkc ;h 0 1 ; ;h 0 c g denes an [[nc;k;d;c]] EAQEC code. We summarize the above result in the following theorem and provide a lower bound on c. Theorem 5.2.1. Suppose H = [H X jH Z ] is the check matrix of an [[n;k;d]] standard stabilizer code. After Gaussian elimination, H can be written in the following standard form: H = 2 6 6 6 6 4 A I ss D 0 C 0 B I ss E 0 F 0 3 7 7 7 7 5 ; (5.4) where 0 s nk. Then there exists an [[nc;k;d;c]] AB EAQEC code for some 0 c s such that the code can correctb d1 2 c-qubit errors on either Alice's or Bob's qubits. In the case of nondegenerate quantum codes, s is bounded by d 1sb nk 2 c: Proof. An [[nc;k;d;c]] AB EAQEC code is dened by the check matrixH with the last c columns of both H X and H Z being removed. If d = 1, the result is trivial. Assume d 2. We perform Gaussian elimination on the check matrix H to pair the rows in the form (5.4). When the process of Gaussian 99 elimination stops, either all the rows of H are paired or some rows are left unpaired. In the rst case, H = 2 4 A I ss D 0 C 0 B I ss 3 5 : We have 2s =nk 2(d 1) by the quantum singleton bound, and thus sd 1. In the second case, we can nd a column in H X (or H Z ) that has a `1' at the (2s + 1)-th position and 0's elsewhere, and the corresponding column in H Z (or H X ) is of the form [u v w 0 0] T ; whereu andv are two binarys-tuples andw is 0 or 1. ThenH is in the following form: H = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 A 0 0 1 0 . . . 0 1 D 0 j u T j 0 C 0 0 0 B 0 j v T j 1 0 . . . 0 1 E 0 1 0 . . . 0 0 F 0 w 0 . . . 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Note that a subset of the last (s + 1) columns of H X and the last (s + 1) columns of H Z are linearly dependent. Thus we can nd an element of weight at most (s + 1) corresponding to these columns that is in the normalizer group of the stabilizer group. In the case of nondegenerate quantum codes, the minimum distance is the minimum 100 weight of an element in the normalizer group of the stabilizer group. Since the minimum distance of the quantum code is d, this implies that sd 1: By this theorem, we can \move" some ancilla qubits to Bob's side for any nonde- generate stabilizer codes and obtain an EAQEC code. The case c =d 1 of the above theorem is also observed by Wilde and Hsieh [WH11] from the viewpoint of purica- tion and tracing qubits [Pre99]. Note that Shor's [[9; 1; 3]] degenerate code [Sho95] and Bacon's [[n 2 ; 1;n]] degenerate codes [Bac06] also satisfy this theorem (by pairing up S X i withS Z i ). It is conjectured that all degenerate codes satisfy the lower bound in Theorem 5.2.1. We have checked that all the optimal quantum codes in Grassl's table [Gra] satisfy the lower bound in this theorem. EAQEC codes that are equivalent to standard stabilizer codes can correct errors on the qubits of both Alice and Bob. The decoder of the corresponding standard stabilizer code can be adopted to decode these EAQEC codes. These codes may perform better in practice than their corresponding standard stabilizer codes, for there are fewer physical qubits transmitted through the noisy channel, and the storage error rate is generally lower than the noisy channel error rate. In the case of CSS codes, the standard form of a parity check matrix is H = 2 4 A I ss 0 0 0 0 B I (nks)(nks) 3 5 : Consequently, we have the following theorem. Theorem 5.2.2. An [[n;k;d]] CSS code, obtained from an [n;k 0 ;d] classical dual- containing code with k = 2k 0 n, gives [[nc;k;d;c]] AB EAQEC codes for 0 c nk 2 =nk 0 . 101 This shows that any CSS codes can be transformed into EAQEC codes that correct errors on both Alice's and Bob's qubits. The decoding method in this scheme is exactly the same as that of standard CSS codes. Since many stabilizer codes are based on the CSS construction, we can take advantage of these codes while also having the power of entanglement. Example 5.2.1. From the [[7; 1; 3]] Steane code, we obtain a [[4; 1; 3; 3]] AB , a [[5; 1; 3; 2]] AB , and a [[6; 1; 3; 1]] AB EAQEC code. The [[4; 1; 3; 3]] AB and the [[6; 1; 3; 1]] AB EAQEC codes were also found by Wilde and Fattal [WF10] and Shaw et al [SWO + 08], respec- tively: X X I X X I I X X X I I X I X I X X I I X Z Z I Z Z I I Z Z Z I I Z I Z I Z Z I I Z : The quantum singleton bound says that for an [[n;k;d]] quantum code, nk 2(d 1): If the parameters n;k;d of a standard stabilizer code achieve the quantum singleton bound, or nk = 2(d 1); then EAQEC codes equivalent to these standard stabilizer codes will achieve the singleton bound for EAQEC codes: n +ck 2(d 1); and we have the following theorem. 102 Theorem 5.2.3. A standard stabilizer code that achieves the quantum singleton bound gives an [[nc;k;d;c]] AB EAQEC code for some c that achieves the singleton bound for EAQEC codes. Example 5.2.2. The [[3; 1; 3; 2]] AB and [[4; 1; 3; 1]] AB EAQEC codes derived from the ve qubit code achieve the singleton bound for EAQEC codes. Example 5.2.3. Using MAGMA [BCP97]to nd optimal standard stabilizer codes, we obtain several optimal EAQEC codes that achieve the linear programming bounds in [LB13b] by Theorem 5.2.1: [[15; 10; 4; 5]] AB ; [[14; 11; 3; 3]] AB ; [[13; 9; 4; 4]] AB ; [[13; 10; 3; 3]] AB ; [[12; 9; 3; 3]] AB ; [[11; 8; 3; 3]] AB ; [[10; 6; 4; 4]] AB ; [[10; 7; 3; 3]] AB ; [[9; 6; 3; 3]] AB ; [[7; 4; 3; 3]] AB ; [[8; 4; 4; 4]] AB ; [[6; 2; 4; 4]] AB ; [[7; 3; 3; 1]] AB ; [[6; 3; 3; 2]] AB ; [[6; 1; 5; 5]] AB ; [[4; 1; 3; 1]] AB ; [[4; 1; 3; 3]] AB ; [[3; 1; 3; 2]] AB : We also found the following EAQEC codes that have the highest minimum distance for xed n and k to the best of our knowledge: [[15; 4; 8; 11]] AB ; [[13; 5; 6; 8]] AB ; [[12; 6; 5; 6]] AB : 5.3 Quantum Codes with Two Encoders In the previous section, we discussed EAQEC codes that are equivalent to standard sta- bilizer codes. Most optimal EAQEC codes are not equivalent to any standard stabilizer codes, such as the entanglement-assisted repetition codes [LB13a, LB13b]. We would like to exploit the high error-correcting ability of these quantum codes even in the presence 103 of storage errors on Bob's side. This can be achieved by using another quantum code to protect Bob's qubits. Assume Alice uses the encoding operator U A of an [[n;k;d A ;c]] EAQEC code to protect her information qubits. Suppose also that there are mc> 0 ancilla qubits on Bob's side, and that Bob applies the encoding operator U B of an [[m;c;d B ]] standard stabilizer code to protect his c qubits. The encoding operator on the whole is U A U B . We use the notation [[n;k;d A ;c]] + [[m;c;d B ]] to represent such a composite quantum code. If there are no ancillas on Bob's side, the set of stabilizer generators is equivalent to that of an EAQEC code for some encoding operatorU 0 A I B after Gaussian elimination. The initial state isj i j0i nck j + i c j0i mc : The encoded state has the following stabilizer generators: g 1 u 1 , h 1 v 1 , g 2 u 2 , h 2 v 2 , , g c u c , h c v c , g c+1 I, ,g nkc I,I u c+1 , ,I u m , whereU B Z B i U y B =u i andU B X B j U y B =v j . A straightforward decoding process of the [[n;k;d A ;c]] + [[m;c;d B ]] quantum code is that Bob rst decodes his c ebits in the [[m;c;d B ]] quantum code, and then he decodes the k information qubits hiding in the [[n;k;d A ;c]] EAQEC code. Or we can treat the combination code as an [[n +m;k;d c ]] code, which has a more complicated decoding circuit but a potentially higher error-correcting ability. Under what condition is an [[n;k;d;c]] EAQEC code not equivalent to an [[n + c;k;d]] standard stabilizer code? If the parameters do not satisfy the following quantum Hamming bound for (nondegenerate) quantum codes P t j=0 3 j n+c j 2 nk+c , an [[n + c;k;d]] code does not exist. Example 5.3.1. Consider the [[7; 1; 5; 2]] code in [LB13a]. We have P t j=0 3 j n j = 211< 2 nk+c = 256: However, P t j=0 3 j n+c j = 352> 2 n+ck = 256: Hence there is no [[9; 1; 5]] code. However, the Hamming bound is not tight. For example, the [[7; 1; 5; 3]] code is not equivalent to any standard code, but the parameters [[10; 1; 5]] satisfy the quantum Hamming bound. A better bound, such as the linear programming bound, can be applied here. If the parameters n +c;k;d violate any upper bound on the minimum distance 104 of the quantum code, then such an [[n +c;k;d]] standard stabilizer code does not exist, and the [[n;k;d;c]] EAQEC code is not equivalent to any standard stabilizer code. We can check the result for n +c 30 from the tables of stabilizer codes in [CRSS98] and [Gra]. Several EAQEC codes that not equivalent to standard codes were found in [LB13a]: [[n; 1;n;n 1]] for n odd, [[n; 1;n 1;n 1]] for n even, [[7; 1; 5; 2]], [[7; 1; 5; 3]], [[7; 2; 5; 5]], [[9; 1; 7; 4]], [[9; 1; 7; 5]], [[9; 1; 7; 6]], [[9; 1; 7; 7]], [[8; 2; 5; 4]], [[8; 3; 5; 5]], [[13; 3; 9; 10]], [[13; 1; 11; 11]], [[13; 1; 11; 10]], [[13; 1; 9; 9]], [[13; 1; 9; 8]]. Also, the [[15; 7; 6; 8]], [[15; 8; 6; 7]], [[15; 9; 5; 6]] EAQEC codes are not equivalent to any known standard quantum stabilizer code. Example 5.3.2. The [[n; 1;n;n 1]] EAQEC codes for n odd saturate the quantum Hamming bound with equality as follows. The number of correctable X or Z errors is n1 2 and the number of correctable error syndromes is n 0 + n 1 + + n b n1 2 c 2 = 2 2(n1) : Suppose Alice uses the [[5; 1; 5; 4]] entanglement-assisted repetition code and Bob applies the optimal [[10; 4; 3]] quantum code to protect his 4 ebits. Then the whole quantum code [[5; 1; 5; 4]] + [[10; 4; 3]] can protect two channel noise errors on the 5 qubits that Alice sends through the channel, and one storage error on Bob's 10. On the other hand, the optimal quantum code using 15 qubits to encode one information qubit is the [[15; 1; 5]] quantum code [Gra], and it can correct an arbitrary two qubit error. Compared to an [[n +m;k;d]] standard stabilizer code, the number of qubits going through the noisy channel is much less using the [[n;k;d A ;c]] + [[m;c;d B ]] quantum code. Consequently, as long as the storage error rate is reasonably small compared to the channel error rate, the [[n;k;d A ;c]]+[[m;c;d B ]] quantum code has better eciency, while keeping the same error-correcting ability against the channel noise. The performances of dierent coding schemes will be compared in Section 5.5. 105 Figure 5.2: Illustration of the weight distributions of the syndrome representatives of the two decoding method. For combination EAQEC codes, the syndrome representatives can be chosen as in the case of standard stabilizer codes if we treat the code as a single stabilizer code. If we use two sequential decoders, we choose two sets of syndrome representatives T A , T B as in the case of standard stabilizer codes. Observe that T = T A T B is the set of syndrome representatives of the combination code. The weight distributions of the syndrome representatives of the two decoding method are illustrated in Fig. 5.2. The x-axis and y-axis represent the weights on Bob's and Alice's qubits, respectively. The weight distribution of the syndrome representatives using two sequential decoders is always a rectangle. If the error probabilities are the same on Alice's and Bob's qubits, the weight distribution of the syndrome representatives using a single decoder looks like a triangle. If the error probabilities are dierent on Alice's and Bob's qubits, the shape varies according to the error probabilities. Note that the area of the triangle is equal to the area of the rectangle. 5.4 Channel Fidelity 5.4.1 Formula for the Channel Fidelity over the Depolarizing Channel LetE be a quantum channel operating on the input state , which lies in a state space H 0 of dimension m, and the output stateE() also lies inH 0 . Suppose the quantum channelE has the operator-sum representationE() = P i E i E y i , where the operation 106 elementsfE i g satisfy P i E y i E i = I. Letj i = 1 m 1=2 P j jjijji, wherefjjig is a basis of the input spaceH 0 . The channel delity [Sch96, RW06] ofE is dened as F C (E) =h j (I Ej ih j)j i = 1 m 2 X i jtr(E i )j 2 : The channel delity can be used as a measure of the performance of a quantum error- correcting code over a noisy channel. Suppose the k-qubit information statej i, after being encoded by the encoding channelU :H k !H n , is transmitted through a noisy channelN :H n !H n , and then decoded by the decoding channelD :H n !H k . ThenF C (DNU) serves as a merit function of the quantum code with encoding-decoding pairfU;Dg over the noisy channelN . The encoding channelU can be written as U(j ih j) =U E (I k j0i nk )j ih j(I k h0j nk )U y E ; for some unitary Cliord encoder U E . We assume there are (nk) ancillas and we implicitly use the equation I k j0i nk j i =j i j0i nk for any k-qubit statej i. The decoding channelD consists of the following steps: syn- drome measurementfP s g, correctionfC s g, decodingU y E , and partial trace of the ancillas. The overall process of the decoding channel is D( 0 ) =tr A X s U y E C s P s 0 P s C y r U E ! = X l X s (I k hlj)U y E C s P s 0 P s C y r U E (I k jli); wherefjlig is the standard basis for the state space of the (nk) ancilla qubits, and jli has a binary representationjl 1 l 2 l nk i. ThusD has operation elementsf(I k 107 hlj)U y E C s P s g l;s . Suppose the noisy channelN is the independent n-fold depolarizing channel T n p with operation elementsf p p i E i g, where p i is dened in (1.2). The composite channelDNU has operation elements n W l;i;s , (I k hlj)U y E C s P s p p i E i U E (I k j0i nk ) o l;i;s and its channel delity is F C (DNU) = 1 2 2k X l;s;i jW l;i;s j 2 : There are a total of 2 nk 2 nk 4 n = 4 2nk terms in the sum, and each term is a product of one (2 k 2 n ) matrix, ve (2 n 2 n ) matrices, and one (2 n 2 k ) matrix. Thus the complexity of the calculation of the channel delity is (n 4 3n2k ) (the complexity of multiplication of (2 n 2 n ) matrices is (n2 n )), which is almost impossible to calculate forn> 10. We will show how to reduce the complexity to something more manageable. Lemma 5.4.1. If the error syndrome of E i is s i , then jtr (W l;i;s )j 2 = 0 for s6=s i . In addition, the measurement result is s i with certainty. It is straightforward to check that P s E i U E (I k j0i nk ) = 0 from the facts that P s 's are orthogonal to each other and P 0 is the projector onto the code space, and the above lemma follows naturally. Next we show that only the error operators in TS have nonzero contribution to the channel delity. Lemma 5.4.2. 1 2 2k X l jtr (W l;i;s i )j 2 = 8 < : p i ; if E i 2TS; 0; otherwise. 108 Proof. It is straightforward to verify the lemma for the case thatE i 2TS. Now assume E i = 2 TS and we have C s i E i 2N (S)nS. Since U E is a Cliord unitary operator, M 1 M 2 ,U y E C s i E i U E 2G n is a Pauli operator, whereM 1 2G k andM 2 2G nk . Since E i = 2 TS, we have M 1 not equal to the identity and tr(M 1 ) = 0. Let the matrix representations of M 1 and M 2 be [a i;j ] and [b i;j ], respectively. Then X l tr (I k hlj)U y E C s i P s i E i U E (I k j0i nk ) 2 = X l tr (I k hlj)M 1 M 2 (I k j0i nk ) 2 = X l jb l;1 tr(M 1 )j 2 = 0; where the second equality follows by explicitly writing down the matrix multiplications of (I k hlj)M 1 M 2 (I k j0i nk ). We derive the following theorem with the help of Lemma 5.4.1 and Lemma 5.4.2. Theorem 5.4.1. The channel delity of a quantum code, with a stabilizer group S and a set of syndrome representatives T , over the depolarizing channel, is the weight enumerator of the probability distributionfq w g of the elements in TS. Proof. F C (DNU) = 1 2 2k X l;i jtr (W l;i;s i )j 2 = X E i 2TS p i = n X w=0 a w q w ; (5.5) where a w is the number of E i 2TS of weight w, and q w is dened in (1.2). The rst equality follows by Lemma 5.4.1 and the second equality follows by Lemma 5.4.2. There are 4 nk terms in (5.5) and each term is generated by vector addition. The complexity is now reduced to (n 4 nk ). From the above theorem, we nd that the 109 Type of codes Channel delity bit- ip code 1 3 2 p + 9 8 p 2 3 8 p 3 [[4; 2; 2]] code 1 3 2 p + 3 4 p 2 [[5; 1; 3]] code 1 45 8 p 2 + 75 8 p 3 45 8 p 4 + 9 8 p 5 [[7; 1; 3]] code 1 147 16 p 2 + 189 8 p 3 1785 64 p 4 + 1155 64 p 5 399 64 p 6 + 57 64 p 7 [[8; 3; 3]] code 1 245 16 p 2 + 1449 32 p 3 8029 128 p 4 + 12743 256 p 5 2961 128 p 6 + 763 128 p 7 21 32 p 8 [[9; 1; 3]] code 1 9p 2 + 195 8 p 3 1071 32 p 4 + 945 32 p 5 567 32 p 6 + 225 32 p 7 27 16 p 8 + 1545 8192 p 9 Table 5.1: Channel delity of dierent quantum codes in the depolarizing channel channel delity for the depolarizing channel is the probability of correctable errors. For a single information-qubit code, the channel delity is the probability that the information qubit can be correctly recovered. The formula for the channel delity of several quantum codes is shown in Table. 5.1. The channel delity of the quantum code dened by a stabilizer groupS over a depolarizing channel depends on the syndrome representativesT and the stabilizer group S, and it can be optimized over the choices of T . (This is to optimize over dierent decoding schemes.) 5.4.2 Channel Fidelity for EAQEC Codes Suppose that Alice uses a noisy channelN A to communicate with Bob and Bob's ebits suer from a storage error channelN B . Denition 5.4.1. The channel delity [Sch96, RW06] of an EAQEC code with encoding and decoding processesfU;Dg over the noisy channelN A N B is F C (D(N A N B )U): Suppose an [[n;k;d;c]] EAQEC code is dened by a simplied stabilizer groupS 0 = hg 0 1 ; ;g 0 nk ;h 0 1 ; ;h 0 c i2G n . We extend the simplied stabilizer group to a stabilizer 110 groupS =hg 1 ; ;g nk ; h 1 ; ;h c i2G n+c , where g i = g 0 i Z i , h i = h 0 i X i , for i = 1; ;c, and g j = g 0 j I for j = c + 1; ;nk. We choose a set T of syndrome representatives for the noisy channelN A N B . Let p a and p b be the depolarizing rate ofN A andN B , respectively. Now we can apply Theorem 5.4.1 to nd a formula for the channel delity of EAQEC codes. Theorem 5.4.2. The channel delity of an EAQEC code with a stabilizer groupS and the set of syndrome representativesT over the depolarizing channelN A N B is the weight enumerator of the probability distributionfq w r w 0g of the elements in TS, where p w and r w 0 are dened in (5.1) and (5.2). Note that the channel delity can be optimized over the choice of T . Since the two noisy channelsN A andN B have dierent error rates, error operators on these two channels should be dierently weighted. In the extreme case of p b = 0 and r 0 = 1, the error-correction condition for EAQEC codes says thatfE i g is a set of correctable error operators if E y i E j = 2N (S 0 )n ~ S I . Note that r 0 = 1, and r i = 0 for i6= 0. We can similarly formulate the channel delity. Corollary 5.4.1. The channel delity of an EAQEC code with a stabilizer groupS and a set of syndrome representatives T over the depolarizing channelN A I B is the weight enumerator of TS I , which is a polynomial infq w g. Therefore, we would like to choose T to consist of likely error operators from the noisy channelN A . Example 5.4.1. We compare the channel delities of the [[3; 1; 3; 2]] entanglement-assisted (EA) repetition code [LB13a],F C (D re 3 (T 3 pa T 2 p b )U re 3 ), and Bowen's [[3; 1; 3; 2]] AB quan- tum code [Bow02], F C (D AB 3 (T 3 pa T 2 p b )U AB 3 ). Bowen's [[3; 1; 3; 2]] AB quantum code is equivalent to the ve qubit code and has T =fI; X 1 ; ;X 5 ; Z 1 ; ;Z 5 ; Y 1 ; ;Y 5 g, while the [[3; 1; 3; 2]] EA repetition code is designed under the assumption that Bob's 111 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 P a P b B A 0 0.005 0.01 0.015 0 0.1 0.2 P a P b A B Figure 5.3: Comparison of two [[3; 1; 3; 2]] EAQEC codes in terms of channel delity. The [[3; 1; 3; 2]] AB code performs better in Region B, while the repetition code performs better in Region A. The region for p a < 0:015 and p b < 0:2 is enlarged on the right. Figure 5.4: (Color online) Contour plot of the dierence between the two delities: F C (D re 3 (T 3 pa T 2 p b )U re 3 ) F C (D AB 3 (T 3 pa T 2 p b )U AB 3 ). The shaded region is where F C (D re 3 (T 3 pa T 2 p b )U re 3 )F C (D AB 3 (T 3 pa T 2 p b )U AB 3 )> 0. qubits are perfect and has T = fI;X 1 ; ;X 3 ; Z 1 ; ;Z 3 ; Y 1 ; ;Y 3 ; X 1 Z 2 ;X 1 Z 3 ; Z 1 X 2 ;X 2 Z 3 ; Z 1 X 3 ;Z 2 X 3 g. We have F C (D AB 3 (T 3 pa T 2 p b )U AB 3 ) =q 0 r 0 + 9q 1 r 0 + 6q 3 r 0 + 6q 0 r 1 + 36q 2 r 1 + 54q 3 r 1 + 18q 1 r 2 + 81q 2 r 2 + 45q 3 r 2 ; 112 and F C (D re 3 (T 3 pa T 2 p b )U re 3 ) =q 0 r 0 + 9q 1 r 0 + 6q 2 r 0 + 18q 1 r 1 + 38q 2 r 1 + 40q 3 r 1 + 18q 1 r 2 + 55q 2 r 2 + 71q 3 r 2 : The channel delities of these two EAQEC codes are compared in Fig. 5.3 and Fig. 5.4. In Fig. 5.3, the curve of the boundary between the two regions passes the origin. The region A in Fig. 5.3 corresponds to the shaded part in Fig. 5.4. In region B, Bowen's code is better than the EA repetition code. In the extreme case of p b = 0, we have F C (D AB 3 (T 3 pa I 2 )U AB 3 ) =q 0 + 9q 1 + 6q 3 ; and F C (D re 3 (T 3 pa I 2 )U re 3 ) =q 0 + 9q 1 + 6q 2 : The EA repetition code corrects more lower-weight errors, and hence it has higher chan- nel delity. The channel delity of the [[n;k;d A ;c]] + [[m;c;d B ]] quantum code depends on the decoding process. If we treat the combination code as an [[n + m;k;d C ]] code, its channel delity can be computed as Theorem 5.4.2. If we use two sequential decoders, it's dierent. First, the 4 m possible error operators on Bob's side collapse to 4 c logical errors. Each can be obtained from 4 mc error operators, after the decoding process of the [[m;c;d B ]] quantum code. If the 4 m errors occur uniformly, the decoded errors also occur uniformly. However, this is not the case for the depolarizing channelN B : the 4 c errors on Bob's ebits occur according to a distributionfr 0 w 0 g that depends on the decoding process. We can nd the channel delity as in Theorem 5.4.2, except that the errors on Bob's ebits follow a new distributionfq w r 0 w 0 g. We will compare the channel delities of these two decoding methods in Section 5.5. 113 5.4.3 Approximation of Channel Fidelity The number of terms in the formula for channel delity over the depolarizing channel is 4 n+ck , which grows exponentially in n +ck: It is dicult even to build a look-up table for decoding when n is large. However, it is possible to approximate the channel delity eciently. The channel delity of a quantum code over a depolarizing channel is the probability that the decoder output is correct, which can be lower bounded by Pr(fcorrectable errorsg) Pr(fsyndrome representativesg); or Pr(fcorrectable errorsg) Pr(ferrors of weight less than or equal tob d1 2 cg); whered is the minimum distance of the quantum code. When the depolarizing rate is low (< 0:2), these bounds are fairly tight and are good approximations of the true channel delity. Dong et al. dened an \indelity" function to characterize the performance of quantum codes [DXJ + 09], which is an approximation of the channel delity in the case of k = 1. We can also apply Monte Carlo methods to approximate the channel delity [MU49], especially when the number of physical qubits involved is large. The steps of the simu- lations are as follows: (i) Fix a depolarizing rate p. (ii) Randomly generate an error operator E according to the probability distribution of a depolarizing channel. (iii) Compute the error syndrome and apply the correction operator and decoding oper- ator. 114 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 Depolarizing rate p Channel Fidelity Channel fidelity of [[11,1,5]] code MC approximation N=10 3 MC approximation N=10 4 MC approximation N=10 5 MC approximation N=10 6 Figure 5.5: (Color online) The approximations of the channel delity of the [[11; 1; 5]] quantum code. (iv) If there is no logical error after decoding, E is correctable. (v) Repeat steps 2 to 4 N times. (vi) Output the channel delity as the number of correctable errors in the experiment divided by N. Two applications of the Monte Carlo method to the [[11; 1; 5]] and [[24; 1; 8]] codes are shown in Fig. 5.5 and Fig. 5.6. In Fig. 5.5, the exact channel delity of the [[11; 1; 5]] code is also plotted, and it can be observed that the simulations quickly converge to the exact channel delity. On the other hand, in Fig. 5.6, only a lower bound of the channel delity of the [[24; 1; 8]] code is given, since computation of the channel delity is dicult (4 23 terms in the formula). However, Monte Carlo simulations also converge quickly from N = 10 4 to 10 6 points, which are much less than 4 23 w 7 10 13 . 115 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Depolarizing rate p Channel Fidelity lower bound of [[24,1,8]] MC simulation of [[24,1,8]], N=10 3 MC simulation of [[24,1,8]], N=10 4 MC simulation of [[24,1,8]], N=10 5 MC simulation of [[24,1,8]], N=10 6 Figure 5.6: (Color online) The approximations of the channel delity of the [[24; 1; 8]] quantum code. 5.5 Performance analysis In this section, we compare the channel delity of the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code with other quantum codes. A comparable standard stabilizer code withn = 15 and k = 1 is a [[15; 1; 5]] stabilizer code, while the smallest quantum code with d = 5 is a [[11; 1; 5]] stabilizer code [Gra]. By Theorem 5.2.1, we can obtain a [[6; 1; 5; 5]] AB EAQEC code from the [[11; 1; 5]]. The [[5; 1; 5; 4]] EAQEC code is also shown as a reference. All the quantum codes above encode k = 1 information qubit and hence can be compared with no ambiguity. The channel delity of these quantum codes are plotted in Fig. 5.7. For simplicity, the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code using a single decoder is treated as a standard stabilizer code, which means that the weight distribution of the syndrome representatives is like a triangle as in Fig. 5.2. Whenp b = 0:99p a , the [[6; 1; 5; 5]] AB EAQEC code is the best code forp a < 0:38, and [[5; 1; 5; 4]]+[[10; 4; 3]] quantum code with sequential decoders and the [[5; 1; 5; 4]] EAQEC code have better performance for p a > 0:38. The [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code with a single decoder has a comparable delity to the [[6; 1; 5; 5]] AB EAQEC code. 116 As the ratio of p b to p a decreases, the performance of the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code and the [[6; 1; 5; 5]] AB , [[5; 1; 5; 4]] EAQEC codes get better. When p b = 0:5p a , the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code, using either decoding method, performs better than the [[6; 1; 5; 5]] AB EAQEC code and the [[11; 1; 5]] stabilizer code. When the ratio of p b to p a is below 0:35 in the last two cases, the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code is the best choice. The [[5; 1; 5; 4]] EAQEC code beats the [[11; 1; 5]] stabilizer code when p b = 0:01p a as expected. From these plots, the quantum codes that have the best performance are the [[6; 1; 5; 5]] AB EAQEC code or the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code. Fig. 5.8 plots the dierence (F single F seq ) between the delity of a single decoder (F single ) and the delity of two sequential decoders (F seq ) of the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code. For low noise rates sequential decoding is not as good; however, it is easier to be implemented. Note that, in the simulations, the syndrome representatives using a single decoder are chosen for the same error probabilities on Alice's and Bob's qubits. The channel delity of the combination using a single decoder will be better if we optimize it over the choices of the syndrome representatives according to the dierent error probabilities. 5.6 EAQEC and Entanglement Distillation Up to now, we have assumed that Alice and Bob have perfect ebits before the com- munication, and any noise results from storage errors. In a more general situation, the exchange of perfect maximally-entangled states might not be possible, and entanglement distillation is needed. Our next direction is to nd good strategies against this problem. We rst introduce the entanglement distillation protocol. Suppose Alice has the ability to prepare n pairs of maximally-entangled states (Bell states) j + i m = 1 p 2 (j00i +j11i) m = 1 p 2 m 2 m 1 X i=0 jii A jii B ; 117 where i is the binary representation of the numbers between 0 and 2 m 1. The state j + i m has the following property. Lemma 5.6.1. For any operator M on an m-qubit maximally entangled state, M A I B j + i m =I A (M T ) B j + i m ; where M T is the transpose of M. Suppose Alice uses an [[m;c]] stabilizer code dened by a stabilizer groupS with generatorsf 1 ; ;f mc , andT is a set of syndrome representatives corresponding toS. Let U be a Cliord encoder of the stabilizer code, andjji L for j = 0; ; 2 c 1 be the logical states. The encodedj + i c is j + i c L =U A U B j + i c = 1 p 2 c 2 c 1 X j=0 jji A L jji B L : We know thathij L E s 1 E s 2 jji L = s 1 ;s 2 i;j andfE s jii L g is a set of orthonormal basis vectors ofH n . In the case of UU T =I, we have j + i m = (UU T ) A I B j + i m =U A U B j + i m = 1 p 2 m 2 m 1 X i=0 U A U B jii A jii B = 1 p 2 m X Es2T E A s E B s 0 @ 2 c 1 X j=0 jji A L jji B L 1 A : If UU T 6=I, Alice applies the operator UU T on half of the ebits. From Wilde's encod- ing circuit algorithm [Wil08], an encoding operator can be implemented by a series of Hadamard gates, CNOT gates, SWAP gates, and phase gates. If phase gates are not used in the circuit, the circuit will satisfy UU T =I. 118 Alice sends half of the ebits to Bob through a noisy channelN C with depolarizing rate equal to p c . The corrupted state is E B i j + i m = 1 p 2 m 2 c 1 X j=0 X Es2T E A s E B i E B s jji A L jji B L : After performing a syndrome measurement, Alice obtains a syndromea, which is a binary (mc)-tuple, and she sends a to Bob through a noiseless classical channel. Now the state is E A a E B i E B a j + i c L = 1 p 2 c 2 c 1 X j=0 E A a E B i E B a jji A L jji B L ; whereE a is the syndrome representative ofa. Lets(i) be the error syndrome ofE i . Bob measures the stabilizer generators f 1 ; ;f mc and obtains the syndrome b =a +s(i): The error syndrome s(i) can be retrieved by s(i) = a +b. He applies the correction operator E s(i) and obtains the state E A a E B s(i) E B i E B a j + i c L : Finally, they restore the state to the standard encoded statej + i c L by applying the operator E a E a , followed by the decoding circuit U y U y and obtain the state j + i c j0i mc j0i mc ; 119 if E i is correctable. This is called an [[m;c]] entanglement distillation protocol. LetE denote the combination of the above processes. The entanglement delity [Sch96] of this protocol is F (j + i c ;E) =h + j c E j + i c h + j c j + i c : The entanglement delity of the [[m;c]] entanglement distillation protocol when the channelN C is the depolarizing channel is just the channel delity of the corresponding [[m;c]] stabilizer code over the depolarizing channelN C . After performing an [[m;c]] entanglement distillation protocol that produces c ebits with some entanglement delity, Alice can use an [[n;k;d;c]] EAQEC code to send k information qubits to Bob. Assume p b = 0 and p c = p a for simplicity. The channel delity from the combination of an entanglement distillation protocol and an EAQEC code is similar to that of a combination code using two sequential decoders. On the other hand, Alice could directly use an [[n 0 ;k;d 0 ;e]] EAQEC code with the imperfect ebits to sendk information qubits to Bob without entanglement distillation. The channel delity of this process is just the channel delity of the EAQEC code. Example 5.6.1. Alice and Bob use an [[8; 3; 3]] entanglement distillation protocol to pro- duce 3 perfect ebits from 8 noisy ebits. Then Alice can use an [[5; 2; 3; 3]] EAQEC code to send quantum information to Bob. Or she can instead use a [[10; 2; 7; 8]] EAQEC code with the 8 corrupted ebits. Comparison of the channel delity of these two schemes is shown in Fig. 5.9. Example 5.6.2. Alice and Bob use a [[5; 1; 3]] entanglement distillation protocol to pro- duce 1 perfect ebit from 5 noisy ebits. Then Alice can use a [[4; 1; 3; 1]] EAQEC code to send quantum information to Bob. Or she can instead use a [[6; 1; 5; 5]] AB EAQEC code with the 5 noisy ebits. Comparison of the channel delity of these two schemes is shown in Fig. 5.10. In Fig. 5.11, we plot the channel delity of four distillation protocols with the same [[4; 1; 3; 1]] EAQEC code. The channel delities of the 4 schemes with distillation 120 protocols are better than the one without distillation protocol for p < 0:1, but the dierence is small. The [[4; 1; 3; 1]] EAQEC code without distillation dominates the performance for higher p. Note that the channel delity of the [[5; 1; 3]] distillation protocol with the [[4; 1; 3; 1]] EAQEC code is roughly equal to that of the the [[8; 1; 3]] distillation protocol with the [[4; 1; 3; 1]] EAQEC code In Fig. 5.12, we plot the channel delity of several coding schemes that encode k = 1 information qubit. It can be observed that the EAQEC codes without distillation protocols have better performance for p< 0:2. 5.7 Discussion We proposed two coding schemes for EAQEC codes when the ebits of the receiver suer errors. In the rst case we assume the ebits suer storage errors. EAQEC codes that are equivalent to standard stabilizer codes have better performance than their corresponding stabilizer codes. Several such EAQEC codes are found to achieve the linear programming bound, and hence are optimal. However, as long as the storage error rate is small enough, a quantum code with two encoders performs well if we start with an EAQEC code that is not equivalent to any standard stabilizer code. We may choose the best quantum code according to the noise channel rate and the storage error rate in real situations. Any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code by Theorem 5.2.1. Families of quantum codes, such as quantum Reed-Muller codes [Ste99b], quantum BCH codes [GB99, AKS07], quantum cyclic codes [CEL99, Lin04], can be transformed into families of EAQEC codes. It is possible to construct EAQEC codes with a large number of information qubits but a small number of ebits that outperform standard codes. We developed a formula for the channel delity over the depolarizing channel, and used it to evaluate the performance of a variety of quantum codes. For large codes, the channel delity cannot be calculated exactly, but can be lower-bounded or approximated 121 by Monte Carlo simulations. A similar formula for the channel delity can be developed for other channels with only Pauli errors. We also compared EAQEC codes combined with entanglement distillation protocols to EAQEC codes designed to tolerate noisy ebits. It seems that EAQEC codes that can correct errors on both the qubits of sender and receiver can have better performance than the codes combined with an entanglement distillation protocol, at least for modest noise rates. For particular combinations of error rates and applications, it should be possible to optimize the choice of code to maximize the delity. This optimization is the subject of ongoing research. 122 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.99*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.65*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.5*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.35*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.1*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p b =0.01*p a Depolarizing rate p a Channel Fidelity F c 1/k [[11,1,5]] [[5,1,5;4]] [[6,1,5;5]] AB [[5,1,5;4]]+[[10,4,3]] single decoder [[5,1,5;4]]+[[10,4,3]] sequential decoders Figure 5.7: (Color online) Channel delity of dierent quantum codes for p b =p a = 0:99, 0:65, 0:5, 0:35, 0:1, 0:01. 123 Figure 5.8: (Color online) Contour plot of the dierence (F single F seq ) between decoding methods for the [[5; 1; 5; 4]] + [[10; 4; 3]] quantum code. The shaded region is where F single F seq > 0. 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 Depolarizing rate p Channel Fidelity F c 1/k [[10,2,7;8]] EAQEC code [[8,3,3]] distillation protocol +[[5,2,3;3]] EAQEC code Figure 5.9: (Color online) Comparison of the [[8; 3; 3]] distillation protocol plus [[5; 2; 3; 3]] EAQEC code, and the [[10; 2; 7; 8]] EA EAQEC code without distillation. The perfor- mance of the [[10; 2; 7; 8]] AB EAQEC code is better for p a < 0:45. 124 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 p b =1*p a Depolarizing rate p a Channel Fidelity F c 1/k [[6,1,5;5]] AB EAQEC code without distillation [[5,1,5;4]] EAQEC code without distillation [[5,1,3]] distillation protocol +[[4,1,3;1]] EAQEC code Figure 5.10: (Color online) Comparison of the [[5; 1; 3]] distillation protocol plus [[4; 1; 3; 1]] EAQEC code, and the [[6; 1; 5; 5]] EA EAQEC code without distillation. The performance of the [[6; 1; 5; 5]] AB EAQEC code is slightly better for p a < 0:11. 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 Depolarizing rate p Channel Fidelity F [[4,1,3;1]] EAQEC code without distillation protocol [[5,1,3]] distillation protocol +[[4,1,3;1]] EAQEC code [[7,1,3]] distillation protocal+[[4,1,3;1]] EAQEC code [[8,1,3]] distillation protocol +[[4,1,3;1]] EAQEC code [[11,1,5]] distillation protocol+ [[4,1,3;1]] EAQEC code Figure 5.11: (Color online) Comparison of dierent distillation protocols with the same [[4; 1; 3; 1]] EAQEC code. 125 0 0.05 0.1 0.15 0.2 0.25 0.3 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Depolarizing rate p Channel Fidelity F c [[6,1,5;5]] AB EAQEC code [[5,1,3]] dist. protocol +[[4,1,3;1]] EAQECC [[9,1,6;7]] AB [[7,1,3]] dist. protocol+[[4,1,3;1]] EAQECC [[9,1,7;8]] AB [[8,1,3]] dist. protocol +[[4,1,3;1]] EAQECC MC of [[13,1,8;11]] AB EAQECC [[11,1,5]] dist. protocol+ [[4,1,3;1]] EAQECC Figure 5.12: (Color online) Comparison of dierent combinations of distillation protocols and EAQEC codes that encode one information qubit. 126 Part III Fault-Tolerant Quantum Computation 127 Chapter 6 Performance and Error Analysis of Knill's Postselection Scheme in a Two-Dimensional Architecture Quantum error correction [Sho96, ABO97, DS96, KLZ96, Got97, Got98, Ste97, Ste98, Pre98, Ste99a] is necessary to build reliable quantum computers using unreliable com- ponents. Quantum computation can be performed with arbitrary accuracy as long as the error rates of physical gates are below a threshold [ABO97]. The error thresholds for several schemes have been estimated, and they range from O(10 5 ) to as high as 3% [Ste97, Ste98, Ste03, AGP06, Rei06, Kni05, AC07, CDT09]. Many of these analyses of error thresholds make simplifying assumptions, such as allowing interactions between any two qubits, that are not possible in real physical architectures. Svore, DiVincenzo, and Terhal designed a two-dimensional qubit layout for quantum computation [SDT07], using the concatenated Steane code [Ste96a]. They assumed that two-qubit gates can be applied only to adjacent qubits, and that qubit movement is done by SWAP gates. Under these assumptions, they showed that the error threshold of the Steane code is 1:85 10 5 . It decreases by roughly a factor of two due to the locality constraints. Sim- ilar work for the concatenated Bacon-Shor code [Bac06, Sho95] was studied by Spedalieri and Roychowdhury [SR09]. The error threshold reported in [SR09] is also O(10 5 ). 128 Knill demonstrated a fault-tolerant quantum computation scheme based on concate- nated error-detecting codes (C 4 andC 6 ) and postselection with a simulated error thresh- old of 3% over the depolarizing channel. Stephens and Evans analyzed a fault-tolerant quantum computation scheme based on the concatenated error-detecting code C 4 with locality constraints in one dimension, and they reported a threshold of O(10 5 ) [SE09]. In this chapter we demonstrate that a two-dimensional layout of the error-detecting code has a signicantly better threshold. To this end, we design the optimal qubit move- ments required to perform quantum computation in a two-dimensional architecture for the concatenated error-detecting code C 4 with postselection [Kni05, AGP08] which has the highest known error threshold without locality constraints. We embed one logical qubit in a 5 5 qubit tile layout. Our tile has a recursive structure, meaning that each qubit is embedded in a 5 5 tile consisting of lower-level qubits. As in [SDT07, SR09], we assume that two-qubit gates can only be performed locally on adjacent qubits, and that additional SWAP gates are needed to move the qubits that are far apart. Each tile contains not only the physical qubits required to maintain the state of a single logical qubit, but also dummy qubits to aid qubit movement by SWAP gates and ancilla qubit preparation for error detection. In this chapter we demonstrate only the tile operations of the error detection block; tile operations for the other gates are available online at http://mizar.usc.edu/~tbrun/Data/KnillTileOps/ We use both analytical and simulation methods to estimate the error threshold. The analytical method counts malignant pairs in the extended rectangle of the CNOT gate in a local adversarial stochastic noise model [AGP08]. In a local adversarial stochastic noise model, arbitrary Pauli errors can be chosen to attack a given set of gates and we may consider the error threshold obtained from this model to be the lower bound on the threshold for a more realistic error model. We calculate the thresholds for dierent ratios of memory error rate to the worst gate error rate. Assuming that all gates have 129 the same error rates, we obtain a threshold of 3:0610 4 in a local adversarial stochastic noise model, which is the highest known error threshold for concatenated codes in 2D. Our second method estimates the threshold by a Monte Carlo simulation of the 2D architecture with depolarizing noise. We calculate a pseudo-threshold of about 0:1%. As expected, the pseudo-thresholds are generally higher than the thresholds obtained in adversarial noise models. By setting the memory error rate to be one-tenth of the worst gate error rate, the error threshold with the adversarial noise model is 4:0610 4 , while the pseudo-threshold with depolarizing noise is about 0:2%. 6.1 Basics of the Knill C 4 =C 6 Scheme with Postselection In his original scheme, Knill concatenated two error-detecting codes C 4 and C 6 which alternate. We follow the simpler version, using only the C 4 code as in [AGP08], which has a high error threshold. In addition, we concatenate M levels of the quantum error- detecting code C 4 with a quantum error-correcting code C ec at the top-level. We use the notation C m 4 to denote the Level-m encoding of the C 4 code and the notation U (m) to denote the gate operation U of C m 4 . We use the notationjvi to denote the statejvi at a higher-level of encoding. The quantum error-detecting code C 4 belongs to the class of stabilizer codes [Got97, CRSS97] and can be dened by the stabilizer group with 2 generators XXXX and ZZZZ, where X = 0 @ 0 1 1 0 1 A and Z = 0 @ 1 0 0 1 1 A are Pauli matrices. The matrix representation of a single-qubit operator is shown in the computational basisfj0i;j1ig. This code encodes two logical qubits in four physical qubits and can simultaneously detect any single-qubit bit- ip error X and any single-qubit phase- ip error Z. How- ever, in Knill's scheme, we use only one of the logical qubits and treat the other as a spectator qubit. The logical operators areX L =XXII,Z L =ZIZI,X S =IXIX, and Z S =IIZZ, where the superscripts L andS are labels for the logical and the spectator qubits, respectively. 130 The top-level quantum error-correcting code C ec can be the Steane code [Ste96a] or the Bacon-Shor code [Bac06, Sho95]. We use the tiled qubit architecture of these two codes studied in [SDT07, SR09] on top of the tiled qubit architecture of the C M 4 code developed in the next section. We can use the Steane or Shor error correction method, or we can use Knill's syndrome extraction in Fig. 6.1 at the top-level of concatenation. This choice does not aect the error threshold of the scheme. In Knill's syndrome extraction, if an error is detected at any error detection step at any level of concatenation, the preparation of the logical EPR pairj + i = j00+11i 2 should be restarted. jQi X jAi P j+i Z jBi P j0i X Z | {z } jQi Preparingj + i Figure 6.1: Knill Syndrome extraction . Now we describe the basic fault-tolerant logical circuits of the C 4 code. We use an ancilla factory model to prepare the high quality ancillas required to execute the phase gate S = 0 @ 1 0 0 i 1 A and the =8 gate T = 0 @ 1 0 0 e i=4 1 A . These gates are then performed by a logical teleportation circuit. The logical statesj 0i =j 0i L j +i S andj +i =j +i L j 0i S can be fault-tolerantly prepared by choosing appropriate spectator qubits as in Fig. 6.2, where P j 0i and P j +i denote the preparation circuits of the logical qubitj 0i andj +i, respectively. To perform fault-tolerant error detection (ED) of C 4 , the two circuits in Fig. 6.3 are used depending on the state of the spectator qubit: we choose ED 0 or ED + when the spectator qubit isj +i S orj 0i S , respectively. This is because the state of the spectator qubit alternates betweenj +i S andj 0i S after each error detection block. As discussed in [Kni05, ?], the ED 0 gate is better suited for detecting Z errors, while the ED + gate is better suited for detecting X errors. 131 If the parity of the X or Z measurement outcomes in ED 0 and ED + is not zero, which means that errors are detected, the ancilla qubits are discarded and the circuit restarts. If there are no errors detected, the measurement outcomes of the the rst two code blocks determine the logical Pauli operators to be applied to the second ancilla block to complete the quantum teleportation. These operations are represented by the decision block in Fig. 6.3. Each single-qubit gate other than measurements is followed by an ED routine, and the two-qubit CNOT gate is followed by an ED on each of the two qubits. As a general rule we shall assume the presence of the input and output error detection routines before and after every logical gate, and this should be understood for every circuit shown. Measurements have quantum ED routines at the input, but classical ED routines at the output, while ancilla preparations typically have only quantum ED routines at the output. The combination of a gate and its following ED(s) is called a rectangle (1-Rec). j+i P j0i j+i j0iLj+iS j0i j0i 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; j+i P j+i j0i j+iLj0iS j+i j0i 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; Figure 6.2: State preparation . The logical controlled-NOT (CNOT) gates between dierent code blocks of C 4 can be done transversally by applying bitwise CNOT gates. The swap of qubits 2 and 3 implements the SWAP gate of the logical qubit and the spectator qubit, and we call this an inner SWAP gate. The logical Hadamard gateH = 1 p 2 0 @ 1 1 1 1 1 A is implemented by transversally applying the Hadamard gates, followed by an inner SWAP gate. The inner SWAP gate does not need to be applied; instead, we switch the labels of the qubits and keep track of them. We assume this can be done eciently. 132 j i L = X j0i L j+i S = P j+i Z j+i L j0i S = P j0i Decision = j i L j0i S j i L = X j+i L j0i S = P j+i Z j0i L j+i S = P j0i Decision = j i L j+i S Figure 6.3: Circuits for fault-tolerant quantum error detection. Top: ED0. Bottom: ED+. To enable universal quantum computation, it remains to prepare the level-M ancilla statej+ii = 1 p 2 j0i +ij1i , which is the +1 eigenstate of Y =iXZ at levelM, and the level-M magic state Tj +i. The phase gate S and the =8 gate T can be implemented with the help of the ancilla statej+ii and Tj +i as shown in Fig. 6.4 and Fig. 6.5, respectively. The logical statej+ii can be non-fault-tolerantly prepared by the circuit in Fig. 6.6. To prepare the physical statej +ii =SHj0i at level 0, we sequentially apply the faulty gates H and S on a physical qubitj0i. After several iterations of distillation, we obtain aj+ii with high delity. The decoding gateD is shown in Fig. 6.7. The output state j+ii can be distilled to one with higher delity by the circuit in Fig. 6.8, where the twirl operation is shown in Fig. 6.9. The statej+ii at level M can be prepared by recursively applying the circuit in Fig. 6.6 or by using a level-M to level-0 decoderD in the teleportation at level M. A level-M to level-0 decoder can be implemented by recursively applying the decoding gateD at each level. j i Sj i j+ii Z j+ii Figure 6.4: The circuit for implementing the logical S gate. 133 j i = Z = Tj+i S EC Tj i Figure 6.5: The circuit for implementing the logical T gate. j+ii X = P j+i D Z = P j0i X Z = j+ii Figure 6.6: The circuit for preparing the logical statej+ii. jq1i j i j iLj0iS jq2i jq3i jq4i 8 > > > > < > > > > : jq1i j i j iLj+iS jq2i jq3i jq4i 8 > > > > < > > > > : Figure 6.7: The decoding circuit for C 4 . j+ii twirl j+ii j+ii twirl Z X Figure 6.8: The distillation circuit for the statej +ii. j+i Z j+ii Y j+ii Figure 6.9: The twirl operation for the statej +ii. 134 The realization of a fault-tolerant T gate is shown in Fig. 6.5. This gate sequence was originally constructed in [ZLC00] using one-bit teleportation. The gate sequence teleports the statej i from the data block to the ancilla and applies the T gate to the state. The ancilla state Tj +i is prepared using the state injection method described before, as in Fig. 6.6, followed by several rounds of distillation. The distillation and twirl procedures of Tj+i are complicated and they are described in [BK05]. 6.2 The Two-Dimensional Qubit Layout of the C 4 Code We now describe the two-dimensional qubit layout for the C 4 code and estimate the number of each physical gate operation required for each logical operation. We assume that two-qubit interactions are available only for the nearest neighbors. That is, we apply horizontal or vertical CNOT gates (hCNOT/vCNOT) only to two neighboring qubits on the same horizontal or vertical line. Similarly, we assume movements of the qubits are accomplished by SWAP gates in two directions: horizontal and vertical SWAP gates (hSWAP/vSWAP). Following the tile structures presented in [SDT07, SR09], we design a two-dimensional 5 5 lattice architecture of physical qubits to represent a logical qubit of the C 4 code. A tile is initialized as one of the following two structures: Structure I : 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 O O O O O O d 1 O O d 3 O O O O O O O O O O O d 2 O O d 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; 135 Structure II : 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 O O O O O O O O O O O O d 1 d 3 O O O d 2 d 4 O O O O O O 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : The four data qubits of the C 4 code are denoted by d 1 , d 2 , d 3 , and d 4 . The O's are dummy qubits used for ancilla preparation or for swapping with data or ancilla qubits in communication, and their states are irrelevant to computation. Each qubit in the tile is encoded in a lower-level tile structure. The following operations are performed on the C4 code: (i) Error detection (ED). (ii) Horizontal and vertical CNOT gates (hCNOT/vCNOT). (iii) Horizontal and vertical SWAP gates (hSWAP/vSWAP). (iv) Measurement in the X basis or the Z basis (M X and M Z ). (v) The Pauli operators X, Y , Z, and the Hadamard gate (H). (vi) Preparation of the ancilla qubitsj+i orj0i (P j+i and P j0i ). (vii) The phase gate S and the =8 gate T . For simplicity, all lower-level gates are assumed to take one time step, which is the longest execution time among all gates. In reality, we may think that a qubit idles for the rest of the time step after it completes a fast gate, and the error rate of this operation is the physical gate error rate plus the memory error rate for the idle time. To achieve 136 favorable error-correction properties and low overhead, our design attempts to minimize the number of SWAPs, idle qubits, and the total number of time steps. Since error detection is performed constantly, extra space is need for preparation of the logical EPR pairs used in Knill's syndrome extraction. In structure I, the data qubits lie on the \corners" and the logical EPR pairs for error detection are prepared inside the data qubits. By contrast, the data qubits are located at the \center" of structure II and the ancilla qubits surround the data qubits, as will be shown in the following. Error detection is designed in each of the two tiles precisely so that the data qubits are transferred between the \center" and the \corners." Therefore, the tile alternates between structures I and II after each error detection. This alternation avoids extra SWAP gates. For a SWAP operation to be fault-tolerant, we only swap a data or ancilla qubit with a dummy qubit. Because of this, only one tile requires an ED circuit. The topmost row and leftmost column of each tile is reserved for transportation of the lower level qubits. This allows realization of the horizontal or vertical CNOT gate without aecting the EDs. Fig. 6.3 demonstrates the ED + block for structure I. Due to space constraints, we have made the other tile operations available online. In the following, \a ! b" means applying a CNOT gate witha being the control qubit andb being the target qubit. 137 Time step 1: O O O O O O d1 O O d3 O P j+i (a1) P j+i (a5) P j0i (a7) P j+i (a3) O P j0i (a2) P j+i (a6) P j0i (a8) P j0i (a4) O d2 O O d4 Time step 2: O O O O O O d1 O O d3 O a1 a5 ! a7 a3 # # O a2 a6 ! a8 a4 O d2 O O d4: Time step 3: O O O O O O d1 O O d3 O a1 ! a5 a7 a3 O a2 ! a6 a8 a4 O d2 O O d4 138 Time step 4: O O O O O O d1 O O d3 # # O a1 a5 a7 a3 O a2 a6 a8 a4 " " O d2 O O d4 Time step 5: O O O O O O MX (d1) O O MX (d3) O MZ (a1) a5 a7 MZ (a3) O MZ (a2) a6 a8 MZ (a4) O MX (d2) O O MX (d4) At the end of time step 5: O O O O O O O O O O O O d1 d3 O O O d2 d4 O O O O O O 139 The ancilla qubits a 1 ; ; a 8 are prepared at time step 1, and logical EPR pairs are made at time steps 2 and 3. Quantum teleportations are completed in the subsequent time steps. We choose the index such that a quantum teleportation occurs on the qubits d i ;a i ;a i+4 for i = 1; 2; 3; 4. Observe that the data qubits d 1 ;d 2 ;d 3 ;d 4 are transferred to the center after teleportation and no SWAPs are needed here. However, the error detection ED + for structure II needs two SWAPs and it takes one more step. Its rst time step is initialized as follows: O O O O O O O P j+i (a1) P j0i (a3) O O P j+i (a5) d1 d3 P j+i (a7) O P j0i (a6) d2 d4 P j0i (a8) O O P j+i (a2) P j0i (a4) O : In addition, applying the logical Pauli operators X or Z to complete the teleportation may take one or two more steps, but this is not shown. In many cases it suces to track these Pauli operators without correcting them. The ED 0 for structure I at time step 1 is as follows and the rest of the steps are similar to those of the above ED + : O O O O O O d1 P j+i (a1) P j0i (a3) d3 O O P j+i (a5) P j+i (a7) O O O P j0i (a6) P j0i (a8) O O d2 P j+i (a2) P j0i (a4) d4 : Note that the operations and required time of ED 0 are the same as those of ED + . Remark: after a logical Hadamard gate, the labels of data qubits 2 and 3 are switched. This can be xed by applying appropriate SWAPs and it takes two more time steps in 140 ED (1) vCNOT (1) hCNOT (1) vSWAP (1) hSWAP (1) P j+i(1) P j0i(1) ED (1) 2 2 1 1 1 1 vCNOT (0) 6 4 2 hCNOT (0) 6 4 2 vSWAP (0) * 40 8 20 4 hSWAP (0) * 8 40 20 4 P j+i(0) 4 2 2 P j0i(0) 4 2 2 M Z(0) 4 M X(0) 4 Z (0) X (0) H (0) M Z(1) M X(1) Z (1) X (1) H (1) S (1) T (1) ED (1) 1 1 1 1 2 vCNOT (0) hCNOT (0) 8 12 vSWAP (0) 20 40 hSWAP (0) 4 12 P j+i(0) P j0i(0) M Z(0) 4 M X(0) 4 4 Z (0) 2 X (0) 2 H (0) 4 8 8 ( The number of SWAPs in the ED is zero in structure I but 4 in structure II.) Table 6.1: The numbers of the quantum operations contained in each higher-level quan- tum operation of the C 4 code and its following error detection (1-Rec). Each entry rep- resents the number of the elementary gate (U (0) ) corresponding to that row contained in the higher-level gate (U (1) ) corresponding to that column. structures I or II. However, we don't adjust it until a CNOT gate acts on two tiles with dierent labels. Based on the tiled operations, we build the recursive relations resulting from the concatenated code structure in order to quantify the total number of gates and total time required for each logical gate. The recursive relations of a 1-Rec for each logical gate in terms of lower-level gates are listed in Table 6.1. For example, the vertical SWAP operation vSWAP requires 20 vertical swap operations at the next lower concatenation level, followed by the error detection operation ED. To allow universal quantum computation, we implement the S and T gates by the ancilla factory method, which uses the decoding circuits in Fig. 6.7. The overhead of the ancilla factories and their decoding circuits are not included in Table 6.1. 141 Remark: It is possible to combine a gate operation with the following error detection and save several time steps. 6.3 Error Analysis of the 2-Dimensional Knill Postselec- tion Scheme 6.3.1 Error Threshold Here we estimate the error threshold of Knill's postselection scheme in the 2-dimensional tile for the local stochastic, adversarial noise model. Following the procedure presented in [AGP08, SDT07, SR09], we count the number of malignant pairs of locations in the 1-exRec of the CNOT gate. The 1-exRec of the CNOT gate includes the bitwise CNOTs on two logical qubits, together with two following and two preceding error detection blocks. As shown in Fig. 6.10, we assume the preceding EDs are ED + s and the following EDs are ED 0 s. Note that the logical Pauli operators to complete teleportation are assumed to be error-free, since they can be tracked in the Pauli frame and hence be deferred until a non-Cliord gate occurs. We also assume that classical computations are perfect, and that any quantum operations depending on the classical results can be applied without delay. ED + ED 0 ED + ED 0 Figure 6.10: The 1-exRec of the CNOT gate. There are seven types of locations in the 1-exRec of the CNOT gate: (1)P j+i ; (2)P j0i ; (3)M X ; (4) M Z ; (5) hSWAP/vSWAP; (6) hCNOT/vCNOT; (7) idle qubits. A set of locations is called malignant if errors happening in these locations could make the calculation of the rectangle incorrect. Since an error at any single location can be 142 detected, errors at two locations dominate the source of logical errors. To determine whether a pair of locations is malignant, we check whether there is any logical error in the output of two perfect ED + s following the 1-exRec of CNOT, as shown in Fig. 6.11. The simulation procedure in the stabilizer formalism proposed in [AG04] can be used to track the logical operators through the circuit in Fig. 6.11. ED + ED 0 ideal ED + ED + ED 0 ideal ED + Figure 6.11: The 1-exRec of the CNOT gate followed by two perfect ED + . Remark: in general the error rate of a SWAP gate is higher than a CNOT gate, since it is implemented by a series of gate operations, such as three CNOT gates. However, in the two-dimensional tile we only swap a data or ancilla qubit with a dummy qubit, and the cost of such a SWAP gate is less than the cost of a CNOT gate. As for the S andT gates, Aliferis, Gottesman, and Preskill showed that the distillation method for ancilla preparations has a higher threshold than the code itself [AGP08]. To maximize the error threshold, we optimized the tile operations of the extended rectangle of the CNOT gate. The animations showing this are available online. There are 196 locations in the extended rectangle of the CNOT gate: 32 idle qubits and 154 gates, of which 38 gates are SWAPs. We assume that the error detection blocks begin before the time step that the data qubits come in, and thus there are no idle qubits at 143 time steps 1, 2, and 3 in the preceding ED. We nd that the numbers of malignant pairs of locations of each kind are given by = 0 B B B B B B B B B B B B @ 4 8 8 0 0 32 16 0 0 14 96 80 32 16 0 96 104 32 16 96 112 32 442 672 268 322 288 106 1 C C C C C C C C C C C C A ; where i;j represents the number of malignant pairs at locations of types i and j. Let (m) j be the error rates of type j at level m. For error correction to be eective, we require (m+1) 6 = X ij i;j (m) i (m) j +O(( (m) max ) 3 ) (m) 6 ; (6.1) where i;j is the number of malignant pairs of types i and j and m max is the maximum of the seven types of error rate. We assume all errors of weight 3 or larger are malignant and the eect of errors of weight higher than three can be ignored. (This might still be an overestimate of higher- order terms.) Let be the ratio of the memory error rate of the idle qubits to the gate error rate. Let B = 6 X i;j;=1 i>j i;j + 6 X i=1 i;7 + 2 7;7 be the eective number of malignant pairs and A = 164 3 + 164 2 32 1 + 164 1 32 2 2 + 32 3 3 be the eective number of errors of weight 3, where the a b 's are binomial coecients. Then Eq. (6.1) reduces to A( (m) 6 ) 2 +B (m) 6 < 1: 144 scheme Steane code Bacon-Shor code Knill's postselection scheme nonlocal 2:73 10 5 1:94 10 4 1:04 10 3 2D( = 1:0) 1:1 10 5 1:3 10 5 3:06 10 4 2D( = 0:1) 1:85 10 5 2:02 10 5 4:06 10 4 Table 6.2: Comparison of the error thresholds of three concatenated codes. If we assume the error rates are the same for all types of locations ( = 1), we have B = 2; 892 and A = 196 3 , and Eq. (6.1) gives an error threshold of ( = 1)< 3:06 10 4 : We compare our results with those of the Steane code and the Bacon-Shor code for = 1:0 and 0:1 in Table 6.2. The rigorous error thresholds obtained in [AGP06, AGP08, AC07] are also listed as a reference. Knill's postselection scheme has the highest error threshold ofO(10 4 ), as expected. Remark: we obtain 714 malignant pairs and calculate a threshold of 1:05 10 3 if we assume no SWAP or memory errors. 6.3.2 Pseudo-Threshold Knill reported a simulated pseudo-threshold of 3% by his postselection scheme over unbiased and independent depolarizing noise [Kni05]. We now present a Monte Carlo simulation of the circuit in Fig. 6.11 over depolarizing noise to obtain pseudo-thresholds for the Knill scheme in two dimensions, as in [CDT09]. In our model, we add depolarizing errors as quantum operations after gates or before measurements in the circuit. Letp be the depolarizing rate. Any single qubit location (other than measurements) undergoesX, Y , orZ with probability p 3 . Any binary measurement outcome is ipped with probability p. CNOT gates are modied by one of the 15 non-identity two-qubit Pauli operators (IX, IZ, , YY ) with probability p 15 . We obtain the pseudo-thresholds by calculating the logical error rate of the circuit in Fig. 6.11. The logical error rate e(p) for a given depolarizing rate p (the worst gate 145 error rate) is dened as the number of samples without logical errors at the output of the circuit in Fig. 6.11, divided by the number of samples without any errors being detected. If an error-detecting code works, it is clear that e(p)<p for p small enough and e(p) is an increasing function ofp. The pseudo-threshold ~ is the value ofp such thate(~ )< ~ and e(~ + )> ~ . If we assume all locations have the same depolarizing rate p ( = 1:0), we nd a pseudo-threshold of about 0:1%. If = 0:1, we obtain a pseudo-threshold of about 0:2%. These values are higher than the error thresholds estimated in the adversarial noise model, as expected, since the adversarial noise model is the worst case. As a comparison, we calculated a pseudo-threshold of about 0:8% for the Knill scheme without locality constraints. Remark: we can reduce the depolarizing rates on the measurements and ancilla preparations as Knill did in [Kni05] by choosing these error rates to be 4=15 of the worst gate error rate. We obtain a pseudo-threshold of about 0:35% by choosing = 0:1 for the Knill scheme in two dimension. For the Knill scheme without locality constraints, we obtain a pseudo-threshold of about 2:5%. 6.4 Discussion We designed a two-dimensional 5 5 qubit tile for quantum computation using the concatenatedC 4 code with postselection. Although we didn't prove the optimality of our design, we believe that a substantial improvement within our architectural framework is unlikely. We demonstrated the tile operations of the ED + block for structure I. Dierent combinations of error detection (ED + or ED ) and tile structures (I or II) require small modications to the logical gates involving two tiles, such as vSWAP, hSWAP, vCNOT, and hCNOT. These modications can be done by slightly changing the locations of d i ;a i ;a i+4 for i = 1; 2; 3; 4 in our demonstration. For example, we have to modify the 146 1-exRec of the CNOT gate error threshold tile SWAPs idle qubits times steps = 1 = 0:1 = 0:0 4 4 38 74 16 1:47 10 4 2:22 10 4 4:89 10 4 5 5 48 32 14 3:06 10 4 4:06 10 4 4:14 10 4 Table 6.3: Comparison of the 4 4 and 5 5 tiles. ancilla preparation of an ED + that follows a vSWAP, and it takes one more time step and four more lower-level SWAPs than a vSWAP followed by an ED 0 . It is desirable to reduce the size of the tile, and hence the number of SWAPs, for physical architectures with very low memory error rates, such as superconducting qubits. To that end, we have also designed a 4 4 tile, and its performance is compared with the 5 5 tile in Table 6.3 for dierent ratios of memory error to gate error rate. The 4 4 tile has a higher threshold with no memory error ( = 0). Surprisingly, the error threshold of the 44 tile decreases by a factor of about two for = 0:1. This is probably because there are many more idle qubits in the 4 4 tile, and the operations in the two code blocks of the 4 4 tile are not parallel: one block is delayed by one time step as shown in the tile operations online. However, the error thresholds of the 5 5 tile for = 0:1 and = 0 are about the same. The eects of some errors may cancel each other due to the symmetry in the 1-exRec of the CNOT gate in the 5 5 tile. Under the realistic assumption that one- and two-qubit quantum gates are local, our threshold analyses establish that Knill's postselection scheme has better error correction capabilities than other concatenated error-correcting codes. This makes our proposed two-dimensional architecture a practical choice for quantum error correction. In addition to the postselection scheme based on error detection, Knill also proposed a Fibonacci scheme to further reduce the overhead of the postselection scheme [Kni05]. He calculated a pseudo-threshold of about 1%. It uses the fact that the concatenated error-detecting code C 4 can correct located errors. Aliferis and Preskill showed that the error threshold of the Fibonacci scheme is slightly lower than the postselection scheme 147 over the adversarial noise model [AP09]. Nonrecursive versions of the CNOT gates or the measurements in the Fibonacci scheme would take many time steps without error detection or correction in our two-dimensional architecture. This might lead to a much worse error threshold. However, we still consider nding the threshold of the Fibonacci scheme, combined with the \soft decision" decoder in [ES12], an interesting question for future work. 148 Chapter 7 Encoded Quantum Syndrome Measurement An important issue in the implementation of a quantum computer is to protect quantum information from decoherence. Concatenated quantum codes and topological quantum codes are extensively studied for fault-tolerant quantum computation. However, there is not much research on large block codes in any fault-tolerant scheme. In this chapter, we are interested in eliminating the eect of faulty syndrome mea- surement and we can have better idea about errors occurred in the data qubits. Currently Shor's syndrome extraction [Sho96], Steane's syndrome extraction [Ste97], and Knill's syndrome extraction [Kni05] are three methods adopted in fault-tolerant quantum com- putation. We will generalize Shor and Steane's methods using the idea of classical error-correcting codes in dierent directions. 7.1 Encoded Quantum Syndrome Measurement In Shors syndrome extraction, a stabilizer generator is measured three times and a syn- drome bit is determined by taking the majority vote of the three measurement outcomes. Repetitive syndrome measurement is similar to the idea of classical repetition codes. We generalize the syndrome measurement to the application of a classical linear block code. Letp m be the error rate of the measurement in theZ basis. That is, the measurement outcome is ipped from 0 to 1 or 1 to 0 with probability p m . Assume all quantum gates (such as Hadamard gate H, controlled-Not gate CNOT , ancilla preparation P j0i ) have depolarizing errors and let p g be the highest error rate among all quantum gates. 149 In fault-tolerant quantum computation, we require that errors that occur in these gates are still correctable by a quantum error-correcting code during each error correction cycle. As shown in Fig. 1.3, these gates are faulty and they may introduce new errors in the data qubits so that the syndrome bit may be false. For examples, errors on the Hadamard gates after the CNOT gates. We rst consider the case that p g = 0 in Shor's syndrome extraction circuit. Lets =s 1 s nk be the correct syndrome measurement. Let ^ s = ^ s 1 ^ s nk be the syndrome bit output by Shor's syndrome extraction. Then Pr(^ s j 6=s j ) = X i:odd wt(g j ) i p i m (1p m ) wt(g j )i : (7.1) Let P se = Pr(s6= ^ s) be the syndrome error rate. The standard approach for reduction ofP se is the repetitive syndrome measurement. That is, repeat the syndrome measurement several times and take a majority vote. This is the idea of classical repetition codes. We can generalize this to any linear classical codes. Suppose we are using an [[n;k;d]] stabilizer code with independent stabilizer gener- ators g 1 ;g 2 ; ;g nk 2G n , the n-fold Pauli group. We choose an [m;nk;d 0 ] classical linear binary codeC(G) with a (nk)m generator matrixG. We dene a new set of stabilizers h l = nk Y j=1 g G(j;l) j ; l = 1; ;m; (7.2) where G(j;l) represents the matrix entry of G with coordinate (j;l). By measuring these m stabilizers, we can correctb d 0 1 2 c measurement errors. We call C the syndrome measurement code. Now we have new syndrome measurement results ^ s = ^ s 1 ^ s m . Let 150 ~ s be the decoded syndrome by the syndrome measurement code. Similarly, we can dene P se = Pr(s6= ~ s) be the syndrome error rate of a syndrome measurement code. Remark: using one redundant stabilizer generator, we can detect any single measure- ment error by the classical [nk + 1;nk; 2] code. Example 7.1.1. Consider the stabilizer generators of the three qubit bit- ip code: g 1 =ZZI g 2 =IZZ: If we obtained the error syndrome 2 4 1 0 3 5 , it means anX error occurs on the rst qubit or a measurement error occurs on the rst syndrome bit. Consider again the stabilizers of the three qubit bit- ip code: g 1 =ZZI g 2 =IZZ g 3 =ZIZ; where we add a nonindependent third stabilizer g 3 = g 1 g 2 . We measure these three stabilizers to obtain the 3-bit error syndrome for X errors. Any single qubit bit- ip error will anti-commute with two of the stabilizers. If a syndrome of odd weight is observed, it means a measurement error occurs. Hence the redundant version of the stabilizers can detect any single measurement error. Example 7.1.2. The [7; 3; 4] simplex code has a weight enumerator B(z) = 1 + 7z 4 and is a constant-weight code. It can be observed that the X stabilizers (g 1 ;g 2 ;g 3 ) and Z stabilizers (g 4 ;g 5 ;g 6 ) of the Steane code can be dened by the [7; 3; 4] simplex code and any linear combination of the X stabilizers or theZ stabilizers has weight 4. We use an [m; 3] syndrome measurement code to protect the measurement of theX stabilizers. The 151 Z stabilizers are treated in the same way. As a result, the encoded stabilizer generators h l = Q nk j=1 g G(j;l) j 's have weight 4 by Eq. (7.2) for any choice of classical syndrome measurement code C(G). Thus Pr(^ s l 6=s l ) = X i2f1;3g 4 i p i m (1p m ) 4i (7.3) for any l and any C(G). Let q l = Pr(^ s l 6=s l ) for convenience. Now we can compare the symbol error rates P se of dierent codes with Eq. (7.3) being the channel error rate. We dene a correctable-error weight enumeratorW (x;y) = P n i=0 w i x i y ni for a classical linear code such thatw i is the number of correctable errors of weight i. Then the probability of symbol error of an [m;r] syndrome measurement code with a correctable-error weight enumerator W (x;y) is P se = m X i=0 m i q i l (1q l ) mi W (q l ; 1q l ): The [7; 3; 4] simplex code has 16 distinct error syndromes. We can build up a lookup table for this code such that its has a correctable-error weight enumerator W (x;y) =y 7 + 7xy 6 + 7x 2 y 5 +x 3 y 4 : Thus its symbol error rate is P se ([7; 3; 4]) = 7 X i=2 7 i q i l (1q l ) 7i 7q 2 l (1q l ) 5 q 3 l (1q l ) 4 : (7.4) Next we consider the [3; 1; 3] repetition code with three code blocks. A [3; 1; 3] can correct any single bit error and its correctable-error weight enumerator is W (x;y) =y 3 + 3xy 2 : 152 Thus P se ([3; 1; 3]) = 3 X i=2 3 i q i l (1q l ) 3i : (7.5) Since we have three code blocks, P se (3 [3; 1; 3]) = 1 (1P se ([3; 1; 3])) 3 : (7.6) For unencoded syndrome measurement, P se ([1; 1; 1]) =Pr(s6= ~ s) = 1 Pr(s = ~ s) = 1 Y l=1;2;3 Pr(~ s l =s l ) =1 Y l=1;2;3 (1 Pr(~ s l 6=s l )) =1 0 @ X i2f0;2;4g 4 i p i m (1p m ) 4i 1 A 3 : (7.7) Note that ~ s = ^ s for unencoded syndrome measurement. 153 Here we consider one more code: the [10; 3; 5] code with a generator matrix 2 6 6 6 6 4 1001011001 0101101010 0011111100 3 7 7 7 7 5 and a parity-check matrix G = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1000000001 0100000010 0010000100 0001000111 0000100110 0000010101 0000001111 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . We know it can cor- rect any error of weight not greater than 2 and some other higher-weight errors. An upper bound on its symbol error rate is P se ([10; 3; 5]) = 10 X i=3 10 i q i l (1q l ) 10i : (7.8) We plot Eqs. (7.6),(7.4), (7.7), and (7.8) in Fig. 7.1. Observe that the encoded syndrome measurement with the [10; 3; 5] code has better performance than the repetitive syndrome measurement in the lower physical error rate region, since its upper bound on symbol error rate is still below the symbol error rate of the repetitive syndrome measurement. Eqs. (7.6),(7.4), and (7.7) converge as p m goes to 0:5. To sum up, the [7; 3] code has a higher code rate than the [3; 1] code but has a slightly higher P s e. The [10; 3; 5] code has a slight lower rate but better P se . Remark: if a stabilizer group is dened by the generator matrix of a classical constant- weight code, then we can choose any syndrome measurement codes. In this case, we can use any good LDPC codes so that the measurement errors can suppressed. For example, the family of [[2 r 1; 2 r 12r; 3]] quantum Hamming codes are dened by the family of [2 r 1;r; 2 r1 ] simplex codes. Although the minimum distance of this family of quantum codes is only three, we may concatenate it with a high distance code and still have a good weight distribution. 154                                                                                                                                                  Figure 7.1: P se for encoded syndrome measurement, repetitive syndrome measurement, encoded syndrome measurements with [7; 3; 4] simplex code and with [10; 3; 5] code. Note that the plot for the [10; 3; 5] is the upper bound on its P se If we use a large syndrome measurement code, the errors in the quantum state may accumulate so that they become uncorrectable by the quantum code. However, the redundant stabilizers might have high weight, which increases the prob- ability of measurement errors. It neglects the complication of errors occurring in the middle of a round of syndrome measurement, causing measurement errors to be correlated. 155 7.2 Using Fewer Ancillas in Syndrome Measurement When the block size is large, the ancilla preparation is very expensive. We can save some ancilla preparation using the idea of classical error-correcting codes when error rate is not too high. We would like to share the ancillas during syndrome measurement as long as errors do not accumulate seriously. Suppose we have m block of quantum codes. We wish to read out the m error syndromes for bit- ip errors (similar for phase- ip errors) by using onlyn ancillaj+i E 's. Let u 1 ; ;u m be m unknowns in a eld F . Consider the linear system v i = m X j=1 a i;j u j for i = 1; ;n and a i;j 2 F . Question: can we recover u 1 ; ;u m , given v 1 ; ;v n ? We learn from fundamental mathematics or linear algebra that a system of n linear equations in m unknowns over a eld F has more than one solution if n < m. Given v 1 ; ;v n , we cannot recover u 1 ; ;u m uniquely. However, the scenario is dierent if the unknowns u 1 ; ;u m are binary vectors of length N. Suppose u i for i = 1; ;m are random vectors inZ N 2 . Assume the bits of u i are identically and independently distributed, and let p be the probability that a bit is 1. It is possible to recover the vectors u 1 ; ;u n , given v i = m X j=1 a i;j u j for i = 1; ;n and \good" choice of a i;j 2Z 2 with a certain probability of success. In the case of small p, we can handle this problem with a classical linear block code [MS77]. Let k = mn. Suppose G is a kn generator matrix of an [n;k;d] linear binary block codeC. A codeword ofC is a binary vector of lengthn in the row space ofG. The weight of a codeword is the number of its nonzero elements. d is the minimum distance 156 of the code, meaning that the weight of xy for any two codewords x and y is at least d. We choose a i;j as follows: h v 1 v 2 v n i = h u n+1 u n+2 u n+k i G + h u 1 u 2 u n i : (7.9) In the viewpoint of communication system, h v 1 v 2 v n i is the \received vector", h u 1 u 2 u n i is the \noise vector", and h u n+1 u n+2 u n+k i is the \informa- tion vector". We use a decoder to identify the \noise vector" h u 1 u 2 u n i and then recover the \information vector" h u n+1 u n+2 u n+k i . Therefore, if p is small, we can decode u 1 ; ;u m successfully. Note that u i 's, and v j 's are binary vectors of length N and we decode each bit separately. IfN = 1, it reduced to the case of a standard error-correcting code. Therefore we repeat the decoding procedure for N times. Example 7.2.1. Assume we have unknowns u 1 ; ;u 6 2 Z 3 2 . Consider the generator matrix G = [11111] of a [5; 1; 5] repetition code. Let v 1 = u 1 +u 6 , v 2 = u 2 +u 6 , v 3 =u 3 +u 6 , v 4 =u 4 +u 6 , v 5 =u 2 +u 6 , By Eq. (7.9) h v 1 v 2 v 3 v 4 v 5 i = h u 6 i [11111] + h u 1 u 2 u 3 u 4 u 5 i = h u 1 +u 6 u 2 +u 6 u 3 +u 6 u 4 +u 6 u 5 +u 6 i : 157 For example, assume v 1 = 100 v 2 = 100 v 3 = 110 v 4 = 101 v 5 = 001: We take a majority vote on each bit and then we have ^ u 6 = 100. The rest are ^ u 1 = ^ u 2 = 000, ^ u 3 = 010, ^ u 4 = 001, ^ u 5 = 101. We use ^ u i to represent the decoder output of u i . This method works with high probability of success when p is small compared to the error-correcting ability of the error-correcting code. The properties of the theory of classical error-correcting codes can be applied here. However, errors \propagate" in this scenario. For example, consider Example 7.2.1 again. If u 6 = 000, there is one bit error on u 6 if ^ u 6 = 100. However, we would have u 1 = 100 u 2 = 100 u 3 = 110 u 4 = 101 u 5 = 001; which means that there is one bit error on each of u 1 ; ;u 5 . We can apply the technique in this chapter by connecting the transversal CNOT gates according to Eq. (7.9). Fig. 7.2 demonstrates the circuit for syndrome measurement of four data blocks with three ancilla blocks according to a classical [3; 1; 3] repetition code. 158 = = = = j+i E = = m j+i E = = m j+i E = = m Figure 7.2: The circuit for syndrome measurement of four data blocks with three ancilla blocks by a classical [3; 1; 3] repetition code. 7.3 Error-Correcting Codes for Measurement Errors In a fault-tolerant scheme of any quantum error correction, we have to perform repeated measurements to ensure correct outcomes or the encoded syndrome measurement in the previous section. If we could reduce the number of measurements, we would have a higher error threshold and save a tremendous cost in real physical implementation. Another potential way it to design a quantum error-correcting code to cope with the measurement errors. We assume the CNOT gates before the measurements are perfect and the measure- ment error rate p m is smaller than the data qubit error rate p d . Assume p d and p m are independent. We may think p d as the error rate of an error occurs after the previous error correction cycle and before the syndrome measurement at this cycle. If CNOT gates are not perfect, p d and p m are increased and correlated. An error syndrome is a linear combination of the columns of the parity check matrix H. We would like do design codes that can correct syndrome bit errors. For simplicity, we consider CSS codes. Suppose we are using an [[n;k;d]] CSS code with a check matrix 0 @ H 1 0 0 H 1 1 A . We treat the weight-1 error syndromes as the error syndrome of the syndrome measurement errors. Ex. the vector (10 0) T is the error syndrome of the measurement error on the rst syndrome bit. 159 Now we have a new \parity-check" matrix 0 @ H 1 I 0 0 0 0 H 1 I 1 A , where the last nk 2 bits on both sides represent the syndrome bits. The distance of the overall matrix needs to be determined. If no columns of H 1 are of weight 1, the distance is at least 3. To include the measurement errors in the set of \correctable errors," apparently the error-correcting ability on the data qubits decreases. Example 7.3.1. Consider the Steane code with H 1 = 2 6 6 6 6 4 1000111 0101011 0011101 3 7 7 7 7 5 . A corretable set of Pauli errors isfE x E z :E x 2fI;X 1 ; ;X 7 g;E z 2fI;Z 1 ; ;Z 7 gg up to a phase. If we would like to correct the measurement errors, the correctable set of Pauli errors isfE x E z : E x 2fI;X 7 g;E z 2fI;Z 7 gg up to a phase. Let M = 8 > > > > < > > > > : 0 B B B B @ 0 0 0 1 C C C C A ; 0 B B B B @ 1 0 0 1 C C C C A ; 0 B B B B @ 0 1 0 1 C C C C A ; 0 B B B B @ 0 0 1 1 C C C C A 9 > > > > = > > > > ; . The last three columns correspond to the syndrome measurement errors. We can divide the syndromes into two cosets: 8 > > > > < > > > > : 0 B B B B @ 0 0 0 1 C C C C A ; 0 B B B B @ 1 0 0 1 C C C C A ; 0 B B B B @ 0 1 0 1 C C C C A ; 0 B B B B @ 0 0 1 1 C C C C A 9 > > > > = > > > > ; ; 8 > > > > < > > > > : 0 B B B B @ 1 1 1 1 C C C C A ; 0 B B B B @ 0 1 1 1 C C C C A ; 0 B B B B @ 1 0 1 1 C C C C A ; 0 B B B B @ 1 1 0 1 C C C C A 9 > > > > = > > > > ; . ThusZ 3 2 =M ( 0 B B B B @ 1 1 1 1 C C C C A +M). It is desire to design codes such that the weight-one syndromes correspond to high weight data-qubit errors. A rst observation is that the columns of H have to be of weight higher than one. Theorem 7.3.1. Given an [[n;k;d]] CSS code, there exists an [[2n + 2;n +k 4;d + 1]] CSS code such that the columns of its check matrix are of weight greater than one. 160 Proof. Suppose 0 @ H 1 0 0 H 1 1 A is a check matrix of an [[n;k;d]] CSS code, where H 1 = h T 1 h T 2 h T nk 2 . The following matrix is a desired parity-check matrix of an [[2n + 2;n +k 4;d + 1]] CSS code. H = 0 B B B B B B B B B B B B B B @ 0 0 1 1 1 1 h 1 0 h 1 0 h 2 0 . . . . . . hnk 2 0 1 1 1 1 0 0 1 C C C C C C C C C C C C C C A : Example 7.3.2. The smallest code by the above construction is a [[16; 4; 4]] code obtained from the Steane code. We consider the Pauli X errors only. 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Creator Lai, Ching-Yi (author)

Core Title Quantum error correction and fault-tolerant quantum computation

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School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 09/24/2013

Defense Date 07/11/2013

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Tag entanglement-assisted quantum error correction,OAI-PMH Harvest,quantum error correction

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Advisor Brun, Todd A. (committee chair), Lidar, Daniel A. (committee member), Reichardt, Benjamin (committee member)

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Abstract (if available)

Abstract Quantum computers need to be protected by quantum error-correcting codes against decoherence. One of the most interesting and useful classes of quantum codes is the class of quantum stabilizer codes. Entanglement-assisted (EA) quantum codes are a class of stabilizer codes that make use of preshared entanglement between the sender and the receiver. We provide several code constructions for entanglement-assisted quantum codes. ❧ The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We find new constraints on the simplified stabilizer group and the logical group, which help improve the linear programming bounds on entanglement-assisted quantum codes. The results also can be applied to standard stabilizer codes. ❧ In the real world, quantum gates are faulty. To implement quantum computation fault-tolerantly, quantum codes with certain properties are needed. We first analyze Knill’s postselection scheme in a two-dimensional architecture. The error performance of this scheme is better than other known concatenated codes. Then we propose several methods to protect syndrome extraction against measurement errors.