University of Southern California Dissertations and Theses

Entanglement-assisted coding theory

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Entanglement-assisted coding theory

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Content ENTANGLEMENT-ASSISTED CODING THEORY. by Min-Hsiu Hsieh A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2008 Copyright 2008 Min-Hsiu Hsieh Table of Contents List of Figures iv List of Tables v Chapter 1: Overview 1 Chapter 2: Background knowledge 7 2.1 Single qubit Pauli group . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Multi-qubit Pauli group . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Properties of the symplectic form . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Symplectic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Classical quaternary codes . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Encoding classical information into quantum states . . . . . . . . . . . . 21 2.6.1 Elementary coding . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.1 Superdense coding . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 3: Standard quantum error-correcting codes 26 3.1 Discretization of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Canonical codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Relation to symplectic codes . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 The CSS construction . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5.1 The [[9;1;3]] Shor code. . . . . . . . . . . . . . . . . . . . . . . . 34 3.5.2 The [[7;1;3]] Steane code . . . . . . . . . . . . . . . . . . . . . . 35 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4: Entanglement-assisted quantum error-correcting codes 38 4.1 The channel model: discretization of errors . . . . . . . . . . . . . . . . 39 4.2 The entanglement-assisted canonical code . . . . . . . . . . . . . . . . . 39 4.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Generalized construction from quaternary codes. . . . . . . . . . . . . . 45 4.5 Bounds on performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ii Chapter 5: Operator quantum error-correcting codes 52 5.1 The canonical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 6: Entanglement-assisted operator quantum error-correcting codes 59 6.1 The canonical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.3 Properties of EAOQECCs . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.4.1 EAOQECC from EAQECC . . . . . . . . . . . . . . . . . . . . . 66 6.4.2 EAOQECCs from classical BCH codes . . . . . . . . . . . . . . . 67 6.4.3 EAOQECCs from classical quaternary codes. . . . . . . . . . . . 71 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 7: Quantum quasi-cyclic low-density parity-check codes 75 7.1 Classical low-density parity-check codes . . . . . . . . . . . . . . . . . . 75 7.1.1 Properties of binary circulant matrices . . . . . . . . . . . . . . . 76 7.1.2 Classical quasi-cyclic LDPC codes . . . . . . . . . . . . . . . . . 78 7.1.3 Iterative decoding algorithm . . . . . . . . . . . . . . . . . . . . . 82 7.2 Quantum low-density parity-check codes . . . . . . . . . . . . . . . . . . 86 7.2.1 Quantum quasi-cyclic LDPC codes . . . . . . . . . . . . . . . . . 86 7.3 Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Bibliography 94 iii List of Figures 1 A canonical quantum error-correcting code. . . . . . . . . . . . . . . . 29 2 A standard quantum error-correcting code. . . . . . . . . . . . . . . . 31 3 A generic entanglement assisted quantum code. . . . . . . . . . . . . . 39 4 The entanglement-assisted canonical code. . . . . . . . . . . . . . . . . 40 5 Generalizing the entanglement-assisted canonical code construction. . 44 6 The operator canonical code. . . . . . . . . . . . . . . . . . . . . . . . 53 7 The operator quantum error-correcting code. . . . . . . . . . . . . . . 56 8 The entanglement-assisted operator canonical code. . . . . . . . . . . . 60 9 The entanglement-assisted operator quantum error-correcting code. . . 63 10 Performance of QLDPC with SPA decoding, and 100-iteration . . . . . 91 iv List of Tables 1 The [[9,1,3]] Shor code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 The [[7,1,3]] Steane code. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Highest achievable minimal distance d in any [[n;k;d;c]] EAQECCs. . . 48 4 Summary of error-correcting criteria. . . . . . . . . . . . . . . . . . . . . 66 5 The original [[8,1,3;c=1]] EAQECC encodes one qubit into eight phys- ical qubits with the help of one ebit. . . . . . . . . . . . . . . . . . . . . 67 6 The resulting [[8,1,3;c = 1,r = 2]] EAOQECC encodes one qubit into eight physical qubits with the help of one ebit, and create two gauge qubits for passive error correction. . . . . . . . . . . . . . . . . . . . . . 68 7 Parameters of the EAOQECCs constructed from a classical [63,39,9] BCH code, where r represents the amount of gauge qubits created and c represents the amount of ebits needed. . . . . . . . . . . . . . . . . . . 71 8 Stabilizer generators of the [[15;9;4;c=4]] EAQECC derived from the classical code given by Eq. (60). The size ofS E is equal to 2 2c . . . . . . 72 9 Stabilizer generators of the [[15;9;3;c = 3;r = 1]] EAOQECC derived from the EAQECC given by Table 8. The size ofS E andS G is equal to 2 2c and 2 2r , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 v Chapter 1: Overview The theory of quantum mechanics, founded in the early 1920s, ended the turmoil caused by the classical physics that predicted various absurd results such as electrons spiraling inexorably into the atom nucleus. Though the mathematical framework of quantum mechanics is simple, even geniuses like Albert Einstein found it counter- intuitive. Generations of physicists since put a lot of e®ort to sharpen our intuition about quantum mechanics, and make it more transparent to normal human minds. Severalfundamentalresultsdiscoveredlateron,suchasthefamousno-cloning theorem [58]thatdeniesthepossibilityofusingquantume®ectstosignalfasterthanlight, help us better understand quantum mechanics. Research on quantum mechanics evolved into a interdisciplinary science due to several successful applications of quantum e®ects on classical computation and com- munication problems in 1990s. Among them, Shor proposed a quantum algorithm for the enormously important problem [53] | the problem of ¯nding the prime factors of an integer | showing exponential speed-up over the best known classical algorithm. This result not only attracted broad interest because this problem is believed to have no e±cient solution on classical computers, but also provided strong evidence that quantum computers are more powerful than classical computers. However, the power of quantum computation and communication over classical computation and communication comes from implementing entangled quantum states 1 that are easily spoilt by their vulnerability to errors. Namely, the destructive inter- ference of the omnipresent environment leads to an exponential loss of the probability thatthecomputationrunsinthedesiredway. Uptothatpoint,therewasawidespread belief that decoherence | environmental noise | would doom any chance of building large scale quantum computers or quantum communication protocols. The equally widespread belief that any analogue of classical error correction was impossible in quantum mechanics due to the famous no cloning theorem produced an even stronger pessimistic atmosphere in developing quantum computers. Luckily, the pessimistic atmosphere did not last long. One of the most important discoveries in quantum information science, the existence of quantum error-correcting codes (QECCs), de¯ed those expectations. The ¯rst quantum error-correcting code, considered as a quantum analogue of the classical repetition code, was proposed by Shor in 1995 [52]. The theory of quantum error correction quickly became a popular research topic. The quantum error-correcting conditions were proved independently by Bennett, DiVincenzo, Smolin and Wootters [5], and by Knill and La°amme [34]. The best quantum code that encodes one-qubit information, the ¯ve-qubit code, was discovered by La°amme, Miquel, Paz, and Zurek [39], and independently by [5]. The development of quantum error-correcting theory then became systematic. A construction of Calderbank, Shor, and Steane [16, 55] showed that it was possible to constructquantumcodesfromclassicallinearcodes|theCSScodes|therebydraw- ing on the well-studied theory of classical error correction. Furthermore, Gottesman invented the stabilizer formalism [28], and used it to de¯ne stabilizer codes. In this view, quantum error-correcting codes are simultaneous eigenspaces of a group of com- muting operators, the stabilizer. Independently, Calderbank, Rain, Shor, and Sloane [14] proposed a similar idea to de¯ne quantum codes based on orthogonal geometry in classical coding theory. This result connected quantum codes to classical quaternary 2 codes [15]. The theory of quantum error correction developed so far is called standard quantum error correction. Important as these results were, they fell short of doing everything that one might wish. The connection between classical codes and quantum codes was not universal. Rather, only classical codes which satis¯ed a self-orthogonality constraint could be used to construct quantum codes. While this constraint was not too di±cult to satisfy forrelativelysmallcodes,itisasubstantialbarriertotheuseofhighlye±cientmodern codes, such as Turbo and Low-Density Parity Check (LDPC) codes, in quantum in- formation theory. These codes are capable of achieving the classical capacity; but the di±culty of constructing self-orthogonal versions of them has made progress toward ¯nding quantum versions very slow. These problems can be overcome with pre-existing entanglement. Entanglement plays a central role in almost every quantum computation and communication task. It enables the teleportation of quantum states without physically sending quantum systems[4];itdoublesthecapacityofquantumchannelsforsendingclassicalinformation[6]; itisknowntobenecessaryforthepowerofquantumcomputation[8,31]. Furthermore, descriptionsinquantuminformationtheoryareoftensimpli¯edbytheassumptionthat pre-existing entanglement is available. In the thesis, we show how shared entanglement provides a simpler and more fun- damental theory of quantum error correction, and at the same time greatly generalize the existing theory of quantum error correction. If the CSS construction for quantum codes is applied to a classical code which is not self-orthogonal, the resulting \stabi- lizer" group is not commuting, and thus has no code space. We are able to avoid this problem by making use of pre-existing entanglement. This noncommuting stabilizer groupcanbeembeddedinalargerspace, whichmakesthegroupcommute, andallows a code space to be de¯ned. Moreover, this construction can be applied to any classical 3 quaternary code, not just self-orthogonal ones. The existing theory of quantum error- correcting codes thus becomes a special case of our theory: self-orthogonal classical codes give rise to standard quantum codes, while non-self-orthogonal classical codes give rise to entanglement-assisted codes. Besides the entanglement-assisted formalism [13, 12] we proposed in this thesis, there has been one other major breakthrough in quantum error correction theory: the discovery of operator quantum error-correcting codes (OQECCs) [1, 2, 3, 33, 37, 38, 48, 50], or subsystem codes. Instead of encoding quantum information into a subspace, OQECCs encode quantum information into a subsystem of the subspace. These provide a general theory which combines passive error-avoiding schemes, such as decoherence-free subspaces [61, 40] and noiseless subsystems [35, 32, 59, 60], with conventional(active)quantumerrorcorrection. OQECCsdonotleadtonewcodes,but insteadprovideanewkindofdecodingprocedure: itisnotnecessarytoactivelycorrect all errors, but rather only to perform correction modulo the subsystem structure. One potential bene¯t of the new decoding procedure is to improve the threshold of fault- tolerant quantum computation [2]. The other major contribution of this thesis is the development of the unifying formalism that uni¯es these two extensions of standard QECCs: the operator quan- tum error-correcting codes (OQECCs), and the entanglement-assisted quantum error- correcting codes (EAQECCs). Furthermore, our formalism retains the advantages of both entanglement-assisted and operator quantum error correction. On one hand, OQECCsprovideageneraltheorywhichcombinespassiveerror-avoidingschemeswith standard quantum error correction. On the other hand, EAQECCs provide a general theory which links any classical quaternary code, not just self-orthogonal ones, to a quantum code. In addition to presenting our formal theory, we have given several examples of code construction. These examples demonstrate that our formalism can be used to construct quantum codes tailored to the needs of particular applications. 4 Because classical LDPC codes have such high performance | approaching the channel capacity in the limit of large block size | there has been considerable interest in ¯nding quantum versions of these codes. However, quantum low-density parity- check codes [30, 44, 17, 20] are far less studied than their classical counterparts. The main obstacle comes from the dual-containing constraint of the classical codes that are used to construct the corresponding quantum codes. The second obstacle comes fromthebadperformanceoftheiterativedecodingalgorithmsuchasthefamoussum- product algorithm (SPA). Though the SPA decoding can be directly used to decode the quantum errors, its performance is severely limited by the many 4-cycles, which are usually the by-product of the dual-containing property, in the standard quantum LDPC codes [44]. In the last part of the thesis, we will show that, with the entanglement-assisted formalism [13, 12], these two obstacles of standard quantum LDPC codes can be over- come. By allowing the use of pre-shared entanglement between senders and receivers, the dual-containing constraint can be removed. Constructing quantum LDPC codes from classical LDPC codes becomes transparent. That is, arbitrary classical quater- nary codes can be used to construct quantum codes via the generalized CSS construc- tion [13]. Furthermore, we can easily construct quantum LDPC codes from classical LDPC codes with girth at least 6. We make use of classical quasi-cyclic LDPC codes in our construction, and show that given similar net yield these quantum LDPC codes perform better than the standard quantum LDPC codes by numerical simulation. Thisthesisisorganizedasfollows. Wegivevariousbackgroundmaterialsinchapter 2. In chapter 3, we introduce standard QECCs using the canonical code method and stabilizerformalism. Inchapter4,wepresentour¯rstresult: theentanglement-assisted formalism. In chapter 5, we introduce operator quantum error-correcting codes. In chapter6,wepresentoursecondresult: theentanglement-assistedoperatorformalism. Finally,weshowhowtousetheentanglement-assistedformalismtoconstructquantum 5 LDPC codes with better performance in chapter 7. Notice that we explicitly assume a communicationscenariothroughoutthethesis. Thatis,noiseismodeledasaquantum channel, and it only happens in the channel. Two parties involved in the information processing are called sender and receiver, respectively, and their operations on the quantum states are assumed to be noise-free. 6 Chapter 2: Background knowledge 2.1 Single qubit Pauli group The set of Pauli matrices over a two-dimensional Hilbert spaceH 2 is de¯ned as I = 2 6 4 1 0 0 1 3 7 5; X = 2 6 4 0 1 1 0 3 7 5; Y = 2 6 4 0 ¡i i 0 3 7 5; Z = 2 6 4 1 0 0 ¡1 3 7 5: The Pauli matrices are Hermitian unitary matrices with eigenvalues belonging to the setf1;¡1g. The multiplication table of these matrices is given by: £ I X Y Z I I X Y Z X X I iZ ¡iY Y Y ¡iZ I iX Z Z iY ¡iX I Observe that the Pauli matrices either commute or anticommute. Let [S]=f¯Sj¯2 C;j¯j = 1g be the equivalence class of matrices equal to S up to a phase factor. ¤ Let G be the group generated by the set of Pauli matrices fI;X;Y;Zg with all possible ¤ It makes good physical sense to neglect this overall phase, which has no observable consequence. 7 phases, then the set [G] = f[I];[X];[Y];[Z]g is readily seen to form a commutative group under the multiplication operation de¯ned by [S][T] = [ST]. We called [G] the Pauli group. WeareinterestedinrelatingthePauligrouptotheadditivegroup(Z 2 ) 2 =f00;01;10;11g of binary words of length 2 described by the table: + 00 01 11 10 00 00 01 11 10 01 01 00 10 11 11 11 10 00 01 10 10 11 01 00 This group is also a two-dimensional vector space over the ¯eld Z 2 . A bilinear form canbede¯nedoverthisvectorspace,calledthesymplectic form orsymplectic product y ¯:(Z 2 ) 2 £(Z 2 ) 2 !Z 2 , given by the table ¯ 00 01 11 10 00 0 0 0 0 01 0 0 1 1 11 0 1 0 1 10 0 1 1 0 In what follows we will often write elements of (Z 2 ) 2 as u=(zjx), with z;x2Z 2 . For instance, 01 becomes (0j1). For u = (zjx);v = (z 0 jx 0 )2 (Z 2 ) 2 the symplectic product is equivalently de¯ned by u¯v =zx 0 +z 0 x: y Strictly speaking it is not an inner product. 8 De¯ne the map N :(Z 2 ) 2 !G by the following table: (Z 2 ) 2 G 00 I 01 X 11 Y 10 Z Thismapisde¯nedinsuchawaythatN (zjx) andZ z X x areequaluptoaphasefactor, i.e. [N (zjx) ]=[Z z X x ]: We make two key observations (1). The map [N]:(Z 2 ) 2 ![G] induced by N is an isomorphism: [N u ][N v ]=[N u+v ]: (2). The commutation relations of the Pauli matrices are captured by the symplectic product N u N v =(¡1) u¯v N v N u : Both properties are readily veri¯ed from the tables. 2.2 Multi-qubit Pauli group Consider an n-qubit system corresponding to the tensor product Hilbert space H n 2 . De¯ne an n-qubit Pauli matrix S to be of the form S = S 1 S 2 ¢¢¢S n , where 9 S j 2G. LetG n be the group of all 4 n n-qubit Pauli matrices with all possible phases. De¯ne as before the equivalence class [S]=f¯Sj¯2C;j¯j=1g. Then [S][T]=[S 1 T 1 ][S 2 T 2 ]¢¢¢[S n T n ]=[ST]: Thustheset[G n ]=f[S]:S2G n gisacommutativemultiplicativegroup, andiscalled the n-fold Pauli Group. Now consider the group/vector space (Z 2 ) 2n of binary vectors of length 2n. Its elements may be written as u = (zjx), z = z 1 :::z n 2 (Z 2 ) n , x = x 1 :::x n 2 (Z 2 ) n . We shall think ofu,z andx as row vectors. The symplectic product ofu=(zjx) and v =(z 0 jx 0 ) is given by u¯v T =zx 0 T +z 0 x T : The right hand side are binary inner products and the superscript T denotes the transpose. This should be thought of as a kind of matrix multiplication of a row vector and a column vector. We use u¯v T rather than the more standard uv T to emphasize that the symplectic form is used rather than the binary inner product. Equivalently, u¯v T = X i u i ¯v i where u i = (z i jx i );v i = (z 0 i jx 0 i ) and this sum represents Boolean addition. Observe that if u¯v T = 0, these two vectors are \orthogonal" to each other with respect to the symplectic inner product. The map N :(Z 2 ) 2n !G n is now de¯ned as N u =N u 1 ¢¢¢N u n : Writing X x =X x 1 ¢¢¢X x n ; 10 Z z =Z z 1 ¢¢¢Z z n ; as in the single qubit case, we have [N (zjx) ]=[Z z X x ]: The two observations made for the single qubit case also hold: (1). The map [N]:(Z 2 ) 2n ![G n ] induced by N is an isomorphism: [N u ][N v ]=[N u+v ]: (1) Consequently, iffu 1 ;:::;u m g is a linearly independent set then the elements of the Pauli group subset f[N u 1 ];:::;[N um ]g are independent in the sense that no element can be written as a product of others. (2). The commutation relations of the n-qubit Pauli matrices are captured by the symplectic product N u N v =(¡1) u¯v T N v N u : (2) We de¯ne the weight of a Pauli operator N u , wt(N u ), to be the number of single- qubit Pauli matrices in N u not equal to the identity I. De¯ne the weight of a vector u = (zjx)2 (Z 2 ) 2n by wt sp (u) = wt 2 (z_x). Here_ denotes the bitwise logical \or", and wt 2 (y) is the number of non-zero bits in y2(Z 2 ) n . It is easy to verify that wt(N u )=wt sp (u): 11 2.3 Properties of the symplectic form A subspace V of (Z 2 ) 2n is called symplectic [18] if there is no v2V such that v¯u T =0; 8u2V: (3) (Z 2 ) 2n isitselfasymplecticsubspace. Considerthestandardbasisfor(Z 2 ) 2n ,consisting of g i = (e i j0) and h i = (0je i ) for i = 1;:::;n, where e i = (0;:::;0;1;0;:::;0) [1 in the ith position] are the standard basis vectors of (Z 2 ) n . Observe that g i ¯g T j =0; for all i;j (4) h i ¯h T j =0; for all i;j (5) g i ¯h T j =0; for all i6=j (6) g i ¯h T i =1; for all i: (7) Thus,thebasisvectorscomeinnhyperbolicpairs (g i ;h i )suchthatonlythesymplectic product between hyperbolic partners is nonzero. The matrix J = [g i ¯h T j ] de¯ning the symplectic product with respect to this basis is given by J = 0 B @ 0 n£n I n£n I n£n 0 n£n 1 C A; (8) whereI n£n and0 n£n arethen£nidentityandzeromatrices, respectively. Abasisfor (Z 2 ) 2n whose symplectic product matrix J is given by (8) is called a symplectic basis. In the Pauli picture, the hyperbolic pairs (g i ;h i ) correspond to (Z e i ;X e i ), and are sometimes expressed as (Z i ;X i ), { the anticommuting Z and X Pauli matrices acting on the ith qubit. 12 Incontrast,asubspaceV of(Z 2 ) 2n iscalledisotropic if(3)holdsforall v2V. The largest isotropic subspace of (Z 2 ) 2n is n-dimensional. The span of theg i , i=1;:::;n, is an example of a subspace saturating this bound. A general subspace of (Z 2 ) 2n is neither symplectic nor isotropic. The following theorem,statedin[18]andrediscoveredinPaulilanguagein[24],saysthatanarbitrary subspace V can be decomposed as a direct sum of a symplectic part and an isotropic part. Here, weprovethistheoremconstructively, usingaversionoftheGram-Schmidt procedure. Theorem 1. Let V be an m-dimensional subspace of (Z 2 ) 2n . Then there exists a symplectic basis of (Z 2 ) 2n consisting of hyperbolic pairs (u i ;v i ), i=1;:::;n, such that fu 1 ;:::;u c+` ;v 1 ;:::;v c g is a basis for V, for some c;`¸0 with 2c+`=m. Equivalently, V =symp(V)©iso(V) wheresymp(V)=spanfu 1 ;:::;u c ;v 1 ;:::;v c gissymplecticandiso(V)=spanfu c+1 ;:::;u c+` g is isotropic. Proof. Pickanarbitrarybasisfw 1 ;:::;w m gforV andextendittoabasisfw 1 ;:::;w 2n g for (Z 2 ) 2n . The procedure consists of n rounds. In each round a new hyperbolic pair (u i ;v i ) is generated; the index i is added to the set U (respectively, V) if u i 2 V (v i 2V). Initially set i=1, m 0 =m, andU =V =;. The ith round reads as follows. (1). We start with vectors w 1 ;:::;w 2(n¡i+1) , and u 1 ;:::u i¡1 ;v 1 ;:::v i¡1 , such that (a) w 1 ;:::;w 2(n¡i+1) ; u 1 ;:::u i¡1 ;v 1 ;:::v i¡1 is a basis for (Z 2 ) 2n , (b) each ofu 1 ;:::u i¡1 ;v 1 ;:::v i¡1 has vanishing symplectic product with each of w 1 ;:::;w 2(n¡i+1) , (c) V =spanfw j :1·j·m 0 g©spanfu j :j2Ug©spanfv j :j2Vg. 13 Theseconditionsaresatis¯edfor i=1wherewebeginwithvectorsw 1 ;:::;w 2n . In this case, we implicitly assume that (u 0 , v 0 ) is the empty set. (2). De¯ne u i =w 1 . If m 0 ¸1 then and add i toU. Let j¸2 be the smallest index for which w 1 ¯w T j = 1. Such a j exists because of (a), (b) and the fact that there exists a w2(Z 2 ) 2n such that u i ¯w T =1. Set v i =w j . (3). If j·m 0 : This means that there is a hyperbolic partner of u i in V. Add i toV; swap w j with w 2 ; for k =3;:::;2(n¡i+1) perform w 0 k¡2 :=w k ¡(v i ¯w T k )u i ¡(u i ¯w T k )v i ; so that w 0 k¡2 ¯u T i =w 0 k¡2 ¯v T i =0; (9) set m 0 :=m 0 ¡2. If j >m 0 : This means that there is no hyperbolic partner of u i in V. Swap w j with w 2(n¡i+1) ; for k =2;:::;2(n¡i)+1 perform w 0 k¡1 :=w k ¡(v i ¯w T k )u i ¡(u i ¯w T k )v i ; so that w 0 k¡1 ¯u T i =w 0 k¡1 ¯v T i =0; (10) if m 0 ¸1 then set m 0 :=m 0 ¡1. 14 (4). Let w k := w 0 k for 1 · k · 2(n¡ i). We need to show that the conditions from item 1 are satis¯ed for the next round (i := i+1). Condition (a) holds because fu i ;v i ;w 0 1 ;:::w 0 2(n¡i) g are related to the old fw 1 ;:::w 2(n¡i+1) g by an invertible linear transformation. Condition (b) follows from (9) and (10). Re- garding condition (c), if m 0 = 0 then it holds because U and V did not change from the previous round. Otherwise, consider the two cases in item 3. If j·m 0 then fu i ;v i ;w 0 1 ;:::w 0 m 0 ¡2 g are related to the old fw 1 ;:::w m 0g by an invert- ible linear transformation. If j > m 0 then fu i ;w 0 1 ;:::w 0 m 0 ¡1 g are related to the oldfw 1 ;:::w m 0g by an invertible linear transformation (the (u i ¯w T k )v i terms vanish for 1·k·m 0 because there is no hyperbolic partner of u i in V). At the end of the ith round, 0·m 0 ·2(n¡i). Thus m 0 =0 after n rounds and hence V =spanfu j :j2Ug©spanfv j :j2Vg. The theorem follows by suitably reordering the (u j ;v j ). Remark It is readily seen that the space iso(V) is unique, given V. In contrast, symp(V) is not. For instance, replacing v 1 by v 0 1 =v 1 +u c+1 in the above de¯nition of symp(V) does not change its symplectic property. Asymplectomorphism ¨:(Z 2 ) 2n !(Z 2 ) 2n isalinearisomorphismwhichpreserves the symplectic form, namely ¨(u)¯¨(v) T =u¯v T : (11) Thefollowingtheoremrelatessymplectomorphismson(Z 2 ) 2n tounitarymapsonH n 2 . It appears, for instance, in [11]. For completeness, we give an independent proof here. 15 Theorem 2. For any symplectomorphism ¨ on (Z 2 ) 2n there exists a unitary map U ¨ on H n 2 such that for all u2(Z 2 ) 2n , [N ¨(u) ]=[U ¨ N u U ¡1 ¨ ]: Remark. The unitary map U ¨ may be viewed as a map on [G n ] given by [S] 7! [U ¨ SU ¡1 ¨ ]. The theorem says that the following diagram commutes (Z 2 ) 2n ¨ ¡ ¡¡¡ ! (Z 2 ) 2n [N] ? ? y ? ? y [N] [G n ] U ¨ ¡ ¡¡¡ ! [G n ] Proof. Considerthestandardbasisg i =(e i j0),h i =(0je i ). De¯netheunique(uptoa phase factor) statej0i onH n 2 to be the simultaneous +1 eigenstate of the commuting operators N g j , j = 1;:::;n. De¯ne an orthonormal basisfjbi :b =b 1 :::b n 2 (Z 2 ) n g forH n 2 by jbi=N P i b i h i j0i: The orthonormality follows from the observation thatjbi is a simultaneous eigenstate of N g j , j =1;:::;n with respective eigenvalues (¡1) b j : N g j jbi=N g j N P i b i h i j0i =(¡1) b j N P i b i h i N g j j0i =(¡1) b j N P i b i h i j0i =(¡1) b j jbi: (12) The second line is an application of (2). 16 De¯ne ~ g i := ¨(g i ). We repeat the above construction for this new basis. De¯ne the unique (up to a phase factor) statej ~ 0i to be the simultaneous +1 eigenstate of the commuting operators N ~ g i , i=1;:::;n. De¯ne an orthonormal basisfj ~ big by j ~ bi=N P i b i ~ h i j ~ 0i: (13) De¯ning u= P i z i g i +x i h i , ~ u= P i z i ~ g i +x i ~ h i and x=x 1 :::x n , we have N ~ u j ~ bi=N ~ u N P i b i ~ h i j ~ 0i =(¡1) ~ u¯( P i b i ~ h i ) T N P i b i ~ h i N ~ u j ~ 0i =(¡1) ~ u¯( P i b i ~ h i ) T e iµ(~ u) N P i b i ~ h i N P i x i ~ h i N P i z i ~ g i j ~ 0i =(¡1) ~ u¯( P i b i ~ h i ) T e iµ(~ u) N P i (b i +x i ) ~ h i j ~ 0i =(¡1) ~ u¯( P i b i ~ h i ) T e iµ(~ u) j ^ b+xi =(¡1) u¯( P i b i h i ) T e iµ(~ u) j ^ b+xi; (14) where µ(~ u) is a phase factor which is independent ofb. The ¯rst equality follows from (13), the second from (2), the third from (1), the fourth from the de¯nition ofj ~ 0i and the fact that X b X x =X b+x , the ¯fth from (13), and the sixth from (11). Similarly N u jbi=(¡1) u¯( P i b i h i ) T e i'(u) jb+xi; (15) where '(u) is a is a phase factor which is independent of b. De¯ne U ¨ by the change of basis U ¨ = X b j ~ bihbj: 17 Combining (14) and (15) gives for alljbi N ¨(u) U ¨ jbi=(¡1) u¯( P i b i h i ) T e iµ(~ u) U ¨ jb+xi =e i[µ(~ u)¡'(u)] U ¨ N u jbi: (16) Therefore [N ¨(u) ]=[U ¨ N u U ¡1 ¨ ]. 2.4 Symplectic codes An [n;k] symplectic code C sp de¯ned by an (n¡k)£2n parity check matrix H sp is given by C sp =rowspace(H sp ) ? where V ? =fw :w¯u T =0; 8u2Vg: The subscript sp emphasizes that the code is de¯ned with respect to the symplectic product. Note that (V ? ) ? =V. We say that C sp is dual-containing if (C sp ) ? =rowspace(H sp )½C sp ; (17) this is true if H sp is self-orthogonal under the symplectic product. For simplicity, the term \self-orthogonal code" is often referred to a code with a self-orthogonal parity- check matrix. Thenotionofdistance providesaconvenientwaytocharacterizetheerror-correcting properties of a code. An [n;k] symplectic code C sp with a parity check matrix H sp is said to have distance d if for each nonzero u of weight < d, u62 C sp , or equivalently, H sp ¯u T 6=0 T . 18 2.5 Classical quaternary codes Following the presentation of Forney et al. [25], the addition table of the additive group of the quaternary ¯eldF 4 =f0;1;!;!g is given by + 0 ! 1 ! 0 0 ! 1 ! ! ! 0 ! 1 1 1 ! 0 ! ! ! 1 ! 0 Comparing the above to the addition table of (Z 2 ) 2 establishes the isomorphism ° : F 4 !(Z 2 ) 2 , given by the table F 4 (Z 2 ) 2 0 00 ! 01 1 11 ! 10 The multiplication table forF 4 is de¯ned as £ 0 ! 1 ! 0 0 0 0 0 ! 0 ! ! 1 1 0 ! 1 ! ! 0 1 ! ! De¯ne the traces (Tr) of the elementsf0;1;!;!g ofF 4 asf0;0;1;1g, and their conju- gates (\ y ") as f0;1;!;!g. Intuitively, Tra measures the \!-ness" of a2F 4 . Observe that a=0 if and only if both Tr!a=0 and Tr!a=0. The Hermitian inner product 19 of two elements a;b2F 4 is de¯ned as ha;bi = a y b2F 4 . The trace product is de¯ned as Trha;bi2F 2 . The trace product table is readily found to be Trh;i 0 ! 1 ! 0 0 0 0 0 ! 0 0 1 1 1 0 1 0 1 ! 0 1 1 0 Comparing the above to the¯ table of (Z 2 ) 2 establishes the identity Trha;bi=°(a)¯°(b): These notions can be generalized to n-dimensional vector spaces over F 4 . Thus, for a;b2(F 4 ) n , Trha;bi=°(a)¯°(b) T ; (18) where the Hermitian inner product over (F 4 ) n is de¯ned by the componentwise sum ha;bi= P i a y b: Let wt 4 (a) be the number of non-zero bits ina2(F 4 ) n , then we have another identity wt sp (°(a))=wt 4 (a); (19) where °(a)2(Z 2 ) 2n . An [n;k] code C 4 (the subscript 4 emphasizes that the code is over F 4 ) with the parity check matrix H 4 is said to have distance d if for each vector a 2 (F 4 ) n with wt 4 (a)<d, a62C 4 , or equivalently,hH 4 ;ai6=0 T . Proposition 1. Given an [n;k;d] code C 4 with parity check matrix H 4 , there exists a corresponding [n;2k¡n;d] symplectic code C sp . 20 Proof. Consider a classical [n;k;d] 4 code with an (n¡k)£n quaternary parity check matrix H 4 . By de¯nition, for each nonzero a2(F 4 ) n such that wt 4 (a)<d, hH 4 ;ai6=0 T : This is equivalent to the logical statement Trh!H 4 ;ai6=0 T _ Trh¹ !H 4 ;ai6=0 T : This is further equivalent to Trh ~ H 4 ;ai6=0 T ; where ~ H = 0 B @ !H 4 ¹ !H 4 1 C A: (20) De¯ne the (2n¡2k)£2n symplectic matrix H sp = °( ~ H 4 ). By the correspondences (18) and (19), H sp ¯u T 6=0 T ; holds for each nonzero u 2 (Z 2 ) 2n with wt(u) < d. Thus C sp is an [n;2k¡ n;d] symplectic code de¯ned by H sp . 2.6 Encoding classical information into quantum states In this section we review two schemes for sending classical information over quantum channels: elementary coding and superdense coding. These will be used later in the contextofquantumerrorcorrection toconveyinformationtothe decoderabout which error happened. 21 2.6.1 Elementary coding In the ¯rst scheme, Alice and Bob are connected by a perfect qubit channel. Alice can send an arbitrary bit a2Z 2 over the qubit channel in the following way: ² Alice locally prepares a state j0i in H 2 . This state is the +1 eigenstate of the Z operator. Based on her message a, she performs the encoding operation X a , producing the statejai=X a j0i. ² Alice sends the encoded state to Bob through the qubit channel. ² BobdecodesbyperformingthevonNeumannmeasurementinthefj0i;j1igbasis. As this is the unique eigenbasis of the Z operator, this is equivalently called \measuring the Z observable". We call this protocol \elementary coding" and write it symbolically as a resource inequality [22, 21, 23] z [q!q]¸[c!c]: Here[q!q]representsaperfectqubitchanneland[c!c]representsaperfectclassical bit channel. The inequality ¸ signi¯es that the resource on the left hand side can be used in a protocol to simulate the resource on the right hand side. Elementary coding immediately extends to m qubits. Alice prepares the simul- taneous +1 eigenstate of the Z e 1 ;:::;Z e m operators j0i, and encodes the message a 2 (Z 2 ) m by applying X a , producing the encoded state jai = X a j0i. Bob decodes by simultaneously measuring the Z e 1 ;:::;Z em observables. We could symbolically represent this protocol by m[q!q]¸m[c!c]: z In [21] resource inequalities were used in the asymptotic sense. Here they refer to ¯nite protocols, and are thus slightly abusing their original intent. 22 2.6.2 Superdense coding In the second scheme, Alice and Bob share the ebit state j©i= 1 p 2 (j0ij0i+j1ij1i) (21) in addition to being connected by the qubit channel. In (21) Alice's state is to the left and Bob's is to the right of the symbol. The statej©i is the simultaneous (+1;+1) eigenstate of the commuting operators ZZ and XX. Again, the operator to the left of the symbol acts on Alice's system and the operator to the right of the symbol acts on Bob's system. Alice can send a two-bit message (a 1 ;a 2 )2(Z 2 ) 2 to Bob using \superdense coding" [6]: ² Basedonhermessage(a 1 ;a 2 ), Aliceperformstheencodingoperation Z a 1 X a 2 on her part of the statej©i, producing the stateja 1 ;a 2 i=(Z a 1 X a 2 I B )j©i. ² Alice sends her part of the encoded state to Bob through the perfect qubit chan- nel. ² Bob decodes by performing the von Neumann measurement in the f(Z a 1 X a 2 I)j©i : (a 1 ;a 2 )2 (Z 2 ) 2 g basis, i.e., by simultaneously measuring the ZZ and XX observables. The protocol is represented by the resource inequality [q!q]+[qq]¸2[c!c]; (22) where [qq] now represents the shared ebit. It can also be extended to m copies. Alice and Bob share the state j©i m which is the simultaneous +1 eigenstate of the Z e 1 Z e 1 ;:::;Z em Z em andX e 1 X e 1 ;:::;X em X em operators. Aliceencodesthe message(a 1 ;a 2 )2(Z 2 ) 2m byapplyingZ a 1 X a 2 ,producingtheencodedstateja 1 ;a 2 i= 23 (Z a 1 X a 2 I)j©i. Bob decodes by simultaneously measuring the Z e 1 Z e 1 ;:::;Z em Z e m andX e 1 X e 1 ;:::;X e m X e m observables. Thecorrespondingresourceinequality is m[q!q]+m[qq]¸2m[c!c]: Superdense coding provides the simplest illustration of how entanglement can increase the power of information processing. 2.7 Useful lemmas Lemma 1. LetV be an arbitrary subgroup ofG n with size 2 m . Then there exists a set of generators f ¹ Z 1 ;¢¢¢ ; ¹ Z p+q ; ¹ X p+1 ;¢¢¢ ; ¹ X p+q g that generates V such that the ¹ Z's and ¹ X's obey the same commutation relations as in (23), for some p;q¸0 and p+2q =m. [ ¹ Z i ; ¹ Z j ]=0 8i;j [ ¹ X i ; ¹ X j ]=0 8i;j [ ¹ X i ; ¹ Z j ]=0 8i6=j f ¹ X i ; ¹ Z i g=0 8i: (23) Proof. Though the proof can be found in [24]; however, a new proof can be easily obtained by combining Theorem 1 and the isomorphic map [N]:(Z 2 ) 2n ![G n ]. The following lemma is a simply result from group theory, and a new proof can be obtained from Theorem 2 and [N]:(Z 2 ) 2n ![G n ]. Lemma 2. If there is a one-to-one map between V and S which preserves their com- mutation relations, which we denote V » S, then there exists a unitary U such that for each V i 2V, there is a corresponding S i 2S such that V i =US i U ¡1 , up to a phase which can di®er for each generator. 24 Lemma 3. If C 0 is a simultaneous eigenspace of Pauli operators from the set S 0 0 , then C = U ¡1 (C 0 ) is a simultaneous eigenspace of Pauli operators from the set S = fU ¡1 AU :A2S 0 0 g. Proof. Observe that if AjÃi=®jÃi; then (U ¡1 AU)U ¡1 jÃi=®U ¡1 jÃi: Lemma 4. Performing U followed by measuring the operator A is equivalent to mea- suring the operator U ¡1 AU followed by performing U. Proof. Let ¦ i be a projector onto the eigenspace corresponding to eigenvalue ¸ i ofA. Performing U followed by measuring the operator A is equivalent to the instrument (generalizedmeasurement)givenbythesetofoperatorsf¦ i Ug. TheoperatorU ¡1 AU has the same eigenvalues as A, and the projector onto the eigenspace corresponding to eigenvalue ¸ i is U ¡1 ¦ i U. Measuring the operator U ¡1 AU followed by performing U is equivalent to the instrumentfU(U ¡1 ¦ i U)g=f¦ i Ug. 25 Chapter 3: Standard quantum error-correcting codes 3.1 Discretization of errors It is well known that for standard quantum error correction (i.e., that unassisted by entanglement) it su±ces to consider errors from the Pauli group (see e.g. [47].) We will review this result here. DenotebyLthespaceoflinearoperatorsde¯nedonthequbitHilbertspaceH 2 . In general, a noisy channel is de¯ned by a completely positive, trace preserving (CPTP) map N : L n ! L n taking n-qubit density operators on Alice's system to density operators on Bob's system. We will often encounter isometric operators U :H n 1 2 ! H n 2 2 . Thecorrespondingsuperoperator,orCPTPmap,ismarkedbyahat ^ U :L n 1 ! L n 2 and de¯ned by ^ U(½)=U½U y : Observe that ^ U is independent of any phases factors multiplying U. Thus, for a Pauli operator N u , ^ N u only depends on the equivalence class [N u ]. Our communication scenario involves two spatially separated parties, Alice and Bob, connected by a noise channel N. Alice wishes to send k qubits perfectly to Bob throughN. An [[n;k]] QECC consists of ² An encoding isometryE = ^ U enc :L k !L n ² A decoding CPTP mapD :L n !L k 26 such that D±N ± ^ U enc =id k ; where id:L!L is the identity map on a single qubit. To make contact with classical error correction it is necessary to discretize the errors introduced by N. This is done in two steps. First, the CPTP map N may be (non-uniquely) written in terms of its Kraus representation N(½)= X i A i ½A y i : Second, each A i may be expanded in the Pauli operators A i = X u2(Z 2 ) 2n ® i;u N u : De¯ne the support of N by supp(N) = fu 2 (Z 2 ) 2n : 9i;® i;u 6= 0g. The following theorem allows us to replace the continuous mapN by the error set S =supp(N). Theorem 3. If D± ^ N u ± ^ U enc =id k for all u2supp(N), then D±N ± ^ U enc =id k . Proof. We may extend the map D to its Stinespring dilation [56] { an isometric map ^ U dec with a larger target Hilbert spaceL n L 0 , such that D(½)=Tr L 0 ^ U dec (½): If for all u2supp(N) and all pure statesjÃi inH n 2 , the following equation holds U dec N u U enc jÃi=jÃijui for some pure statejuihuj onL 0 , then by linearity, we have U dec A i U enc jÃi=jÃijii; 27 with the unnormalized statejii= P u ® i;u jui. Furthermore, ( ^ U dec ±N ± ^ U enc )(jÃihÃj)=U dec Ã X i A i U enc jÃihÃjU y enc A y i ! U y dec =jÃihÃj X i jiihij; (24) where the second subsystem corresponds toL 0 . Tracing out the latter gives (D±N ± ^ U enc )(jÃihÃj)=jÃihÃj; concluding the proof. 3.2 Canonical codes We¯rstintroducethesimplestformofstandardquantumerror-correctingcodes(QECCs), the canonical codes. The canonical codeC 0 is de¯ned by the following trivial encoding operationE 0 = ^ U 0 , where U 0 :j'i7!j0ij'i: (25) In other words, the register containingj0i (of sizes=n¡k qubits) is appended to the registerscontainingj'i(ofsizek qubits). Wecalltheencodedstatein(25)a codeword of C 0 . What errors can this canonical code C 0 correct with such a simple-minded encoding? Proposition2. The encoding given byE 0 and a suitably-de¯ned decoding mapD 0 can correct the error set E 0 =fX a Z b X ®(a) Z ¯(a) :a;b2(Z 2 ) s g; (26) for any ¯xed functions ®;¯ :(Z 2 ) s !(Z 2 ) k . 28 Encoding α β X Z φ X Z a b { } a a α β - - X Z 0 s ⊗ a ' φ Decoding φ φ a Figure 1: A canonical quantum error-correcting code. Proof. TheprotocolisshowninFigure1. Afterapplyinganerror E2E 0 , thechannel output becomes (up to a phase factor): E(j0ij'i)=(X a Z b )j0i(X ®(a) Z ¯(a) )j'i=jaij' 0 i (27) wherejai=X a j0i, andj' 0 i=(X ®(a) Z ¯(a) )j'i. As the vector (a;b) completely speci¯es the error operator E, it is called the error syndrome. However, in order to correct this error, only the reduced syndrome, a, matters. Ine®ect,ahasbeenencodedusingelementarycoding(seesection2.6.1), and the receiver Bob can identify the reduced syndrome by simultaneously measuring the Z e 1 ;¢¢¢ ;Z e s observables. He then performs X ¡®(a) Z ¡¯(a) on the remaining k-qubit systemj' 0 i, returning it to the original statej'i. Since the goal is the transmission of quantum information, no actual measurement is necessary. Instead, Bob can perform the CPTP decoding operationD 0 consisting of the controlled unitary U 0;dec = X a jaihajX ¡®(a) Z ¡¯(a) ; (28) which is constructed based on the reduced syndrome, and is also known as collective measurement, followed by discarding the unwanted systems. 29 We can rephrase the above error-correcting procedure in terms of the stabilizer formalism. Let S 0 =hZ 1 ;¢¢¢ ;Z s i be an Abelian group of size 2 s . Group S 0 is called the stabilizer for C 0 , since every element of S 0 ¯xes the codewords of C 0 . Notice that we have used Z i to represent Z e i here for simplicity. Proposition 3. The QECC C 0 de¯ned by S 0 can correct an error set E 0 if for all E 1 ;E 2 2E 0 , E y 2 E 1 2S 0 S (G n ¡Z(S 0 )), where Z(S) is the normalizer of group S. Proof. Since the vector (a;b) completely speci¯es the error operator E, we consider the following two di®erent cases: ² If two error operators E 1 and E 2 have the same reduced syndrome a, then the error operator E y 2 E 1 gives us the all-zero reduced syndrome. Therefore, E y 2 E 1 2 S 0 . This error E y 2 E 1 has no e®ect on the codeword. ² If two error operators E 1 and E 2 have di®erent reduced syndromes, and let a be the reduced syndrome of E y 2 E 1 , then E y 2 E 1 62Z(S 0 ). This error E y 2 E 1 can be corrected by the decoding operation given in (28). 3.3 The general case Theorem 4. Given an Abelian group S I of size 2 n¡k that does not contain ¡I, there exists an [[n;k]] quantum error-correcting codeC de¯ned by the encoding and decoding pair (E;D) with the following properties: (1). The code C can correct the error set E if for all E 1 ;E 2 2E, E y 2 E 1 2S I S (G n ¡ Z(S I )). (2). The codespace C is a simultaneous eigenspace of the S I . 30 α β X Z X Z a b U U -1 { } a a α β - - X Z Encoding Decoding D Error Set 1 0 U U - = E E 0 s ⊗ φ φ φ E 0 E 0 D U -1 U Figure 2: A standard quantum error-correcting code. (3). To decode, the reduced error syndrome is obtained by simultaneously measuring the observables from S I . Proof. TheprotocolisshowninFigure2. SinceS I hasthesamecommutationrelations withthestabilizerS 0 ofthecanonicalcodeC 0 givenintheprevioussection, byLemma 2, there exists an unitary matrix U such thatS 0 =US I U ¡1 . De¯neE =U ¡1 ±E 0 and D =D 0 ±U, whereE 0 andD 0 are given in (25) and (28), respectively. (1). Let E 0 be the error set that can be corrected byC 0 . Then by Proposition 2, D 0 ±E 0 ±E 0 =id k foranyE 0 2E 0 . LetE=fU ¡1 E 0 U :8E 0 2E 0 g. Itfollowsthat, foranyE2E, D±E±E =id k : Thus,theencodinganddecodingpair(E;D)correctsE. FollowingProposition3, thecorrectableerrorsetEcontainsallE 1 ;E 2 suchthatE y 2 E 1 2S I S (G n ¡Z(S I ). (2). Since C 0 is the simultaneous +1 eigenspace of S 0 , and S I = U ¡1 S 0 U, Lemma 3 guarantees that the codespace C after encoding E is a simultaneous eigenspace ofS I . (3). The decoding operationD 0 involves 31 i. measuring the set of generators ofS 0 , yielding the error syndrome according to the error E 0 . ii. performing a recovering operation E 0 again to undo the error. By Lemma 4, performingD =D 0 ±U is equivalent to measuringS I =U ¡1 S 0 U, followedbyperformingtherecoveringoperationU ¡1 E 0 U ,followedbyU toundo the encoding. We said an [[n;k]] QECC de¯ned by S I to have distance d, if for all operators E 1 and E 2 with weigh <d and E 1 6=E 2 , either i. E y 2 E 1 = 2G n ¡Z(S I ), or ii. E y 2 E 1 2S I . The code is called non-degenerate if the second condition is not invoked. A QECC with distance d can correct up to t-qubit errors, where t =b(d¡1)=2c. Such code is called an [[n;k;d]] QECC. 3.4 Relation to symplectic codes Proposition 4. Consider an [n;k;d] symplectic code C sp de¯ned by H sp . If C sp is dual-containing, then C sp de¯nes a non-degenerate [[n;k;d]] QECC. Proof. SinceH sp isself-orthogonal,thatmeansthegroupS I generatedbytheoperator g i = N r i , where r i is the i-th row of H sp , is an Abelian group with size 2 n¡k . From Theorem 4,S I de¯nes an [[n;k]] QECCC. For all vectors u 1 ;u 2 with weight <t, where t=b(d¡1)=2c, we have H sp ¯(u 1 ¡u 2 )6=0 T ; 32 or, equivalently, N y u 2 N u 1 62G n ¡Z(S I ): ThereforeC is a non-degenerate QECC with distance d. 3.4.1 The CSS construction Proposition 5. Given a dual-containing classical binary codes [n;k;d] C, there exists an [[n;2k¡n;d]] QECC. Proof. Let H be the parity check matrix of C. Since rowspace(H)=C ? ½C =rowspace(H) ? ; therefore H sp = 0 B @ H 0 0 H 1 C A; (29) is dual-containing, and de¯nes an [n;2k¡n] symplectic code C sp . By de¯nition of classical linear codes, for each nonzero a2(Z 2 ) n such that wt(a)<d, hH;ai6=0 T ; Then H sp ¯u6=0 T ; holds for each nonzerou2(Z 2 ) 2n with wt(u)<d. Thus C sp de¯nes a non-degenerate [[n;2k¡n;d]] QECC by Proposition 4. Actually, instead of using the same code C, one can use two codes C 1 and C 2 , such that C 1 ½ C 2 , in the CSS construction [47]. Furthermore, the CSS code have one 33 interesting property that its generators contain all X's and protect against phase °ips and generators contain all Z's and protect against bit °ips. 3.5 Examples 3.5.1 The [[9;1;3]] Shor code The¯rstquantumerror-correctingcodeconstructedbyShor[52]wasaquantumanalog of the classical repetition code, which stores information redundantly by duplicating each bit several times. We list the stabilizer generators for the [[9;1;3]] Shor code in Table 1. It is easy to verify that it can correct arbitrary single-qubit error. S 1 Z Z I I I I I I I S 2 I Z Z I I I I I I S 3 I I I Z Z I I I I S 4 I I I I Z Z I I I S 5 I I I I I I Z Z I S 6 I I I I I I I Z Z S 7 X X X I I I X X X S 8 X X X X X X I I I ¹ Z Z Z Z Z Z Z Z Z Z ¹ X X X X X X X X X X Table 1: The [[9,1,3]] Shor code. 34 3.5.2 The [[7;1;3]] Steane code The second example, the [[7;1;3]] Steane code, is constructed using the CSS construc- tion from dual-containing [7;4;3] Hamming code with the parity check matrix H = 0 B B B B @ 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 C C C C A : (30) We list the stabilizer generators in Table 2. S 1 I I I Z Z Z Z S 2 I Z Z I I Z Z S 3 Z I Z I Z I Z S 4 I I I X X X X S 5 I X X I I X X S 6 X I X I X I X ¹ Z Z Z Z Z Z Z Z ¹ X X X X X X X X Table 2: The [[7,1,3]] Steane code. 3.6 Discussion We have developed a canonical code method together with the stabilizer formalism [14, 29, 47] to introduce the standard quantum error-correcting codes. The canonical code method provides us essential insight into the error-correcting property. First of all, the canonical code is obtained by attaching some ancillas, initially in the j0i state, to the quantum information we want to preserve. Therefore the codewords of 35 the canonical code can be easily described by a set of commuting Pauli Z operators. Theerrorsyndromeofeachcorrectableerrorcanbeseenasclassicalinformationbeing encoded in the canonical code by elementary coding. Therefore, reading out the error syndrome is equivalent to recovering the classical message. Then we can restore the codewords of the canonical code by performing a correction operation based on the measurement outcome since the outcome tells us which error happens. These two steps, reading out the error syndrome and performing correction operation, are called the decoding operation. For a useful QECC, we expect it to be able to correct at least arbitrary t-qubit errors,forsomet¸1. Inthissense,thecanonicalcodeisnotasatisfactoryQECC,but wecantransformthecanonicalcodetoaQECCwithdesirabledistanceproperty. The mapping(encoding)isdonewithsomeunitarythattakesthecodespaceofthecanonical code to the codespace speci¯ed by the stabilizer of the QECC. Essentially, all QECCs developed to date are stabilizer codes. The problem of ¯nding QECCs was reduced to that of constructing dual-containing symplectic codes, or equivalently, classical dual- containing quaternary codes[14]. When binary codes are viewed as quaternary, this amounts to the well known Calderbank-Shor-Steane (CSS) construction [55, 16]. The requirementthatacodecontainsitsdualisaconsequenceoftheneedforacommuting stabilizergroup. Thevirtueofthisapproachisthatwecandirectlyconstructquantum codes from classical codes with a certain property, rather than having to develop a completely new theory of quantum error correction from scratch. Unfortunately, the need for a self-orthogonal parity check matrix presents a substantial obstacle to importing the classical theory in its entirety, especially in the context of modern codes such as low-density parity check (LDPC) codes [44]. In the next chapter, we will show that actually every quaternary (or binary) classi- callinearcode,notjustdual-containingcodes, canbetransformedintoaQECC,given that the encoder Alice and decoder Bob have access to shared entanglement. If the 36 classicalcodesarenotdual-containing,theycorrespondtoasetofstabilizergenerators that do not commute; however, if shared entanglement is an available resource, these generators may be embedded into larger, commuting generators, giving a well-de¯ned code space. We call this the entanglement-assisted stabilizer formalism, and the codes constructed from it are entanglement-assisted QECCs (EAQECCs). 37 Chapter 4: Entanglement-assisted quantum error-correcting codes We consider the following communication scenario depicted in Figure 3. The protocol involves two spatially separated parties, Alice and Bob, and the resources at their disposal are ² anoisychannelde¯nedbyaCPTPmapN :L n !L n takingdensityoperators on Alice's system to density operators on Bob's system; ² the c-ebit statej©i c shared between Alice and Bob. Alice wishes to send k-qubit quantum information perfectly to Bob using the above re- sources. An[[n;k;c]]entanglement-assistedquantumerrorcorrectingcode(EAQECC) consists of ² An encoding isometryE = ^ U enc :L k L c !L n ² A decoding CPTP mapD :L n L c !L k such that D±N ± ^ U enc =id k ; where U enc is the isometry which appends the statej©i c , U enc j'i=j'ij©i c ; 38 Figure 3: A generic entanglement assisted quantum code. and id : L ! L is the identity map on a single qubit. The protocol thus uses up c ebits of entanglement and generates k perfect qubit channels. We represent it by the resource inequality (with a slight abuse of notation [21]) hNi+c[qq]¸k[q!q]: Even though a qubit channel is a strictly stronger resource than its static analogue, an ebit of entanglement, the parameter k¡c isstill a good (albeit pessimistic) measure of thenetnoiselessquantumresourcesgained. Itshouldbeborneinmindthatanegative value of k still refers to a non-trivial protocol. 4.1 The channel model: discretization of errors Again we need to show that we can discretize the errors introduced by N in the entanglement-assistedcommunicationscenario. Thiscanbedoneusingstepsdescribed in Section 3.1. The continuous map N then can be replaced by the error set S = supp(N) by Theorem 3. 4.2 The entanglement-assisted canonical code The entanglement-assisted quantum error-correcting codes (EAQECCs) come from a simple idea: replacing some portion of the ancillas of the canonical codes (25) by the 39 Figure 4: The entanglement-assisted canonical code. maximally entangled states shared between the sender and receiver. We can construct theentanglement-assisted(EA)canonicalcodeC EA 0 withthefollowingtrivialencoding operationE 0 = ^ U 0 de¯ned by U 0 :j'i!j0ij©i c j'i: (31) The operation simply appends ` ancilla qubits in the state j0i, and c copies of j©i (the maximally entangled state shared between sender Alice and receiver Bob), to the initial register containing the statej'i of size k qubits, where `+k+c=n. Proposition6. The encoding given byE 0 and a suitably-de¯ned decoding mapD 0 can correct the error set E 0 =fX a Z b Z a 1 X a 2 X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) :a;b2(Z 2 ) ` ;a 1 ;a 2 2(Z 2 ) c g; (32) for any ¯xed functions ®;¯ :(Z 2 ) ` £(Z 2 ) c £(Z 2 ) c !(Z 2 ) k . Proof. TheprotocolisshowninFigure4. Afterapplyinganerror E2E 0 , thechannel output becomes (up to a phase factor): (X a Z b )j0i(Z a 1 X a 2 I B )j©i c (X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) )j'i=jaija 1 ;a 2 ij' 0 i (33) 40 where jai = X a Z b j0i=X a j0i (34) ja 1 ;a 2 i = (Z a 1 X a 2 I B )j©i c ; (35) j' 0 i = X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) j'i: (36) (37) As the vector (a;a 1 ;a 2 ;b) completely speci¯es the error E, it is called the error syn- drome. The state (33) only depends on the reduced syndrome r=(a;a 1 ;a 2 ). In e®ect, aand(a 1 ;a 2 )havebeenencodedusingelementaryandsuperdensecoding,respectively. Bob, who holds the entire state (33), can identify the reduced syndrome. Bob simulta- neousmeasurestheZ e 1 ;:::;Z e ` observablestodecodea,theX e 1 X e 1 ;:::;X e c X e c observables to decode a 1 , and the Z e 1 Z e 1 ;:::;Z e c Z e c observables to decode a 2 . He then performs X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) on the remaining k qubit systemj' 0 i, restoring it to the original statej'i. Since the goal is the transmission of quantum information, no actual measurement is necessary. Instead, Bob can perform the following decoding D 0 consisting of the controlled unitary U 0;dec = X a;a 1 ;a 2 jaihajja 1 ;a 2 iha 1 ;a 2 jX ¡®(a;a 1 ;a 2 ) Z ¡¯(a;a 1 ;a 2 ) ; (38) followed by discarding the unwanted subsystems. We can rephrase the above error-correcting procedure in terms of the stabilizer formalism. Let S 0 =hS 0;I ;S 0;E i, where S 0;I =hZ 1 ;¢¢¢ ;Z ` i is the isotropic subgroup of size 2 ` and S 0;E = hZ `+1 ;¢¢¢ ;Z `+c ;X `+1 ;¢¢¢ ;X `+c i is the symplectic subgroup of 41 size 2 2c . We can easily construct an Abelian extension ofS 0 that acts on n+c qubits, by specifying the following generators: Z 1 I; . . . Z ` I; Z `+1 Z 1 ; X `+1 X 1 : . . . Z `+c Z c ; X `+c X c ; (39) where the ¯rst n qubits are on the side of the sender (Alice) and the extra c qubits are taken to be on the side of the receiver (Bob). The operators Z i or X i to the right of the tensor product symbol above is the Pauli operator Z or X acting on Bob's i-th qubit. We denote such an Abelian extension of the group S 0 by e S 0 . It is easy to see that the group e S 0 ¯xes the code spaceC EA 0 (therefore e S 0 is the stabilizer forC EA 0 ), and we will call the groupS 0 the entanglement-assisted stabilizer forC EA 0 . Consider the parameters of the EA canonical code. The number of ancillas ` is equaltothenumberofgeneratorsfortheisotropicsubgroupS 0;I . Thenumberofebits c is equal to the number of symplectic pairs that generate the entanglement subgroup S 0;E . Finally, the number of logical qubits k that can be encoded in C EA 0 is equal to n¡`¡c. To sum up, C EA 0 de¯ned by S 0 is an [[n;k;c]] EAQECC that ¯xes a 2 k -dimensional code space. Proposition 7. The EAQECC C EA 0 de¯ned by S 0 =hS 0;I ;S 0;E i can correct an error set E 0 if for all E 1 ;E 2 2E 0 , E y 2 E 1 2S 0;I S (G n ¡Z(hS 0;I ;S 0;E i)). 42 Proof. Since the vector (a;a 1 ;a 2 ;b) completely speci¯es the error operator E, we consider the following two di®erent cases: ² IftwoerroroperatorsE 1 andE 2 havethesamereducedsyndrome(a;a 1 ;a 2 ),then the error operator E y 2 E 1 gives us the all-zero syndrome. Therefore, E y 2 E 1 2S 0;I . This error E y 2 E 1 has no e®ect on the codewords ofC EA 0 . ² If two error operators E 1 and E 2 have di®erent reduced syndromes, and let (a;a 1 ;a 2 ) be the reduced syndrome of E y 2 E 1 , then E y 2 E 1 62 Z(hS 0;I ;S 0;E i). This error E y 2 E 1 can be corrected by the decoding operation given in (38). 4.3 The general case Theorem 5. Given a general group S = hS I ;S E i with the sizes of S I and S E being 2 n¡k¡c and 2 2c , respectively, there exists an [[n;k;c]] EAQECC C EA de¯ned by the encoding and decoding pair (E;D) with the following properties: (1). The codeC EA can correct the error setE if for all E 1 ;E 2 2E, E y 2 E 1 2S I S (G n ¡ Z(hS I ;S E i)). (2). The codespace C EA is a simultaneous eigenspace of the Abelian extension of S, ~ S. (3). To decode, the reduced error syndrome is obtained by simultaneously measuring the observables from e S. Proof. Since the commutation relations of S are the same as the EA stabilizer S 0 for the EA canonical code C EA 0 in the previous section, by Lemma 2, there exists an unitary matrix U such that S 0 = USU ¡1 . The protocol is shown in Figure 5. De¯ne 43 Figure 5: Generalizing the entanglement-assisted canonical code construction. E =U ¡1 ±E 0 andD =D 0 ± ^ ¹ U, where ¹ U is the trivial extension of U are Bob's Hilbert space, andE 0 andD 0 are given in (31) and (38), respectively. (1). Since D 0 ±E 0 ±E 0 =id k for any E 0 2E 0 , then D±E±E =id k follows for any E2E. Thus, the encoding and decoding pair (E;D) corrects E. Following Proposition 7, the correctable error setE contains all E 1 ;E 2 such that E y 2 E 1 2S I S (G n ¡Z(hS I ;S E i)). (2). Since C EA 0 is the simultaneous +1 eigenspace of e S 0 , S = U ¡1 S 0 U, and by de¯- nition C EA = ¹ U ¡1 (C EA 0 ), we conclude that C EA is a simultaneous eigenspace of e S. (3). The decoding operationD 0 involves i. measuring the set of generators of e S 0 , yielding the error syndrome according to the error E 0 . ii. performing a recovering operation E 0 again to undo the error. 44 By Lemma 4, performing D =D 0 ± ^ ¹ U is equivalent to measuring e S = U ¡1 e S 0 U, followed by performing the recovering operation U ¡1 E 0 U based on the measure- ment outcome, followed by ^ U to undo the encoding. 4.4 Generalized construction from quaternary codes Proposition 8. If a classical [n;k;d] code C 4 exists then an [[n;2k ¡ n + c;d;c]] EAQECC exists for some non-negative integer c. Proof. LetH 4 bethe(n¡k)£nquaternaryparitycheckmatrixforC 4 . ByProposition 1, there exists an [n;2k¡n;d] symplectic code C sp with parity check matrix H sp = °( ~ H 4 ), where ~ H 4 = 0 B @ !H 4 ¹ !H 4 1 C A: (40) Notice that even if 2k¡n<0, the following still holds H sp ¯u T 6=0 T ; for each nonzero u2(Z 2 ) 2n with wt(u)<d. For simplicity, let V = rowspace(H sp ). Theorem 1 shows that there exists a symplectic basis consisting of hyperbolic pairs (u i ;v i ), i = 1;2;¢¢¢ ;n, such that fu 1 ;¢¢¢ ;u c+` ;v 1 ;¢¢¢ ;v c g is a basis for V. Then by the map N : (Z 2 ) 2n ! G n , the groupS =hS I ;S E i, de¯nes an [[n;2k¡n+c;d;c]] EAQECC by Theorem 5, where S E = hN u 1 ;N v 1 ;¢¢¢ ;N u c ;N v c i S I = hN u c+1 ;¢¢¢ ;N u (c+`) i 45 and c= 1 2 dimsympV: When c = 0, V is dual-containing. The above construction will give us standard quantum error-correcting codes. Any classical binary [n;k;d] code may be viewed as a quaternary [n;k;d] 4 code. In this case, the above construction gives rise to a CSS-type code. 4.5 Bounds on performance In this section we shall see that the performance of EAQECCs is comparable to the performance of QECCs (which are a special case of EAQECCs). ThetwomostimportantouterboundsforQECCsarethequantumSingletonbound [34, 51] and the quantum Hamming bound [28]. Given an [[n;k;d]] QECC (which is an [[n;k;d;0]] EAQECC), the quantum Singleton bound reads n¡k¸2(d¡1): The quantum Hamming bound holds only for non-degenerate codes and reads b d¡1 2 c X j=0 3 j µ n j ¶ ·2 n¡k : The proofs of these bounds [28, 51] are easily adapted to EAQECCs. This was ¯rst noted by Bowen [10] in the case of the quantum Hamming bound. Consequently, an [[n;k;d;c]] EAQECC satis¯es both bounds for any value of c. Note that the F 4 construction connects the quantum Singleton bound to the classical Singleton bound n¡k ¸ d¡1. An [n;k;d] quaternary code saturating the classical Singleton bound 46 implies an [[n;2k¡n+c;d;c]] EAQECC saturating the quantum Singleton bound, that is n¡(k¡c)¸2(d¡1). It is instructive to examine the asymptotic performance of quantum codes on a particular channel. A popular choice is the tensor power channel N n , where N is the depolarizing channel with Kraus operators f p p 0 I; p p 1 X; p p 2 Y; p p 3 Zg, for some probability vector p=(p 0 ;p 1 ;p 2 ;p 3 ). It is well known that the maximal transmission rate R=k=n achievable by a non- degenerate QECC (in the sense of vanishing error for large n on the channelN n ) is equal to the hashing bound R = 1¡H(p). Here H(p) is the Shannon entropy of the probabilitydistributionp. Thisboundisattainedbypickingarandomself-orthogonal code. However no explicit constructions are known which achieve this bound. Interestingly, theF 4 construction also connects the hashing bound to the Shannon bound for quaternary channels. Consider the quaternary channel a 7! a+c, where c takes on values 0;!;1;¹ !, with respective probabilities p 0 ;p 1 ;p 2 ;p 3 . The maximal achievablerateR=k=nforthischannelwasprovedbyShannontoequalR=2¡H(p). An [n;k] quaternary code saturating the Shannon bound implies an [[n;2k¡n+c;c]] EAQECC, achieving the hashing bound! 4.6 Table of codes In [15] a table of best known QECCs was given. Below we show an updated table which includes EAQECCs. The entries with an asterisk mark the improvements over the table from [15]. All these are obtained from Proposition 3.1. The corresponding classical quaternary code is available online at http://www.win.tue.nl/»aeb/voorlincod.html. 47 nnk¡c 0 1 2 3 4 5 6 7 8 9 10 3 2 2 ¤ 1 1 4 3 ¤ 2 2 1 1 5 3 3 2 2 ¤ 1 1 6 4 3 2 2 2 1 1 7 3 3 2 2 2 2 ¤ 1 1 8 4 3 3 3 2 2 2 1 1 9 4 4 ¤ 3 3 2 2 2 2 ¤ 1 1 10 5 ¤ 4 4 3 3 2 2 2 2 1 1 Table 3: Highest achievable minimal distance d in any [[n;k;d;c]] EAQECCs. The general methods from [15] for constructing new codes from old also apply here. Moreover, new constructions are possible since the self-orthogonality condition is removed. An example is given by the following Theorem. Theorem 6. (a) Suppose an [[n;k;d;c]] code exists, then an [[n+1;k¡1;d 0 ;c 0 ]] code exists for some c 0 and d 0 ¸d; (b) Suppose a non-degenerate [[n;k;d;c]] code exists, then an [[n¡1;k+1;d¡1;c 0 ]] code exists for some c 0 . Proof. (a)Recallthatthenetyieldis ^ k =k¡c. LetH bethe(n¡ ^ k£2n)paritycheck matrix of the [[n;k;d;c]] code. The parity check matrix of the new [[n+1; ^ k¡1;d 0 ;c 0 ]] is then H 0 = 0 B B B B B B B B B B B @ 0¢¢¢ 0 0 1¢¢¢ 1 1 1¢¢¢ 1 1 0¢¢¢ 0 0 0 0 H Z . . . H X . . . 0 0 1 C C C C C C C C C C C A : (41) 48 This corresponds to the classical construction of adding a parity check at the end of the codeword [46]. The additional rows ensure that errors involving the last qubit are detected. Sometimes the distance actually increases: for instance, the [[8;0;4]] is obtained from the [[7;1;3]] code in this way. (b) We mimic the classical \puncturing" method [46]. Let C be the (n + ^ k)- dimensional subspace of (Z 2 ) 2n corresponding to the [[n;k;d;c]] EAQEC code. Punc- turing C by deleting the ¯rst Z and X coordinate, we obtain a new \code" C 0 which is an (n + ^ k)-dimensional subspace of (Z 2 ) 2(n¡1) . This corresponds to an [[n¡1;k+1;d¡1;c 0 ]]EAQECcode,astheminimumdistancebetweenthe\codewords" of C decreases by at most 1. 4.7 Discussion Motivated by recent developments in quantum Shannon theory, we have introduced a generalization of the stabilizer formalism to the setting in which the encoder Alice and decoder Bob pre-share entanglement (EAQECCs). The powerful canonical code technique again provides us essential insight into the error-correcting property. First of all, the entanglement-assisted canonical code is obtained by replacing some ancillas of the standard canonical code with maximally entangled states. The codewords of the entanglement-assisted canonical code then can be described by a set of commuting operators (see (39)). The error syndrome of each correctable error can be seen as clas- sical information being encoded in the entanglement-assisted canonical code by either elementary coding or superdense coding. Therefore, reading out the error syndrome is equivalent to recovering the classical message. Then we can restore the codewords of the entanglement-assisted canonical code by performing a correction operation based on the measurement outcome since the outcome tells us which error happens. These 49 two steps, reading out the error syndrome and performing correction operation, are called the decoding operation. Uptothispoint,theentanglement-assistcanonicalcodeisnothingbutthestabilizer formalism. Whatmakestheentanglement-assistedcanonicalcodedi®erentiswhenhalf ofthemaximallyentangledstatesareassumedtobeoriginallypossessedbythereceiver Bob (These half of ebits do not go through the noisy channel). Then the operators on Alice's sie form a non-commuting set of generators, allowing us to map arbitrary classical quaternary codes to EAQECCs. There are two practical advantages of EAQECCs over standard QECCs: (1). Theyaremucheasiertoconstructfromclassicalcodesbecauseself-orthogonality is not required. This allows us to import the classical theory of error correction wholesale, including capacity-achieving modern codes. The attraction of these moderncodescomesfromtheexistenceofe±cientdecodingalgorithmsthatpro- vide excellent trade-o® between decoding complexity and decoding performance. In fact, these decoding algorithms, such as sum-product algorithm, can be mod- i¯ed to decode the error syndromes e®ectively [44]. The only problem of using theseiterativedecodingalgorithmsonquantumLDPCactuallycomesfromthose shortest 4-cycles that were introduced inevitably because of self-orthogonality constrain. However, we have demonstrated recently that by allowing assisted entanglement, those 4-cycles can be eliminated completely, and the performance of the iterative decoding improves dramatically by our numerically simulation results (see Chapter 7). This ¯nding further con¯rms the contributionof our EA formalism. (2). Comparing [[n;k;d;c]] EAQECCs to [[n;k;d;0]] QECCs is not being entirely fair to former, since the entanglement used in the protocol is a strictly weaker resource than quantum communication. However, by using an EAQECC, we 50 typically achieve a higher rate for the same distance, or a higher distance for the samerate,thanaQECC;andbecauseentanglementisa\cheaper"resources,this is often a worthwhile trade-o®. Or to think of it a di®erent way, if we construct an EAQECC and a QECC from two classical codes with the same parameters [n;k;d], the EAQECC will have a higher rate; or by using an EAQECC derived from a classical code with higher distance and lower rate, we can achieve the same rate and a higher distance than a QECC. Ifoneisinterestedinapplicationstofaulttolerantquantumcomputation,wherethe resource of entanglement is meaningless, high values of c are unwelcome because they require a long seed QECCs. We expect this obstacle to be overcome by bootstrapping. Anotherfruitfullineofinvestigationconnectstoquantumcryptography. Quantum cryptographic protocols, such as BB84, are intimately related to CSS QECCs. In [41] itisshownthatEAQECCsanaloguesofCSScodesgiverisetokeyexpansionprotocols which do not rely on the existence of long self-orthogonal codes. 51 Chapter 5: Operator quantum error-correcting codes Inthischapter,wewillbrie°yreviewthewell-knownoperatorquantumerror-correcting codes (OQECCs), using the canonical code method and linking to the operator stabi- lizer formalism. 5.1 The canonical code The idea of OQECCs also comes from a simple idea: replacing some portion of the ancillas of the canonical code (25) by some garbage states. We can construct the operator canonical code C OP 0 with the following trivial encoding operation E 0 de¯ned by E 0 :j'ih'j!j0ih0j¾j'ih'j: (42) The operation simply appends s ancilla qubits in the statej0i, and an arbitrary state ¾ of size r qubits, to the initial register containing the state j'i of size k qubits, where s+k+r = n. These r extra garbage qubits are called the gauge qubits. Two states of this form which di®er only in ¾ are considered to encode the same quantum information. Proposition9. The encoding given byE 0 and a suitably-de¯ned decoding mapD 0 can correct the error set E 0 =fX a Z b X c Z d X ®(a) Z ¯(a) :a;b2(Z 2 ) s ;c;d2(Z 2 ) r g; (43) 52 k+s+r=n ' σ X Z c d X Z a b α β X Z s 0 0 ⊗ σ { } a a α β - - X Z φ φ s r k a a φ φ ' ' 0 D φ φ φ φ a 0 E Figure 6: The operator canonical code. for any ¯xed functions ®;¯ :(Z 2 ) s !(Z 2 ) k . Proof. TheprotocolisshowninFigure6. Afterapplyinganerror E2E 0 , thechannel output becomes (up to a phase factor): (X a Z b )j0ih0j(X a Z b ) y (X c Z d )¾(X c Z d ) y (X ®(a) Z ¯(a) )j'ih'j(X ®(a) Z ¯(a) ) y =jaihaj¾ 0 j' 0 ih' 0 j (44) where jai = X a j0i; (45) ¾ 0 = (X c Z d )¾(X c Z d ) y ; (46) j' 0 i = (X ®(a) Z ¯(a) )j'i: (47) As the vector (a;b;c;d) completely speci¯es the error operator E, it is called the error syndrome. However, in order to correct this error, only the reduced syndrome a matters. Here two kinds of passive error correction are involved. The errors that come from vector b are passively corrected because they do not a®ect the encoded state given in (42). The errors that come from vector (c;d) are passively corrected because of the subsystem structure inside the code space: ½¾ and ½¾ 0 represent 53 thesameinformation, di®eringonlybyagaugeoperation. Thoughtheseerrorschange the encoded states, they do not damage the information encoded in the states. The decoding operation D 0 is constructed based on the reduced syndrome, and is also known as collective measurement. Bob can recover the state j'i by performing the decodingD 0 : D 0 = X a jaihajIX ¡®(a) Z ¡¯(a) ; (48) followed by discarding the unwanted systems. We can rephrase the above error-correcting procedure in terms of the stabilizer formalism. Let S 0 = (S 0;I ;S 0;G ), where S 0;I =hZ 1 ;¢¢¢ ;Z s i is the isotropic subgroup of size 2 s and S 0;G =hZ s+1 ;¢¢¢ ;Z s+r ;X s+1 ;¢¢¢ ;X s+r i is the symplectic subgroup of size 2 2r . It follows that the two subgroups (S 0;I ;S 0;G ) de¯ne the canonical OQECC C OP 0 given in (42). The subgroup S 0;I de¯nes a 2 k+r -dimensional code space C OP 0 , and the gauge subgroup S 0;G speci¯es all possible operations that can happen on the gauge qubits. Thus we can useS 0;G to de¯ne an equivalence class between two states in the code space of the form: ½¾ and ½¾ 0 , where ½ is a state on H k 2 , and ¾;¾ 0 are states onH r 2 . Consider the parameters of the canonical code. The number of ancillas s is equal to the number of generators for the isotropic subgroup S 0;I . The number of gauge qubits r is equal to the number of symplectic pairs for the gauge subgroup S 0;G . Finally, the number of logical qubits k that can be encoded in C OP 0 is equal to n¡s¡r. To sum up,C OP 0 de¯ned by (S 0;I ;S 0;G ) is an [[n;k;r]] OQECC that ¯xes a 2 k+r -dimensional code space, within which ½¾ and½¾ 0 are considered to carry the same information. Notice that there is a tradeo® between the number of encoded bits and gauge bits, in that we can reduce the rate by improving the error-avoiding ability or vice versa. 54 Proposition 10. The OQECCC OP 0 de¯ned by (S 0;I ;S 0;G ) can correct an error setE 0 if for all E 1 ;E 2 2E 0 , E y 2 E 1 2hS 0;I ;S 0;G i S (G n ¡Z(S 0;I )). Proof. Since the vector (a;b;c;d) completely speci¯es the error operator E, we con- sider the following two di®erent cases: ² IftwoerroroperatorsE 1 andE 2 havethesamereducedsyndromea,thentheer- ror operator E y 2 E 1 gives us all-zero reduced syndrome with some vector (b;c;d). Therefore, E y 2 E 1 2hS 0;I ;S 0;G i. ThiserrorE y 2 E 1 hasnoe®ectonthelogicalstate j'ih'j. ² If two error operators E 1 and E 2 have di®erent reduced syndromes, and leta be the reduced syndrome of E y 2 E 1 , then E y 2 E 1 62 Z(S 0;I ). This error E y 2 E 1 can be corrected by the decoding operation given in (48). 5.2 The general case Theorem 7. Given a general group S = hS I ;S G i with the sizes of S I and S G being 2 n¡k¡r and 2 2r , respectively, there exists an [[n;k;r]] OQECC C OP de¯ned by the encoding and decoding pair (E;D) with the following properties: (1). ThecodeC OP cancorrecttheerrorsetEifforallE 1 ;E 2 2E,E y 2 E 1 2hS I ;S G i S (G n ¡ Z(S I )). (2). The codespace C OP is a simultaneous eigenspace of S I . (3). To decode, the reduced error syndrome is obtained by simultaneously measuring the observables from S I . Proof. Since the commutation relations of S = (S I ;S G ) are the same as the OP sta- bilizer S 0 = (S 0;I ;S 0;G ) for the OP canonical code C OP 0 in the previous section, by 55 s 0 0 ⊗ σ { } a a α β - - X Z φ φ X Z c d X Z a b α β X Z 1 U - U U 1 U - U - = 1 0 E E 1 0 U U - = E E U = 0 D D φ φ φ φ Figure 7: The operator quantum error-correcting code. Lemma 2, there exists an unitary matrix U such that S 0 = USU ¡1 . The protocol is shown in Figure 7. De¯neE = ^ U ¡1 ±E 0 andD =D 0 ± ^ U, andE 0 andD 0 are given in (42) and (48), respectively. (1). Since D 0 ±E 0 ±E 0 =id k for any E 0 2E 0 , then D±E±E =id k follows for any E 2 E. Thus, the encoding and decoding pair (E;D) corrects E. Following Proposition 10, the correctable error setE contains all E 1 ;E 2 such that E y 2 E 1 2hS I ;S G i S (G n ¡Z(S I )). (2). Since C OP 0 is the simultaneous +1 eigenspace of S 0;I , S = U ¡1 S 0 U, and by de¯nition C OP = U ¡1 (C OP 0 ), we conclude that C OP is a simultaneous eigenspace ofS I . (3). The decoding operationD 0 involves i. measuring the set of generators ofS 0 , yielding the error syndrome according to the error E 0 . ii. performing a recovering operation E 0 again to undo the error. 56 By Lemma 4, performing D =D 0 ± ^ U is equivalent to measuring S = U ¡1 S 0 U, followed by performing the recovering operation U ¡1 E 0 U based on the measure- ment outcome, followed by ^ U to undo the encoding. 5.3 Discussion The idea of the operator canonical code comes from replacing some portion of ancillas of the standard canonical code with an arbitrary garbage state that we do not care about. In terms of the operator stabilizer formalism, the codespace of the operator canonical code is described by a set of commuting Pauli Z operators together with a set of anti-commuting operators specifying all possible operations that can occur on the garbage state. These operations on the garbage state do not a®ect our quantum information, therefore no correction is needed, and thus the passive error-correcting power is increased. The error syndrome of each correctable error can be seen as clas- sical information being encoded in the operator canonical code by elementary coding. Therefore,readingouttheerrorsyndromeisequivalenttorecoveringtheclassicalmes- sage. Thenwecanrestorethecodewordsoftheoperatorcanonicalcodebyperforming a correction operation based on the measurement outcome since the outcome tells us whicherrorhappens. Thesetwosteps, readingouttheerrorsyndromeandperforming correction operation, are called the decoding operation. The operator quantum error-correcting codes are a combination of standard quan- tum error-correcting codes (active error correction) and the passive passive error- avoiding schemes, such as decoherence-free subspaces and noiseless subsystems. The operator stabilizer is generated by a set of non-commuting generators. Therefore, we can map arbitrary classical quaternary codes to OQECCs, though the distance of the 57 OQECCs is not always guaranteed. There has been a couple of clever construction of OQECCs whose distance is inherited from their classical counterpart [1, 38]. The advantage of OQECCs comes from the fact that it is not necessary to actively correct all errors, but rather only to perform correction modulo the subsystem struc- ture. One potential bene¯t of the new decoding procedure is to improve the threshold of fault-tolerant quantum computation. This research direction remains a hot topic in quantum computation. 58 Chapter 6: Entanglement-assisted operator quantum error-correcting codes Now it becomes clear how to combine the idea of entanglement-assisted and operator formalism, to construct the entanglement-assisted operator quantum error-correcting codes (EAQECCs). We will begin with its canonical code. 6.1 The canonical code We illustrate the idea of EAOQECCs by the following canonical code. Consider the trivial encoding operationE 0 de¯ned by E 0 :j'ih'j!j0ih0j s j©ih©j c ¾j'ih'j: (49) The operation simply appends s ancilla qubits in the state j0i, c copies of j©i (a maximally entangled state shared between sender Alice and receiver Bob), and an arbitrary state ¾ of size r qubits, to the initial register containing the statej'i of size k qubits, where s+k +r+c = n. These r extra qubits are the gauge qubits. Two states of this form which di®er only in ¾ are considered to encode the same quantum information. 59 s+c+r+k=n ' σ φ X Z c d X Z a b α β X Z s 0 0 ⊗ σ { } a a α β - - X Z s r k a a φ φ ' ' c { } 1 2 1 2 a a a a 1 2 1 2 a a a a c ⊗ + Φ 1 2 X Z a a φ φ φ φ = 0 E 0 D a 1 2 a a Figure 8: The entanglement-assisted operator canonical code. Proposition 11. The encoding given by E 0 and a suitably-de¯ned decoding map D 0 can correct the error set E 0 =fX a Z b Z a 1 X a 2 X c Z d X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) : a;b2(Z 2 ) s ;a 1 ;a 2 2(Z 2 ) c ;c;d2(Z 2 ) r g; (50) for any ¯xed functions ®;¯ :(Z 2 ) s £(Z 2 ) c £(Z 2 ) c !(Z 2 ) k . Proof. TheprotocolisshowninFigure8. Afterapplyinganerror E2E 0 , thechannel output becomes (up to a phase factor): jaihajja 1 ;a 2 iha 1 ;a 2 j¾ 0 j' 0 ih' 0 j; (51) where jai = X a j0i; (52) ja 1 ;a 2 i = (Z a 1 X a 2 I B )j©i c ; (53) ¾ 0 = (X c Z d )¾(X c Z d ) y ; (54) j' 0 i = (X ®(a;a 1 ;a 2 ) Z ¯(a;a 1 ;a 2 ) )j'i: (55) 60 Asthevector(a;a 1 ;a 2 ;b;c;d)completelyspeci¯estheerroroperatorE,itiscalled theerror syndrome. However, inordertocorrectthiserror, onlythe reduced syndrome (a;a 1 ;a 2 ) matters. The entanglement-assisted operator canonical code C EAO 0 keeps advantages of both EAQECCs and OQECCs. On one hand, the two kinds of passive error correction are preserved. On the other hand, the power of active error correction is increased by the use of pure entanglement. The decoding operation D 0 is constructed based on the reduced syndrome. Bob can recover the statej'i by performing the decodingD 0 : D 0 = X a;a 1 ;a 2 jaihajja 1 ;a 2 iha 1 ;a 2 jI X ¡®(a;a 1 ;a 2 ) Z ¡¯(a;a 1 ;a 2 ) ; (56) followed by discarding the unwanted systems. We can rephrase the above error-correcting procedure in terms of the stabilizer formalism. LetS 0 =hS 0;I ;S 0;S i,whereS 0;I =hZ 1 ;¢¢¢ ;Z s iistheisotropicsubgroupof size 2 s andS 0;S =hZ s+1 ;¢¢¢ ;Z s+c+r ;X s+1 ;¢¢¢ ;X s+c+r i is the symplectic subgroup of size 2 2(c+r) . We can further divide the symplectic subgroupS 0;S into an entanglement subgroup S 0;E =hZ s+1 ;¢¢¢ ;Z s+c ;X s+1 ;¢¢¢ ;X s+c i of size 2 2c and a gauge subgroup S 0;G =hZ s+c+1 ;¢¢¢ ;Z s+c+r ;X s+c+1 ;¢¢¢ ;X s+c+r i 61 of size 2 2r , respectively. The generators of (S 0;I ;S 0;E ;S 0;G ) are arranged in the follow- ing form: Z e i I I I I Z e j I I I X e j I I I I Z e l I I I X e l I Ã ! s Ã ! c Ã ! r Ã ! k (57) where fe i g i2[s] , fe j g j2[c] , and fe l g l2[r] are the set of standard bases in (Z 2 ) s , (Z 2 ) c , and (Z 2 ) r , respectively, and [k]´f1;¢¢¢ ;kg. Itfollowsthatthethreesubgroups(S 0;I ;S 0;E ;S 0;G )de¯nethecanonicalcodeC EAO 0 given in (49). The subgroups S 0;I and S 0;E de¯ne a 2 k+r -dimensional code space C EAO 0 ½ H (n+c) , and the gauge subgroup S 0;G speci¯es all possible operations that can happen on the gauge qubits. Thus we can use S 0;G to de¯ne an equivalence class betweentwostatesinthecodespaceoftheform: ½¾ and½¾ 0 ,where½isastateon H k , and ¾;¾ 0 are states onH r . Consider the parameters of the canonical code. The number of ancillas s is equal to the number of generators for the isotropic subgroup S 0;I . The number of ebits c is equal to the number of symplectic pairs that generate theentanglementsubgroupS 0;E . Thenumberofgaugequbitsr isequaltothenumber of symplectic pairs for the gauge subgroupS 0;G . Finally, the number of logical qubits k that can be encoded inC EAO is equal to n¡s¡c¡r. To sum up,C EAO de¯ned by (S 0;I ;S 0;E ;S 0;G )isan[[n;k;r;c]]EAOQECCthat¯xesa2 k+r -dimensionalcodespace, within which ½¾ and ½¾ 0 are considered to carry the same information. Proposition 12. The EAOQECC C EAO de¯ned by (S 0;I ;S 0;E ;S 0;G ) can correct an error set E 0 if for all E 1 ;E 2 2E 0 , E y 2 E 1 2hS 0;I ;S 0;G i S (G n ¡Z(hS 0;I ;S 0;E i)). Proof. Since the vector (a;a 1 ;a 2 ;b;c;d) completely speci¯es the error operator E, we consider the following two di®erent cases: 62 E D 1 0 U U - = E E ' σ φ X Z c d X Z a b α β X Z s 0 0 ⊗ σ { } a a α β - - X Z { } 1 2 1 2 a a a a c ⊗ + Φ 1 2 X Z a a φ φ φ φ = U -1 U U -1 U a 1 2 a a Figure 9: The entanglement-assisted operator quantum error-correcting code. ² If two error operators E 1 and E 2 have the same reduced syndrome (a;a 1 ;a 2 ), thentheerroroperatorE y 2 E 1 givesusall-zeroreducedsyndromewithsomevector (b;c;d). Therefore, E y 2 E 1 2 hS 0;I ;S 0;G i. This error E y 2 E 1 has no e®ect on the logical statej'ih'j. ² If two error operators E 1 and E 2 have di®erent reduced syndromes, and let (a;a 1 ;a 2 ) be the reduced syndrome of E y 2 E 1 , then E y 2 E 1 62 Z(hS 0;I ;S 0;E i). This error E y 2 E 1 can be corrected by the decoding operation given in (56). 6.2 The general case Theorem8. Giventhesubgroups(S I ;S E ;S G ),thereexistsan[[n;k;r;c]]entanglement- assisted operator quantum error-correcting code C eao de¯ned by the encoding and de- coding pair: (E;D). The code C EAO can correct the error set E if for all E 1 ;E 2 2E, E y 2 E 1 2hS I ;S G i S (G n ¡Z(hS I ;S E i)). 63 Proof. SinceS »S 0 , thereexistsanunitarymatrix U thatpreservesthecommutation relations. The protocol is shown in Figure 9. De¯ne E = U ¡1 ±E 0 and D =D 0 ±U, whereE 0 andD 0 are given in (49) and (56), respectivley. Since D 0 ±E 0 ±E 0 =id k for any E 0 2E 0 , then D±E±E =id k follows for any E2E. Thus, the encoding and decoding pair (E;D) corrects E. We say that the [[n;k;d;r;c]] EAOQECC C eao has distance d if it can correct any error set E such that for each operator E2E, the weight t of E satis¯es 2t+1·d. 6.3 Properties of EAOQECCs In the description earlier in this chapter, we assumed that the gauge subgroup was generated by a set of symplectic pairs of generators. In some cases, it may make sense to start with a gauge subgroup which itself has both an isotropic (i.e., commuting) and a symplectic subgroup. In this case, we can arbitrarily add a symplectic partner for each generator in the isotropic subgroup of the gauge group. This can be useful in constructingEAOQECCsfromEAQECCs,inawayanalogoustohowOQECCscanbe constructed by starting from standard QECCs. Poulin shows in [50] that it is possible to move generators from the stabilizer group into the gauge subgroup, together with their symplectic partners, without changing the essential features of the original code. We provide an example of such a construction in section 6.4.2. There is further °exibility in trading between active error correction ability and passive noise avoiding ability [1]. This is captured by the following theorem: 64 Theorem 9. We can transform any [[n;k+r;d 1 ;0;c]] code C 1 into an [[n;k;d 2 ;r;c]] codeC 2 , andtransformthe[[n;k;d 2 ;r;c]]codeC 2 intoan[[n;k;d 3 ;0;c]]codeC 3 , where d 1 ·d 2 ·d 3 . Proof. ThereexistsanisotropicsubgroupS I andanentanglementsubgroupS E associ- atedwithC 1 ofsize2 s and2 2c ,respectively. Theseparameterssatisfys+c+k+r =n. ThiscodeC 1 correspondstoan [[n;k+r;d 1 ;0;c]]EAQECC for somed 1 . If weadd the gauge subgroupS G of size 2 2r , then (S I ;S E ;S G ) de¯nes an [[n;k;d 2 ;r;c]] EAOQECC C 2 for some d 2 , which follows from Theorem 8. Let E 1 be the error set that can be corrected byC 1 , andE 2 be the error set that can be corrected byC 2 . Clearly,E 1 ½E 2 (see the following table), soC 2 can correct more errors thanC 1 . By sacri¯cing part of the transmission rate, we have gained additional passive correction, and d 2 ¸d 1 . If we now throw away half of each symplectic pair inS G and include the remaining generators in S I , which becomes S 0 I , the size of the isotropic subgroup increases by a factor of 2 r . Then (S 0 I ;S E ) de¯nes an [[n;k;d 3 ;0;c]] EAQECC C 3 . Let E 3 be the error set that can be corrected by C 3 . Let E 2 E 2 , then either E 2 hS I ;S G i or E62Z(hS I ;S E i). ² If E2hS I ;S G i, then either E2S 0 I or E2hS I ;S G i=S 0 I . If E2hS I ;S G i=S 0 I , this implies E62Z(S 0 I ). Thus, E2E 3 . ² SincehS I ;S E i½hS 0 I ;S E i,wehaveZ(hS 0 I ;S E i)½Z(hS I ;S E i). IfE62Z(hS I ;S E i), then E62Z(hS 0 I ;S E i). Thus, E2E 3 . Putting these together we get E 2 ½E 3 . Therefore d 3 ¸d 2 . To conclude this section, we list the di®erent error-correcting criteria of a conven- tionalstabilizercode(QECC),anEAQECC,anOQECC,andanEAOQECCinTable . 65 QECC EAQECC E y 2 E 1 62Z(S I ) E y 2 E 1 62Z(hS I ;S E i) E y 2 E 1 2S I E y 2 E 1 2S I OQECC EAOQECC E y 2 E 1 62Z(S I ) E y 2 E 1 62Z(hS I ;S E i) E y 2 E 1 2hS I ;S G i E y 2 E 1 2hS I ;S G i Table 4: Summary of error-correcting criteria. 6.4 Examples 6.4.1 EAOQECC from EAQECC Our ¯rst example constructs an [[8;1;3;c = 1;r = 2]] EAOQECC from an [[8,1,3;1]] EAQECC. Consider the EAQECC code de¯ned by the group S generated by the operators in Table 5. Here ¹ Z and ¹ X refer to the logical Z and X operation on the codeword, respectively. The isotropic subgroup is S I = hS 1 ;S 2 ;S 3 ;S 4 ;S 5 ;S 8 i, the entanglement subgroup is S E = hS 6 ;S 7 i, and together they generate the full group S =hS I ;S E i. This code C(S I ;S E ) encodes one qubit into eight physical qubits with the help of one ebit, and therefore is an [[8;1;1]] code. It can be easily checked that this code can correct an arbitrary single-qubit error, and it is degenerate. By inspecting the group structure ofS, we can recombine the ¯rst four stabilizers ofthecodetogivetwoisotropicgenerators(whichweretaininS I ),andtwogenerators whichweinclude, togetherwiththeirsymplecticpartners, in thesubgroupS G , fortwo qubits of gauge symmetry. This yields an [[8;1;3;c = 1;r = 2]] EAOQECC whose generators are given in Table 6. where S I = hS 0 1 ;S 0 2 ;S 0 3 ;S 0 6 i, S E = hS 0 4 ;S 0 5 i, and S G =hg z 1 ;g x 1 ;g z 2 ;g x 2 i. 66 Alice Bob S 1 Z Z I I I I I I I S 2 Z I Z I I I I I I S 3 I I I Z Z I I I I S 4 I I I Z I Z I I I S 5 I I I I I I Z Z I S 6 I I I I I I I Z Z S 7 X X X I I I X X X S 8 X X X X X X I I I ¹ Z Z I I Z I I I Z I ¹ X I I I X X X I I I Table 5: The original [[8,1,3;c = 1]] EAQECC encodes one qubit into eight physical qubits with the help of one ebit. 6.4.2 EAOQECCs from classical BCH codes EAOQECCs can also be constructed directly from classical binary codes. Before we give examples, however, we need one more theorem: Theorem 10. Let H be any binary parity check matrix with dimension (n¡k)£n. We can obtain the corresponding [[n;2k¡n+c;c]] EAQECC, where c = rank(HH T ) is the number of ebits needed. Proof. By the CSS construction, let ~ H be ~ H = 0 B @ H 0 0 H 1 C A: (58) 67 Alice Bob S 0 1 Z Z I Z Z I I I I S 0 2 Z I Z Z I Z I I I S 0 3 I I I I I I Z Z I S 0 4 I I I I I I I Z Z S 0 5 X X X I I I X X X S 0 6 X X X X X X I I I ¹ Z Z I I Z I I I Z I ¹ X I I I X X X I I I g z 1 Z Z I I I I I I I g x 1 I X I I X I I I I g z 2 I I I Z I Z I I I g x 2 I I X I I X I I I Table 6: The resulting [[8,1,3;c = 1,r = 2]] EAOQECC encodes one qubit into eight physical qubits with the help of one ebit, and create two gauge qubits for passive error correction. 68 LetS bethegroupgeneratedby ~ H,thenS =hZ r 1 ;¢¢¢ ;Z r n¡k ;X r 1 ;¢¢¢ ;X r n¡k i,where r i is the i-th row vector of H. Now we need to determine how many symplectic pairs are in groupS. Since rank(HH T )=c, there exists a matrix P such that PHH T P T = 0 B B B B B B B B @ I p£p 0 0 0 0 0 I q£q 0 0 I q£q 0 0 0 0 0 0 1 C C C C C C C C A (n¡k)£(n¡k) where p+2q = c. Let r 0 i be the i-th row vector of the new matrix PH, then S = hZ r 0 1 ;¢¢¢ ;Z r 0 n¡k ;X r 0 1 ;¢¢¢ ;X r 0 n¡k i. Using the fact that fZ a ;X b g = 0 if and only if a¢b = 1, we know that the operators Z r 0 i ;X r 0 i for 1 · i · p, and the operators Z r 0 p+j ;X r 0 p+q+j for 1 · j · q, generate a symplectic subgroup inS of size 2 2c . De¯nition 1. [46] A cyclic code of length n over GF(p m ) is a BCH code of designed distance d if, for some number b¸0; the generator polynomial g(x) is g(x)=lcmfM b (x);M b+1 (x);¢¢¢ ;M b+d¡2 (x)g; where M k (x) is the minimal polynomial of ® k over GF(p m ). I.e. g(x) is the lowest degree monic polynomial over GF(p m ) having ® b ;® b+1 ;¢¢¢ ;® b+d¡2 as zeros. When b = 1, we call such BCH codes narrow-sense BCH codes. When n = p m ¡1, we call such BCH codes primitive. 69 Consider the primitive narrow-sense BCH code over GF(2 6 ). This code has the following parity check matrix H q = 0 B B B B B B B B @ 1 ® ® 2 ¢¢¢ ® n¡1 1 ® 3 ® 6 ¢¢¢ ® 3(n¡1) 1 ® 5 ® 10 ¢¢¢ ® 5(n¡1) 1 ® 7 ® 14 ¢¢¢ ® 7(n¡1) 1 C C C C C C C C A ; (59) where ® 2 GF(2 6 ) satis¯es ® 6 + ® + 1 = 0 and n = 63. Since all ¯nite ¯elds of order p m are isomorphic, there exists a one-to-one correspondence between elements in f® j : j = 0;1;¢¢¢ ;p m ¡2;1g and elements in fa 0 ;a 1 ;¢¢¢ ;a m : a i 2 GF(p)g. If we replace ® j 2 GF(2 6 ) in (59) with its binary representation, this gives us a binary [63;39;9] BCH code whose parity check matrix H 2 is of size 24£63. If we carefully inspect the binary parity check matrix H 2 , we will ¯nd that the ¯rst 18 rows of H 2 give a [63;45;7] dual-containing BCH code. From Theorem 10, it is easy to check that c = rank(H 2 H T 2 ) = 6. Thus by the CSS construction [13], this binary [63;39;9] BCH code will give us a corresponding [[63;21;9;6]] EAQECC. If we further explore the group structure of this EAQECC, we will ¯nd that the 6 symplectic pairs that generate the entanglement subgroup S E come from the last 6 rows of H 2 . (Remember that we are using the CSS construction.) If we remove one symplectic pair at a time from S E and add it to the gauge subgroup S G , we get EAOQECCs with parameters given in Table 7. In general, there could be considerable freedom in which of the symplectic pairs is to be removed. There are plenty of choices in the generators of S E . In fact, it does not matter which symplectic pair we remove ¯rst in this example, due to the algebraic structure of this BCH code. The distance is always lower bounded by 7. 70 n k d r c 63 21 9 0 6 63 21 7 1 5 63 21 7 2 4 63 21 7 3 3 63 21 7 4 2 63 21 7 5 1 63 21 7 6 0 Table 7: Parameters of the EAOQECCs constructed from a classical [63,39,9] BCH code, where r represents the amount of gauge qubits created and c represents the amount of ebits needed. One ¯nal remark: this example gives EAOQECCs with positive net rate, so they could be used as catalytic codes. 6.4.3 EAOQECCs from classical quaternary codes In the following, we will show how to use MAGMA [9] to construct EAOQECCs from classical quaternary codes with positive net yield and without too much distance degradation. Consider the following parity check matrix H 4 of a [15;10;4] quaternary code: H 4 = 0 B B B B B B B B B B B @ 1 0 0 0 1 1 ! 2 0 1 ! 2 0 ! ! 2 1 0 0 1 0 0 1 0 ! ! 2 1 ! 0 0 1 ! 1 0 0 1 0 ! ! 2 1 ! 1 0 0 ! 1 ! 2 ! 0 0 0 1 1 ! 2 0 1 ! 2 ! 0 ! 2 1 0 ! 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 C C C C C C C C C C C A ; (60) 71 where f0;1;!;! 2 g are elements of GF(4) that satisfy: 1 + ! + ! 2 = 0 and ! 3 = 1. This quaternary code has the largest minimum weight among all known [n = 15;k = 10] linear quaternary codes. By the construction given in [13], this code gives a corresponding [[15;9;4;c=4]] EAQECC with the stabilizers given in Table 8. S E I I Y I Z X Y Z Y I I Z Y X Z I Y I I Y I Z X Y Z I I Y Z Y I Z Y I I X Z X X X I Z X I I I I X I Y Z X Y X I I Y X Z Y I I I I I I I I I I Z I I I I I I I I I I I I I I Y I I I I I Z Z Z X I Y I Y I I Z Z Z I I Y Y Y Z I X I X I I Y Y Y I S I Z Z Y I Z Y X X Y Z I Y Z Z I Y Y X I Y X Z Z X Y I X Y Y I Table 8: Stabilizer generators of the [[15;9;4;c = 4]] EAQECC derived from the classical code given by Eq. (60). The size ofS E is equal to 2 2c . The entanglement subgroupS E of this EAQECC has c = 4 symplectic pairs. Our goal is to construct an EAOQECC from this EAQECC such that the power of error correction is largely retained, but the amount of entanglement needed is reduced. In this example, the choice of which symplectic pair is removed strongly a®ects the distancedoftheresultingEAOQECC.ByusingMAGMAtoperformarandomsearch ofallthepossiblesympleticpairsinS E ,andthenputtingthemintothegaugesubgroup S G ,wecanobtaina[[15;9;3;c=3;r =1]]EAOQECCwithstabilizersgiveninTable9. The distance is reduced by one, which still retains the ability to correct all one-qubit errors; the amount of entanglement needed is reduced by one ebit; and we gain some 72 extra power of passive error correction, due to the subsystem structure inside the code space, given by the gauge subgroupS G . S E I I Y I Z X Y Z Y I I Z Y X Z I Y I I Y I Z X Y Z I I Y Z Y I Z Y I I X Z X X X I Z X I I I I X I Y Z X Y X I I Y X Z Y I I I I I I I I I I Z I I I I I I I I I I I I I I Y I I I I S G I Z Z Z X I Y I Y I I Z Z Z I I Y Y Y Z I X I X I I Y Y Y I S I X X Z I X Z Y Y Z X I Z X X I Z Z Y I Z Y X X Y Z I Y Z Z I Table 9: Stabilizer generators of the [[15;9;3;c = 3;r = 1]] EAOQECC derived from the EAQECC given by Table 8. The size of S E and S G is equal to 2 2c and 2 2r , respectively. 6.5 Discussion We have shown a very general quantum error correction scheme that combines two extensions of standard stabilizer codes. This scheme includes the advantages of both entanglement-assisted and operator quantum error correction. In addition to presenting the formal theory of EAOQECCs, we have given several examples of code construction. The methods of constructing OQECCs from standard QECCs can be applied directly to the construction of EAOQECCs from EAQECCs. We can also construct EAOQECCs directly from classical linear codes. We also show that, by exploring the structure of the symplectic subgroup, we can construct versatile classes of EAOQECCs with varying powers of passive versus 73 active error correction. Starting with good classical codes, this entanglement-assisted operator formalism can be used to construct quantum codes tailored to the needs of particular applications. 74 Chapter 7: Quantum quasi-cyclic low-density parity-check codes 7.1 Classical low-density parity-check codes Given a binary parity check matrix H, its density is de¯ned to be the ratio of the number of \1" entries to the total number of entries in H. When the density is less than 1 2 , we call such code \low-density parity-check (LDPC) code". LDPC codes were ¯rst proposed by Gallager [27] in the early 1960s, and were rediscovered [45, 19, 43] in the 90s. It has been shown that these codes can achieve a remarkable performance that is very close to the Shannon limit. Sometimes, they perform even better [42] than their main competitors, the Turbo codes. These two families of codes are called modern codes. A LDPC code is regular, if its parity check matrix H has ¯xed weight for columns and rows; otherwise, it is irregular. A (J;L)-regular LDPC code is de¯ned to be the null space of a Boolean parity check matrix H with the following properties: (1) each column consists of J \ones" (each column has weight J); (2) each row consists of L \ones" (each row has weight L); (3) both J and L are small compared to the length of the code n and the number of rows in H. We de¯ne a cycle in H to be of length 2s if there is an ordered list of 2s matrix elements such that: (1) all 2s elements of H are equal to 1; (2) successive elements in the list are obtained by alternately changing the row or column only (i.e., two 75 consecutive elements will have either the same row and di®erent columns, or the same column and di®erent rows); (3) the positions of all the 2s matrix elements are distinct, except the ¯rst and last ones. We call the cycle of the shortest length the girth of the code. Several methods of constructing good families of regular LDPC codes have been proposed [43, 36, 26]. However, probably the easiest method is based on circulant per- mutation matrices [26], which was inspired by Gallager's original LDPC construction. In the following, we will ¯rst review several relevant properties of binary circulant matrices, and then show the construction of this type of classical LDPC codes using circulant matrices. 7.1.1 Properties of binary circulant matrices Let M be an r£r circulant matrix over F 2 . We can uniquely associate with M a polynomial M(X) with coe±cients given by entries of the ¯rst row of M. If c = (c 0 ;c 1 ;¢¢¢ ;c r¡1 ) is the ¯rst row of the circulant matrix M, then M(X)=c 0 +c 1 X +c 2 X 2 +¢¢¢+c r¡1 X r¡1 : (61) Adding or multiplying two circulant matrices is equivalent to adding or multiplying their associated polynomials modulo X r ¡1. We now give some useful properties of these matrices and polynomials. Proposition 13. The set of binary circulant matrices of size r £ r forms a ring isomorphic to the ring of polynomials of degree less than r: F 2 [X]=hX r ¡1i. Lemma 5. Let M(X) be the polynomial associated with the r£ r binary circulant matrix M. If gcd(M(X);X r ¡1)=K(X), and the degree of K(X) is k, then the rank of M is r¡k. 76 Proof. Let L(X) = (X r ¡1)=K(X), and let b2 (Z 2 ) r be the coe±cient vector asso- ciated with L(X). Since the degree of L(X) is r¡k, b i = 0 for i > r¡k. It follows that L(X)M(X)=0 mod (X r ¡1): (62) Ifr i is the i-th row of M, then (62) gives the following k linearly dependent equations: b 0 r 0 +b 1 r 1 +¢¢¢+b r¡k r r¡k =0 b 0 r 1 +b 1 r 2 +¢¢¢+b r¡k r r¡k+1 =0 . . . b 0 r k¡1 +b 1 r k +¢¢¢+b r¡k r r¡1 =0: (63) Thesetfr r¡k ;¢¢¢ ;r r¡1 gcanthereforebeexpressedaslinearcombinationsoffr 0 ;¢¢¢ ;r r¡k¡1 g, and the rank of M is r¡k. Theorem 11. Let r = p¢q, and let c = (c 0 ;c 1 ;¢¢¢ ;c r¡1 ) be the ¯rst row of an r£r circulant matrix M. If c i is 1 only when i=0 mod p, then rank(M)=p. Proof. LetM(X)= P q¡1 i=0 X pi bethepolynomialassociatedwithM,withdegreer¡p. SinceM(X)j(X r ¡1), thedegreeofK(X)=gcd(M(X);X r ¡1)=M(X)isalsor¡p. Therefore, by lemma 5, the rank of M is p. Theorem 12. Let r = p¢q, and let c = (c 0 ;c 1 ;¢¢¢ ;c r¡1 ) be the ¯rst row of an r£r circulant matrix M. If c i is 1 only when i<p, then rank(M)=r¡p+1. Proof. In this case, M(X)=1+X +¢¢¢X p¡1 has degree p¡1. Since M(X)jX r ¡1, again by lemma 5 the rank of M is r¡p+1. Corollary 1. Let r = p¢q, and let c = (c 0 ;c 1 ;¢¢¢ ;c r¡1 ) be the ¯rst row of an r£r circulant matrix M such that the weight of c is p. If M(X)j(X r ¡1), then the rank · of M is lower-bounded by r¡p+1. 77 Proof. Since the weight of c is p, the lowest possible degree of M(X) is p¡1. Then by the method of Theorem 12, the rank · is at least r¡p+1. 7.1.2 Classical quasi-cyclic LDPC codes De¯nition 2. A binary linear code C(H) of length n = r¢L is called a quasi-cyclic (QC)codewithperiodr ifanycodewordwhichiscyclicallyright-shiftedbyr positionsis againacodeword. Suchacodecanberepresentedbyaparity-checkmatrixH consisting of r£r blocks, each of which is an (in general di®erent) r£r circulant matrix. BytheisomorphismmentionedinProp.13, wecanassociatewitheachquasi-cyclic parity-check matrix H 2 F Jr£Lr 2 a J £ L polynomial parity-check matrix H(X) = [h j;l (X)] j2[J];l2[L] where h j;l (X) is the polynomial, as de¯ned in Eq. (61), representing the r£r circulant submatrix of H, and the notation [J]:=f1;2;¢¢¢ ;Jg. Generally, there are two ways of constructing (J;L)-regular QC-LDPC by using circulant matrices [54]: De¯nition 3. We say that a QC-LDPC code is Type-I if it is given by a polyno- mial parity-check matrix H(X) with all monomials. We say that a QC-LDPC code is Type-II if it is given by a polynomial parity-check matrix H(X) with either binomials, monomials, or zero. 7.1.2.1 Type-I QC-LDPC To give an example, let r = 16, J = 3, and L = 8. The following polynomial parity check matrix H(X)= 2 6 6 6 6 4 X X X X X X X X X 2 X 5 X 3 X 5 X 2 X 5 X 3 X 5 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 3 7 7 7 7 5 (64) 78 gives a Type-I (3;8)-regular QC-LDPC code of length n = 16¢8 = 128. Later on, we will also expressH(X) by its exponent matrix H E . For example, the exponent matrix of (64) is H E = 2 6 6 6 6 4 1 1 1 1 1 1 1 1 2 5 3 5 2 5 3 5 2 3 4 5 6 7 8 9 3 7 7 7 7 5 : (65) The di®erence of arbitrary two rows of the exponent matrix H E is de¯ned as d ij =c i ¡c j =((c i;k ¡c j;k )mod r) k2[L] ; (66) wherec i is the i-th row of H E and r is the size of the circulant matrix. We then have d 21 = (1;4;2;4;1;4;2;4) d 31 = (1;2;3;4;5;6;7;8) d 32 = (0;14;1;0;4;2;5;4): We call an integer sequence d = (d 0 ;d 1 ;¢¢¢ ;d L¡1 ) multiplicity even if each entry appears an even number of times. For example, d 21 is multiplicity even, but d 32 is not, since only 0 and 4 appear an even number of times. We call d multiplicity free if no entry is repeated; for example, d 31 . A simple necessary condition for Type-I (J;L)-regular QC-LDPC codes to give girth g ¸ 6 is given in [26]. However, a stronger result (both su±cient and necessary condition) is shown in [30]. We state these theorems from [30] without proof. Theorem13. AType-IQC-LDPCcodeC(H E )isdual-containingifandonlyifc i ¡c j is multiplicity even for all i and j, wherec i is the i-th row of the exponent matrix H E . Theorem 14. A necessary and su±cient condition for a Type-I QC-LDPC code C(H E ) to have girth g¸6 is c i ¡c j to be multiplicity free for all i and j. 79 Theorem 15. There is no dual-containing Type-I QC-LDPC having girth g¸6. 7.1.2.2 Type-II QC-LDPC Take r =16, J =3, and L=4. The following is an example of a Type-II (3,4)-regular QC-LDPC code: H(X)= 2 6 6 6 6 4 X +X 4 0 X 7 +X 10 0 X 5 X 6 X 11 X 12 0 X 2 +X 9 0 X 7 +X 13 3 7 7 7 7 5 : (67) The exponent matrix of (67) is H E = 2 6 6 6 6 4 (1;4) 1 (7;10) 1 5 6 11 12 1 (2;9) 1 (7;13) 3 7 7 7 7 5 : (68) Here we denote X 1 =0. The di®erence of two arbitrary rows of H E is de¯ned similarly to (66) with the following additional rules: (1) if for some entry c i;k is 1, then the di®erence of c i;k and other arbitrary term is again1; (2) if the entries c i;k and c j;k are both binomial, then the di®erence of c i;k and c j;k contains four terms. In this example, we have d 21 = ((4;1);1;(4;1);1) d 31 = (1;1;1;1) d 32 = (1;(12;3);1;(11;1)) d 11 = ((0;3;13;0);1;(0;3;13;0);1) d 22 = (0;0;0;0) d 33 = (1;(0;9;7;0);1;(0;10;6;0)): 80 The de¯nition of multiplicity even and multiplicity free is the same except that we do not take1 into account. For example,d 32 is multiplicity free, since there is no pairwiththesameentryexcept1. UnlikeType-IQC-LDPCcodeswhosed ii isalways the zero vector, d ii of Type-II QC-LDPC codes can have non-zero entries. Therefore it is possible to have cycles of length 4 in a single layer if d ii is not multiplicity free. Each layer is said to be a set of rows of size r in the original parity check matrix H that corresponds to the row of H E . For example, d 11 is multiplicity even, therefore the ¯rst layer of this Type-II regular QC-LDPC parity check matrix contains 4-cycles. In the following, we will generalize theorems 13-14 given in the previous section to include the Type-II QC-LDPC case. Theorem16. C(H E ) is a dual-containing Type-II regular QC-LDPC code if and only if c i ¡c j is multiplicity even for all i and j. Proof. LetH(X)=[h j;l (X)] j2[J];l2[L] bethepolynomialparitycheckmatrixassociated with a Type-II (J;L)-regular QC-LDPC parity check matrix H. Denote the transpose of H(X) by H(X) T =[h t l;j (X)] l2[L];j2[J] , and we have h t l;j (X)= 8 > > > > > > < > > > > > > : 0 if h j;l (X)=0 X r¡k if h j;l (X)=X k X r¡k 1 +X r¡k 2 if h j;l (X)=X k 1 +X k 2 : (69) Let ^ H(X)=H(X)H(X) T , and let the (i;j)-th component of ^ H(X) be ^ h i;j (X). Then ^ h i;j (X)= X l2[L] h i;l (X)h t l;j (X): (70) Thecondition thatc i ¡c j ismultiplicityevenimplies that ^ h i;j (X)=0 moduloX r ¡1, and vice versa. 81 Theorem 17. A necessary and su±cient condition for a Type-II regular QC-LDPC code C(H E ) to have girth g¸6 is that c i ¡c j be multiplicity free for all i and j. Proof. The condition that c i ¡c j is multiplicity free for all i and j guarantees that there is no 4-cycle between layer i and layer j, and vice versa. Theorem 18. There is no dual-containing QC-LDPC having girth g¸6. Proof. This proof follows directly from theorem 16 and theorem 17. If the Type- II regular QC-LDPC code is dual-containing, then by theorem 16, c i ¡c j must be multiplicity even for all i and j. However, theorem 17 says that this QC-LDPC must contain cycles of length 4. 7.1.3 Iterative decoding algorithm There are various methods for decoding classical LDPC codes [36]. Among them, sum-product algorithm (SPA) decoding [43] provides the best trade-o® between error- correction performance and decoding complexity. Before leaving this section, we will review this SPA decoding procedure for classical LDPC codes. It turns out that the same SPA decoding algorithm can be used in the quantum case to decode the error syndromes e®ectively. Let s;r 2 (Z 2 ) n be the encoded signal and the received signal, respectively, such that hH;si = 0 T ; (71) r = s+n; (72) where n 2 (Z 2 ) n is the noise vector introduced by the binary symmetric channel, and H is the parity check matrix. The decoder's task is to infer s based on the received signal r and the knowledge of the noise n. The optimal decoder, also known 82 as the maximally likelihood decoder, returns the encoded signal s that maximizes the posterior probability P(sjr)= P(rjs)P(s) P(r) : (73) It is known that this optimal decoding is an NP-complete problem [7]. If we assume that the prior probability ofs is uniform, and the noisen is indepen- dentofs, thenitfollowsthatestimatingtheencodedsignalsisthesameasestimating the noise n. This is because once n is known, then the encoded signal is s=r+n: We can further reduce the decoding problem to the task of ¯nding the most probable noise vector n based on the error syndrome vector z since z T =hH;ni=hH;ri: (74) Next,wewillformallyintroducethesum-productalgorithm,alsoknownasa\belief propagation algorithm" [49]. Assume the parity check matrix H is of size m£n. The decodingproblemisto¯ndanoisevectorn(giventhatnisindependentofs)satisfying hH;ni=z T : The elements fn i g, i = 1;2;¢¢¢ ;n, are referred as bits, while the elements fz j g, j = 1;2;¢¢¢ ;m, are referred as checks. Togetherfn i g andfz j g form a belief network, and the network of checks and bits are a bipartite graph: bits only connect to checks and vice versa. The algorithm presented below follows closely from [43]. The goal is to compute the marginal posterior probability P(n i jz;H) for each i. Denote the set of bits that participate in check j by N(j)=fi:H ji =1g. Denote the set of checks in which bit i 83 participatesby M(i)=fj :H ji =1g. Denoteaset N(j)withbitiexcludedby N(j)ni. De¯ne the quantity q x ji to be the probability that bit i of n has the value x2f0;1g, given the probability obtained via checks other than check j, fr x j 0 i : j 0 2 M(i)njg. De¯ne the quantity r x ji to be the probability of check j being satis¯ed if bit i of n is considered ¯xed at the value x and the other bits have a separable distribution given by the probabilities fq ji 0 : i 0 2 N(j)nig. These two quantities q ij and r ij associated with each nonzero element of H are iteratively updated, and would produce the exact marginal posterior probabilities of all the bits after a ¯xed number of iterations if the bipartite graph de¯ned by the matrix H contained no cycle [49]. When cycles exist, the algorithm produces inaccurate probabilities. However, the correct marginal probabilities are not necessary as long as the decoding is correct. Initialization. Denotethepriorprobabilitythatbitn i =0byp 0 i ,andp 1 i =1¡p 0 i . Set p 1 i = f, where f is the crossover probability of binary symmetric channel. The variables q 0 ji and q 1 ji are initialized to the value p 0 i and p 1 i when H ji =1. Horizontal step.The procedure in the horizontal step of the algorithm is to run throughthechecksj andcomputeforeachi2 N(j)twoprobabilitiesr 0 ji andr 1 ji ,where r 0 ji = X n i 0:i 0 2N(j)ni 2 4 P ¡ z j jn i =0;fn i 0 :i 0 2 N(j)nig ¢ Y i 0 2N(j)ni q n i 0 ji 0 3 5 ; (75) r 1 ji = X n i 0:i 0 2N(j)ni 2 4 P ¡ z j jn i =1;fn i 0 :i 0 2 N(j)nig ¢ Y i 0 2N(j)ni q n i 0 ji 0 3 5 : (76) Thequantityr 0 ji istheprobabilityoftheobservedvalueofz j whenn i isassumedtobe 0, given that the other bits fn i 0 : i 0 2 N(j)nig have a separable distribution given by the probabilitiesfq 0 ji 0 ;q 1 ji 0 g. The quantity r 1 ji is de¯ned similarly except n i is assumed to be 1. 84 Vertical step.The procedure in the vertical step of the algorithm is to take the computed values of r 0 ji and r 1 ji and update the values of the probabilities q 0 ji and q 1 ji for each j. q 0 ji = ® ji p 0 i Y j 0 2M(i)nj r 0 j 0 i ; (77) q 1 ji = ® ji p 1 i Y j 0 2M(i)nj r 1 j 0 i ; (78) where ® ji is chosen such that q 0 ji +q 1 ji =1. DecodingThe pseudoposterior probabilities q 0 i and q 1 i are calculated after each iteration of the horizontal and vertical steps, where q 0 i = ® i p 0 i Y j2M(i) r 0 ji ; (79) q 1 i = ® i p 1 i Y j2M(i) r 1 ji : (80) Thesequantitiesareusedtocreateatentativedecoding ^ n. Ifq 1 i >0:5, ^ n i issetto1. If ^ nsatis¯eshH;^ ni=z T ,thedecodingalgorithmstops. Otherwise,thealgorithmrepeats from the horizontal step. If the number of iterations reaches some preset maximum number without successful decoding, we declare a failure. It has been shown that the performance of iterative decoding very much depends on the cycles of shortest length [57]|in particular, cycles of length 4. These shortest cycles make successive decoding iterations highly correlated, and severely limit the decoding performance. Therefore, to use SPA decoding, it is important to design codes without short cycles, especially cycles of length 4. Thesum-productdecodingalgorithmcanbedirectlyappliedtothequantumcodes constructedusingthe(generalized)CSSconstruction. ThisisbecausetheZ errorsand X errors of a CSS-type quantum code can be decoded separately. Therefore, decoding 85 the quantum errors is equivalent to using the SPA separately for each classical code in the CSS construction (though this would throw away some information about the correlations between X errors and Y errors). 7.2 Quantum low-density parity-check codes The quantum versions of low-density parity-check codes [30, 44, 17, 20] are far less studied than their classical counterparts. The main obstacle comes from the dual- containing constraint of the classical codes that are used to construct the correspond- ing quantum codes. While this constraint was not too di±cult to satisfy for relatively small codes, it is a substantial barrier to the use of highly e±cient LDPC codes. How- ever, with the entanglement-assisted formalism, such constrains can be removed, and constructing quantum LDPC codes from classical LDPC codes becomes transparent. The second obstacle to constructing quantum LDPC codes comes from the bad performance of the e±cient decoding algorithm. Though the SPA can be directly used to decode the quantum errors, the performance of SPA decoding was severely limited by the many 4-cycles in the standard quantum LDPC codes. We show in this section that using the entanglement-assisted formalism, we can completely eliminate all the 4-cyclesinthequantumLDPCcodes. WewillfocusonthequantumLDPCcodescon- structed from classical quasi-cyclic LDPC codes, and demonstrate their performance using numerical methods. 7.2.1 Quantum quasi-cyclic LDPC codes Ithasbeenshownthatanyclassicallinearcodecanbeusedtoconstructacorrespond- ing entanglement-assisted quantum error-correcting code. In the following, we will consider conditions that will give us (J;L)-regular QC- LDPCcodesC(H)withgirthg¸6andwiththerankofHH T assmallaspossible. In 86 general, ^ H(X)representsasquareHermitianmatrix ^ H withsizeJr£Jr thatcontains J 2 circulant r£r matrices represented by ^ h i;j (X) as de¯ned in (70). Next, we provide two examples to illustrate two di®erent ways of minimizing the rank of the square Hermitian matrix represented by ^ H(X). The ¯rst method is to make the matrix ^ H =HH T become a circulant matrix with a small rank. This can be achieved by choosing H(X) such that ^ h i;j (X)= ^ h i+1;j+1 (X); for i;j = 0;1;¢¢¢ ;J ¡ 2: The rank · of ^ H can then be read o® by lemma 5. If gcd( ^ H(X);X Jr ¡1) = K(X), and the degree of K(X) = k, then · = Jr¡k. Let's look at an example of this type using a classical Type-I QC-LDPC code. Take r =16, J = 3, and L = 8. The following polynomial parity check matrix H(X) gives the corresponding quantum QC-LDPC code with length 128: H(X)= 2 6 6 6 6 4 X X X X X X X X X X 2 X 3 X 4 X 5 X 6 X 7 X 8 X X 3 X 5 X 7 X 9 X 11 X 13 X 15 3 7 7 7 7 5 : (81) Then ^ h i;j (X)= 8 > > > > > > < > > > > > > : 0; i=j; P 7 k=0 X k ; i=j+1 P 7 k=0 X 2k ; i=j+2 (82) It can be easily veri¯ed that ^ H(X) represents a circulant matrix, and the polynomial associated with ^ H is ^ H(X)=X 16 Ã 7 X k=0 X k ! +X 32 Ã 7 X k=0 X 2k ! : 87 The degree of gcd( ^ H(X);X 48 ¡1) = 30, therefore by lemma 5, the number of ebits that were needed to construct the corresponding quantum code is only 18. Actually, (81) gives us a [[128;48;6;18]] EAQECC, and we will refer to this example as \ex1" later in section 7.3. The second method is to minimize the rank of each circulant matrix inside ^ H. Let the rank of the circulant matrix represented by ^ h i;j (X) be · i;j . Let the rank of ^ H be ·. Then ·· J X i=1 max j2[J] · i;j : (83) This upper bound is not tight for Type-I (J;L)-regular QC-LDPC codes when L is odd. This is because · i;i = r for every i. When L is even, we have · i;i = 0 for every i. We can obtain a tighter upper bound for · by carefully choosing the exponents of H(X) such that the degree of gcd( ^ h i;j (X);X r ¡1) is as large as possible for every i and j. Theorem 19. Given a Type-I (J;L)-regular QC-LDPC code with H(X), if L is even andgcd( ^ h i;j (X);X r ¡1)>1fori6=j, thentherank·isupperboundedbyJ(r¡L+1). Proof. Let ^ h i;j be the circulant matrix associated with the polynomial ^ h i;j (X), then the weight of the coe±cient vector of ^ h i;j is L. By Corollary 1, · i;j · r¡ L + 1. Therefore ·· J X i=1 max j2[J] · i;j ·J(r¡L+1): 88 Our second example comes from a classical Type-II QC-LDPC code. Again take r = 16, J = 3, and L = 8. The following polynomial parity check matrix H(X) gives the corresponding quantum QC-LDPC code with length 128: H(X)= 2 6 6 6 6 4 X +X 2 0 X +X 4 0 X +X 6 0 X +X 8 0 X 5 X 5 X 6 X 6 X 7 X 7 X 8 X 8 0 X +X 2 0 X +X 4 0 X +X 6 0 X +X 8 3 7 7 7 7 5 : (84) Then ^ h i;j (X)= 8 > > > > > > < > > > > > > : 0; (i;j)=(2;2);(1;3);or(3;1) P 7 k=0 X 1+2k ; (i;j)=(1;1);(3;3) P 7 k=0 X k ; (i;j)=(2;1);(2;3) (85) In this example, each layer of the matrix ^ H(X) has rank less than 9. Actually, (84) gives a [[128;48;6;18]] quantum QC-LDPC code, and we will refer to this example as \ex2" in section 7.3. 7.3 Performance In this section, we compare the performance of the QLDPC codes given in Sec. 7.2 to conventional (dual-containing) QLDPC codes that have been derived in the existing literature. The easiest way of constructing a QLDPC is the following technique, pro- posed by MacKay et al. in [44]. Take an n=2£n=2 cyclic matrix C with row weight L=2, and de¯ne H 0 =[C;C T ]: Then we delete some rows from H 0 to obtain a matrix H with m rows. It is easy to verify that H is dual-containing. Therefore by the CSS construction, we can obtain conventional QLDPC codes of length n. The advantage of this construction is that 89 the choice of n;m, and L is completely °exible; however, the column weight J is not ¯xed. We picked n = 128, m = 48, and L = 8, and called this quantum LDPC code \ex-MacKay." The second example of constructing a conventional QLDPC is described in the following theorem [30]: Theorem 20. Let P be an integer which is greater than 2 and ¾ an element ofZ ¤ P := fz : z ¡1 existsg with ord(¾) 6= jZ ¤ P j, where ord(¾) := minfm > 0j¾ m = 1g and jXj means the cardinality of a set X. If we pick any ¿ 2Z ¤ P =f1;¾;¾ 2 ;¢¢¢g, de¯ne c j;l := 8 > > < > > : ¾ ¡j+l 0·l <L=2 ¡¿¾ j¡1+l L=2·l <L d k;l := 8 > > < > > : ¿¾ ¡k¡1+l 0·l <L=2 ¡¾ k+l L=2·l <L ; and de¯ne the exponent matrix H C and H D as H C =[c j;l ] j2[J];l2[L] ; H D =[d k;l ] k2[K];l2[L] ; where L=2 = ord(¾) and 1· J;K · L=2, then H C and H D can be used to construct quantum QC-LDPC codes with girth at least 6. 90 Here,wepickthesetofparameters(J;L;P;¾;¿)tobe(3;8;15;2;3). Theexponent matrices H C and H D described in theorem 20 are H C = 2 6 6 6 6 4 1 2 4 8 6 12 9 3 8 1 2 4 12 9 3 6 4 8 1 2 9 3 6 12 3 7 7 7 7 5 (86) H D = 2 6 6 6 6 4 9 3 6 12 14 13 11 7 12 9 3 6 13 11 7 14 6 12 9 3 11 7 14 13 3 7 7 7 7 5 ; (87) and by the CSS construction, it will give a [[120;38;4]] quantum QC-LDPC code. We will call this code \ex-HI". 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 10 −3 10 −2 10 −1 10 0 Cross Over Probability of Depolarizing Channel, f x Block Error Rate Performace of Quantun LDPC Codes ex1 ex2 ex−MacKay ex−HI Figure 10: Performance of QLDPC with SPA decoding, and 100-iteration 91 We compare the performance of our examples in section 7.2.1 with these two dual- containing quantum LDPC codes in ¯gure 10. In the simulation, we assume the depo- larizing channel and use of sum-product decoding algorithm. The performances of ex1 and ex2 do not di®er much. This is not surprising, since these two codes have similar parameters. The reason that the performance of ex-MacKay is worse than our two examples is because there are so many 4-cycles in ex-MacKay. These cycles impair the decoding performance of sum-product algorithm. Our entanglement-assisted quantum QC-LDPCcodesalsooutperformthequantumQC-LDPCcodeofex-HI,sincetheclas- sical QC-LDPC codes used to construct our examples have better distance properties than the classical QC-LDPC of ex-HI. This simulation result is also consistent with our result in [13]: better classical codes give better quantum codes. Even though the parameters are not exactly the same, our codes have higher rate than the code rate of ex-HI. It is not di±cult to verify that the girth of ex1 is 6, and the girth of ex2 is 8. We numerically investigated the performance of these two examples with various numbers of iterations. According to our simulation results, the performance of ex1 and ex2 is almost the same. The result agrees with the classical result in [26] showing that the increase of girth from 6 to 8 is not of great help. The result is quite interesting since it implies that we do not need to worry about constructing QLDPC with higher girth. 7.4 Conclusions There are two advantages of Type-II QC-LDPCs over Type-I QC-LDPCs. First, ac- cordingto[54]certaincon¯gurationsofType-IIQC-LDPCcodeshavelargerminimum distance than Type-I QC-LDPC. Therefore, we can construct better quantum QC- LDPCs from classical Type-II QC-LDPC codes. Second, it seems likely that Type-II QC-LDPCs will have more °exibility in constructing quantum QC-LDPC codes with 92 small amount of pre-shared entanglement, because of the ability to insert zero subma- trices. However, further investigation of this issue is required. By using the entanglement-assisted error correction formalism, it is possible to construct EAQECCs from any classical linear code. We have shown how to do this for two classes of quasi-cyclic LDPC codes (Type-I and Type-II), and proven a number of theorems that make it possible to bound how much entanglement is required to send a code block for codes of these types. Using these results, we have been able to easily construct examples of quantum QC-LDPC codes that require only a relatively small amount of initial shared entanglement, and that perform better than previously constructed dual-containing QLDPCs. Since in general the performance of quantum codes follows directly from the performance of the classical codes used to construct them, and the evidence of our examples suggests that the iterative decoders can also be made to work e®ectively on the quantum versions of these codes, this should make possible the construction of large-scale e±cient quantum codes. 93 Bibliography [1] S.A. Aly, A. Klappenecker, and P. K. Sarvepalli. Subsystem codes, 2006. quant- ph/0610153. [2] D.Bacon. 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Abstract (if available)

Abstract In my thesis, I present a general method for studying quantum error correction codes (QECCs). This method not only provides us an intuitive way of understanding QECCs, but also leads to several extensions of standard QECCs, including the operator quantum error correction (OQECC),the entanglement-assisted quantum error correction (EAQECC). Furthermore, we can combine both OQECC and EAQECC into a unified formalism, the entanglement-assisted operator formalism. This provides great flexibility of designing QECCs for different applications. Finally, I show that the performance of quantum low-density parity-check codes will be largely improved using entanglement-assisted formalism.

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Creator Hsieh, Min-Hsiu (author)

Core Title Entanglement-assisted coding theory

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/24/2008

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

Tag entanglement,OAI-PMH Harvest,quantum error-correcting code

Language English

Advisor Brun, Todd A. (committee chair), Lidar, Daniel (committee member), Zanardi, Paolo (committee member)

Creator Email minhsiuh@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1390

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Document Type Dissertation

Rights Hsieh, Min-Hsiu

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