Structured Error Recovery for Codeword-Stabilized Quantum Codes
aa r X i v : . [ qu a n t - ph ] M a r Structured Error Recovery for Codeword-Stabilized Quantum Codes
Yunfan Li ∗ and Ilya Dumer † Department of Electrical Engineering, University of California, Riverside, CA, 92521, USA
Markus Grassl ‡ Centre for Quantum Technologies, National University of Singapore, Singapore 117543, SINGAPORE
Leonid P. Pryadko § Department of Physics & Astronomy, University of California, Riverside, CA, 92521, USA (Dated: November 4, 2018)Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correcterrors by an exhaustive search of different error patterns, similar to the way that we decode classicalnon-linear codes. For an n -qubit quantum code correcting errors on up to t qubits, this brute-forceapproach consecutively tests different errors of weight t or less, and employs a separate n -qubitmeasurement in each test. In this paper, we suggest an error grouping technique that allows tosimultaneously test large groups of errors in a single measurement. This structured error recoverytechnique exponentially reduces the number of measurements by about 3 t times. While it stillleaves exponentially many measurements for a generic CWS code, the technique is equivalent tosyndrome-based recovery for the special case of additive CWS codes. I. INTRODUCTION
Quantum computation makes it possible to achievepolynomial complexity for many classical problems thatare believed to be hard [1, 2]. To preserve coherence,quantum operations need to be protected by quantumerror correcting codes (QECCs) [3–5]. With error proba-bilities in elementary gates below a certain threshold, onecan use multiple layers of encoding (concatenation) to re-duce errors at each level and ultimately make arbitrarily-long quantum computation possible [6–14].The actual value of the threshold error probabilitystrongly depends on the assumptions of the error modeland on the chosen architecture, and presently varies from10 − % for a chain of qubits with nearest-neighbor cou-plings [15] and 0 .
7% for qubits with nearest-neighbor cou-plings in two dimensions [14], to 3% with postselection[12], or even above 10% if additional constraints on errorsare imposed [13].The quoted estimates have been made using stabilizercodes, an important class of codes which originate fromadditive quaternary codes, and have a particularly simplestructure based on Abelian groups [16, 17]. Recently, amore general class of codeword stabilized (CWS) quan-tum codes was introduced in Refs. [18–21]. This classincludes stabilizer codes, but is more directly related tonon-linear classical codes.This direct relation to classical codes is, arguably,the most important advantage of the CWS framework.Specifically, the classical code associated with a given ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
CWS quantum code has to correct certain error patternsinduced by a graph associated with the code. The graphalso determines the graph state [22] serving as a startingpoint for an encoding algorithm exploiting the structureof the classical code [20]. With the help of powerful tech-niques from the theory of classical codes, already severalnew families of non-additive codes have been discovered,including codes with parameters proven to be superior toany stabilizer code [18–20, 23–26].Both classical additive codes and additive quantumcodes can be corrected by first finding the syndrome ofa corrupted vector or quantum state, respectively, andthen looking up the corresponding error (coset leader) ina precomputed table [27]. This is not the case for non-linear codes. In fact, even the notions of a syndrome anda coset become invalid for general non-linear codes. Fur-thermore, since quantum error correction must preservethe original quantum state in all intermediate measure-ments, it is more restrictive than many classical algo-rithms. Therefore, the design of a useful CWS code mustbe complemented by an efficient quantum error correc-tion algorithm.The goal of this work is to address this important unre-solved problem for binary CWS codes. First, we designa procedure to detect an error in a narrower class, the union stabilizer (USt) codes, which possess some partialgroup structure [23, 24, 28]. Then, for a general CWScode and a set of graph-induced maps of correctable er-rors forming a group, we construct an auxiliary USt codewhich is the union of the images of the original CWScode shifted by all the elements of the group. Finally,we construct Abelian groups associated with correctableerrors located on certain index sets of qubits. The actualerror is found by first applying error-detection to locatethe index set with the relevant auxiliary USt code, thenusing a collection of smaller USt codes to pinpoint theerror in the group. Since we process large groups of er-rors simultaneously, we make a significant reduction ofthe number of measurements compared with the bruteforce error correction for non-linear (quantum or classi-cal) codes.More precisely, we consider an arbitrary distance- d CWS code (( n, K, d )) that uses n qubits to encode aHilbert space of dimension K and can correct all t -qubiterrors, where t = ⌊ ( d − / ⌋ . In Sec. II we give a briefoverview of the notations and relevant facts from the the-ory of quantum error correction. Then in Sec. III, weconstruct a reference recovery algorithm that deals witherrors individually. This algorithm requires up to B ( n, t )measurements, where B ( n, t ) ≡ t X i =0 (cid:0) ni (cid:1) i (1)is the total number of errors of size up to t (this boundis tight for non-degenerate codes). Each of these mea-surements requires up to n + K O ( n ) two-qubit gates.In order to eventually reduce the overall complexity, weconsider the special case of USt codes in Sec. IV. Herewe design an error-detecting measurement for a USt codewith a translation set of size K that requires O ( Kn )two-qubit gates to identify a single error. Our errorgrouping technique presented in Sec. V utilizes such ameasurement to check for several errors at once. Foradditive CWS codes the technique reduces to stabilizer-based recovery [Sec. V C]. In the case of generic CWScodes [Sec. V D], we can simultaneously check for all er-rors located on size- t qubit clusters; graph-induced mapsof these errors form groups of size up to 2 t . Searchingfor errors in blocks of this size requires up to (cid:0) nt (cid:1) − t addi-tional measurements to locate the error inside the group.In Sec. VI we discuss the obtained results and outlinethe directions of further study. Finally, in Appendix Awe consider some details of the structure of corruptedspaces for the codes discussed in this work.Note that some of the reported results have been pre-viously announced in Ref. [29]. II. BACKGROUNDA. Notations
Throughout the paper, H = C denotes the complexHilbert space that consists of all possible states α | i + β | i of a single qubit, where α, β ∈ C and | α | + | β | = 1 . Correspondingly, we use the space H ⊗ n = ( C ) ⊗ n = C n to represent any n -qubit state. Also, P n ≡ i m { I, X, Y, Z } ⊗ n , m = 0 , . . . , n +2 , where X , Y , Z arethe usual (Hermitian) Pauli matrices and I is the iden-tity matrix. The members of this group are called Pauli operators; the operators in (2) with m = 0 form a basisof the vector space that consists of all operators actingon n -qubit states. The weight wgt( E ) of a Pauli operator E is the number of terms in the tensor product (2) whichare not a scalar multiple of identity. Up to an overallphase, a Pauli operator can be specified in terms of twobinary strings, v and u , U = Z v X u ≡ Z v Z v . . . Z v n n X u X u . . . X u n n . Hermitian operators in P n have eigenvalues equal to 1or −
1. Generally, unitary operators (which can be out-side of the Pauli group) which are also Hermitian, i.e., alleigenvalues are ±
1, will be particularly important in thediscussion of measurements. We will call these operators measurement operators . Indeed, for such an operator M ,a measurement gives a Boolean outcome and can be con-structed with the help of a single ancilla, two Hadamardgates, and a controlled M gate [16] (see Fig. 1). The al-gebra of measurement operators is related to the algebraof projection operators discussed in [30], but the formeroperators, being unitary, are more convenient in circuits. | i H • H | i P M | ψ i + | i Q M | ψ i| ψ i / n M | a i | b i FIG. 1: Measurement of an observable M with all eigenval-ues ±
1. The first Hadamard gate prepares the ancilla inthe state ( | i + | i ) | ψ i / √
2, hence | a i = ( | i + | i ) | ψ i / √ M gate returns | b i = C M | a i = ( | i | ψ i + | i M | ψ i ) / √
2. The second Hadamard gate finishes the in-complete measurement, | c i = | i P M | ψ i + | i Q M | ψ i , wherewe used the projector identities (3). If the outcome of the an-cilla measurement is | i , the result is the projection of the ini-tial n -qubit state | ψ i onto the +1 eigenspace of M ( P M | ψ i ),otherwise it is the projection onto the − M ( Q M | ψ i ). For an input state | i | ψ i with ancilla in the state | i , the circuit returns | i P M | ψ i + | i Q M | ψ i . A measurement of an observable defined by a Pauli op-erator M will be also called Pauli measurement [31]. Forlack of a better term, other measurements will be called non-Pauli ; typically the corresponding circuits are muchmore complicated than those for Pauli measurements.We say that a state | ψ i ∈ H ⊗ n is stabilized (anti-stabilized) by a measurement operator M if M | ψ i = | ψ i ( M | ψ i = −| ψ i ). The corresponding projectors onto thepositive and negative eigenspace are denoted by P M and Q M , respectively; they satisfy the identities M = P M − Q M = 2 P M − − Q M . (3)We say that a space Q is stabilized by a set of operators M if each vector in Q is stabilized by each operator in M .We use P ( M ) to denote the maximum space stabilized by M , and P ⊥ ( M ) to denote the corresponding orthogonalcomplement. For a set M of measurement operators,each state in P ⊥ ( M ) is anti-stabilized by some operatorin M .When discussing complexity, we will quote the two-qubit complexity which just counts the total number oftwo-qubit gates. Thus, we ignore any communicationoverhead, as well as any overhead associated with single-qubit gates. For example, the complexity of the mea-surement in Fig. 1 is just that of the controlled- M gateoperating on n + 1 qubits [39]. For all circuits we discuss,the total number of gates (single- and two-qubit) is of thesame order in n as the two-qubit complexity. B. General QECCs
A general n -qubit quantum code Q encoding K quan-tum states is a K -dimensional subspace of the Hilbertspace H ⊗ n . Let {| i i} Ki =1 be an orthonormal basis of the K -dimensional code Q and let E ⊂ P n be some set ofPauli errors. The overall phase of an error [ i m in Eq. (2)]is irrelevant and will be largely ignored. The code detectsall errors E ∈ E if and only if [2, 16] h j | E | i i = C E δ ij (4)where C E only depends on the error E , but is indepen-dent of the basis vectors [32]. The code has distance d ifit can detect all Pauli errors of weight ( d − d . Such a code is denoted by (( n, K, d )).The necessary and sufficient condition for correctingerrors in E is that all non-trivial combinations of errorsfrom E are detectable. This gives [4, 5] h j | E † E | i i = C E ,E δ ij , (5)where E , E ∈ E and, again, C E ,E is the same for allbasis states i , j . A distance- d code corrects all errors ofweight s such that 2 s ≤ d −
1, that is, s ≤ t ≡ ⌊ ( d − / ⌋ .The code is non-degenerate if linearly independent er-rors from E produce corrupted spaces E ( Q ) ≡ { E | ψ i : | ψ i ∈ Q} whose intersection is trivial (equals to { } );otherwise the code is degenerate [17]. A stricter condi-tion that the code is pure (with respect to E ) requiresthat the corrupted spaces E ( Q ) and E ( Q ) be mutuallyorthogonal for all linearly independent correctable errors E , E ∈ E .For a degenerate code, we call a pair of correctableerrors E , E ∈ E mutually-degenerate if the corruptedspaces E ( Q ) and E ( Q ) coincide. Such errors belong tothe same degeneracy class . For recovery, one only needsto identify the degeneracy class of the error that hap-pened. The operators like E † E , connecting mutually-degenerate correctable errors E and E , have no effecton the code and can be ignored.As shown in Appendix A, for all codes discussed in thiswork, any two correctable errors E , E yield corruptedspaces E ( Q ), E ( Q ) that are either identical or orthog-onal. Then, errors from different degeneracy classes take the code to corrupted spaces that are mutually orthog-onal. Also, for these codes, a non-degenerate code is al-ways pure. In terms of the error correction condition (5),we have C E ,E = 0 for errors E , E in different de-generacy classes and C E ,E = 0 for errors in the samedegeneracy class. C. Stabilizer codes
Stabilizer codes [16] are a well known family of quan-tum error-correcting codes that are analogous to clas-sical linear codes. An [[ n, k, d ]] stabilizer code maps a2 k -dimensional k -qubit state space into a 2 k -dimensionalsubspace of an n -qubit state space.The code is defined as the space stabilized by anAbelian subgroup of the n -qubit Pauli group, S ⊂ P n ,with n − k Hermitian generators, S = h G , . . . , G n − k i .For such a space to exist, it is necessary that − / ∈ S .The Abelian group S is called the stabilizer of Q . Ex-plicitly, Q ≡ {| ψ i : S | ψ i = | ψ i , ∀ S ∈ S } . (6)The code Q is stabilized by S iff it is stabilized by all n − k generators G i . In other words, it is an intersectionof subspaces stabilized by G i , Q = n − k \ i =1 P ( G i ) . (7)The normalizer of S in P n , denoted as N , is thegroup of all Pauli operators U which fix S under con-jugation ( U † SU = S for all S ∈ S ). The term nor-malizer reflects the fact that these operators commutewith S [16]. It is possible to construct 2 k logical op-erators X j , Z j , j = 1 , . . . , k , with the usual commuta-tion relations, that together with the generators of S generate the normalizer (modulo an overall phase fac-tor) [16, 33]. The ( n − k ) generators G i of the stabi-lizer, along with the k operators Z j , generate a subgroup S ≡ h G , . . . , G n − k , Z , . . . Z k i of P n which becomes amaximal Abelian subgroup when including the generator i
1. The group S stabilizes a unique state | s i ≡ | . . . i . (8)The operators X j acting on | s i generate the basis of thecode, | c . . . c k i ≡ X c . . . X c k k | s i . (9)Generally, each detectable Pauli error E j ∈ P n thatacts non-trivially on the code anti-commutes with at leastone generator G i , and errors from different degeneracyclasses anti-commute with different subsets of S . Wecan thus identify a degeneracy class by the set of genera-tors G i which anti-commute with it. The corrupted codespace E j ( Q ) = { E j | ψ i : | ψ i ∈ Q} is anti-stabilized bythose generators G i that anti-commute with E j . Indeed, G i ( E j | ψ i ) = ( − E j G i ) | ψ i = − E j | ψ i , which means that the measurement G i of gives −
1. Bymeasuring all G i , we get the syndrome that consists of n − k numbers 1 or −
1. There are in total 2 n − k possiblesyndromes identifying different error degeneracy classes,including the trivial error E i ( Q ) and E j ( Q ) are mutu-ally orthogonal or identical. The whole 2 n -dimensional n -qubit state space H ⊗ n is thus divided into L ≡ n − k orthogonal 2 k -dimensional subspaces Q j ≡ E j ( Q ), H ⊗ n = L − M j =0 Q j , Q i ⊥ Q j for i = j. (10)The representatives of different error classes can be cho-sen to commute with each other and with the logicaloperations X i . These representatives form an Abeliangroup [34] T ≡ h g , . . . g n − k i whose generators g i can bechosen to anti-commute with only one of the generatorsof the stabilizer each, g i G j = ( − δ ij G j g i (this followsfrom Proposition 10.4 in Ref. [2]). Altogether, the gen-erators { g , . . . , g n − k } can be regarded as a set of Paulioperators forming the basis of the cosets of the normal-izer N of the code Q in P n . Example 1.
The [[5 , , G = XZZXI, G = IXZZX,G = XIXZZ, G = ZXIXZ. (11)For this code, the logical operators can be taken as X = ZZZZZ, Z = XXXXX. (12)A basis of the code space is (up to normalization) | ¯0 i = Y i =1 ( G i ) | i , | ¯1 i = X | ¯0 i . By construction, both basis states are stabilized by thegenerators G i . The corresponding stabilizer group is S = h G , . . . , G i . The group of equivalence classesof correctable errors is generated by the representatives(note the mixed notation, e.g., Z Z ≡ ZIZII ) g = Z Z , g = ZZZZI, g = ZZIZZ, g = Z Z . (13)The g j are chosen to commute with the logical operatorsand also to satisfy G i g j = ( − δ ij g j G i . Note that theoperators of weight one forming the correctable error setdo not by themselves form a group. The generators g i canbe used to map correctable errors to the correspondinggroup elements with the same syndrome. This gives, e.g., Z → g g = Z X , X → g = Z Z , Y → g g g = Z Z Z X = Z Z . (cid:3) D. Union Stabilizer codes
The decomposition (10) can be viewed as a construc-tive definition of the Abelian group T ≡ h g , . . . g n − k i ofall translations of the original stabilizer code Q in P n .(In the following, this stabilizer code is denoted Q .) Anytwo non-equivalent translations t i , t j ∈ T belong to dif-ferent cosets of the normalizer N of the code Q in P n and, therefore, yield mutually orthogonal shifts t i ( Q ) ⊥ t j ( Q ) , i = j. (14)A union stabilizer code [23, 24, 28] (USt) is a direct sum Q ≡ K M i =1 t i ( Q ) (15)of K shifts of the code Q by non-equivalent translations t i ∈ T , i = 1 , . . . K .The basis of the code defined by (15) is the union of thesets of basis vectors of all t i ( Q ). As a result, the dimen-sion of Q is K k . This code is then denoted (( n, K k , d )),where d is the distance of the new code. Generally, thisdistance does not exceed the distance of the original codewith respect to the same error set, d ( Q ) ≤ d ( Q ). How-ever, if the code Q is degenerate and the original code Q is one-dimensional, this need not be true. E. Graph states
The unique state | s i defined by Eq. (8) itself formsa single-state stabilizer code [[ n, , d ′ ]]. Its stabilizer S ≡ h G , . . . , G n − k , Z , . . . Z k i has exactly n mutuallycommuting generators. Note that for stabilizer states wefollow the convention that the distance d ′ is given by theminimum weight of the non-trivial elements of the stabi-lizer S . Generally, such a state stabilized by an Abeliansubgroup in P n of order 2 n (which does not include − stabilizer state [16].A graph state [22] is a stabilizer state of a group whose n mutually commuting generators S i are defined by a(simple) n -vertex graph G with an adjacency matrix R ∈{ , } n × n . Specifically, the generators are S i ≡ X i Z R i Z R i . . . Z R in n ≡ X i Z r i , (16)where r i , i = 1 , . . . n denotes the i th row of the adjacencymatrix R . A graph state can be efficiently generated[20, 35] by first initializing every qubit in the state | + i =( | i + | i ) / √ | i ),and then using a controlled- Z gate C Zi,j on every pair( i, j ) of qubits corresponding to an edge of the graph
G ≡ ( V, E ), | s i = Y ( i,j ) ∈ E C Zi,j H ⊗ n | i n ≡ U G | i n , (17)where | i n is a state with all n qubits in state | i .Any stabilizer state is locally Clifford-equivalent (LC-equivalent) to a graph state [36–38]. That is, any sta-bilizer state can be transformed into a graph state byindividual discrete rotations of the qubits. This definesthe canonical form of a stabilizer state. FIG. 2: Ring graphs with (a) 3 vertices and (b) 5 vertices.
Example 2.
Consider the ring graph for n = 3[Fig. 2(a)] which defines the stabilizer generators S = XZZ , S = ZXZ , S = ZZX . The corresponding sta-bilizer state | s i is an equal superposition of all 2 states[result of the Hadamard gates in Eq. (17)], taken withpositive or negative signs depending on the number ofpairs of ones at positions connected by the edges of thegraph [result of the gates C Zi,j = ( − ij ]. In the followingexpressions we omitted normalization for clarity: | s i = | i + | i + | i − | i + | i − | i − | i − | i (18)= S | s i = | i − | i + | i + | i− | i − | i + | i − | i . (19) F. CWS codes and their standard form
Codeword stabilized (CWS) codes [19, 20] represent ageneral class of non-additive quantum codes that also in-cludes stabilizer codes. They can be viewed as USt codesoriginating from a stabilizer state. Any CWS code is lo-cally Clifford-equivalent to a CWS code which originatesfrom a graph state [20].Specifically, a CWS code (( n, K, d )) in standard form [20] is defined by a graph G with n vertices and a classi-cal code C containing K binary codewords c i . The orig-inating stabilizer state is the graph state | s i defined by G , and the codeword operators [translations in Eq. (15)]have the form W i ≡ Z c i , i = 1 , . . . K . For a CWS codein standard form we use the notation Q = ( G , C ).An important observation made in Ref. [20] is that asingle qubit error X , Z , or Y acting on a code state | w i i ≡ W i | s i (20)is equivalent (up to a phase) to an error composed onlyof Z operators. This establishes the following mappingbetween multi-qubit errors and classical binary errors.First, let E i = Z { , } i X i be an error acting on the i thqubit of W j | s i . Then we see that E i W j | s i = E i W j S i | s i = ± ( E i S i ) W j | s i , where the term E i S i = Z { , } i Z r i consists only of opera-tors Z . The general mapping of an error E = i m Z v X u from the error set E ⊂ P n to a classical error vector in { , } n is defined asCl G ( E ) ≡ v ⊕ n M l =1 u l r l , (21)where u l is the l th component of the vector u . We willrefer to both the binary vector Cl G ( E ) and the operator Z Cl G ( E ) as the graph image of the Pauli operator E .Theorem 3 from Ref. [20] establishes the correspon-dence between the error-correcting properties of a quan-tum code Q and those of the corresponding classical code C . It states that a CWS code (in standard form) given bya graph G and word operators { W l = Z c } c ∈C detects er-rors in the set E if and only if the classical code C detectserrors from the set Cl G ( E ), and for each E ∈ E ,either Cl G ( E ) = , (22)or, for each i , Z c i E = EZ c i . (23)The code Q is non-degenerate (and also pure, see Ap-pendix A) iff condition (22) is satisfied for all errors in E [20, 21]. For a degenerate code, condition (23) needs tobe ensured for errors that do not satisfy Eq. (22).The beauty of the CWS construction is that, for agiven code in standard form, we no longer need to worryabout possible degeneracies. The classical error patternsinduced by the function Cl G ( · ) also separate the errorsinto corresponding degeneracy classes [20, 21].The CWS framework is general enough to also includeall stabilizer codes [20]. Specifically, a stabilizer code[[ n, k, d ]] with the stabilizer S = h G , . . . G n − k i and log-ical operators Z i , X i , corresponds to a CWS code withthe stabilizer S = h G , . . . G n − k , Z , . . . , Z k i (24)and the codeword operator set W = h X , . . . , X k i form-ing a group of size K = 2 k . Generally, an LC transfor-mation is required to obtain standard form of this code.Conversely, an additive CWS code Q = (( n, K, d ))where the codeword operators form an Abelian group(in which case K = 2 k with integer k ) is a stabilizer code[[ n, k, d ]] [20]. In Sec. V B we show that the n − k genera-tors G i of the stabilizer can be obtained from the graph-state generators S i [Eq. (16)] by a symplectic Gram-Schmidt orthogonalization procedure [33] which has noeffect on the codeword operators. Example 3.
Consider a non-degenerate CWS code((5 , , n = 5 generators of the sta-bilizer S are S = XZIIZ and its four cyclic permuta-tions. The corresponding stabilizer state has a structuresimilar to Eq. (18), but with more terms. Word operatorshave the form Z c i , with the classical codewords c = 00000 , c = 01101 , c = 10110 , c = 01011 , c = 10101 , c = 11010 . (25) (cid:3) Example 4.
To express the [[5 , , , , S i of the stabilizer S = h G , G , G , G , Z i [Eq. (24)] tocontain only one X operator each. We obtain S = G G Z = IZXZI and its four cyclic permutations.This does not require any qubit rotations due to aslightly unconventional choice of the logical operators inEq. (12). The corresponding graph is the ring [20], seeFig. 2(b). The codeword operators are W = (cid:8) I, X (cid:9) ,which by Eq. (12) correspond to classical binary code-words { , } . Note that the error map inducedby the graph is different from the mapping to groupelements in Example 1; in particular, Z Cl G ( Z ) = Z , Z Cl G ( X ) = Z Z , Z Cl G ( Y ) = Z Z Z . (cid:3) III. GENERIC RECOVERY FOR CWS CODES
In this section we construct a generic recovery algo-rithm which can be adapted to any non-additive code.To our best knowledge, such an algorithm has not beenexplicitly constructed before.The basic idea is to construct a non-Pauli measurementoperator M Q = 2 P Q −
1, where P Q ≡ P Ki =1 | w i ih w i | isthe projector onto the code Q spanned by the orthonor-mal basis {| w i i} Ki =1 . The measurement operator is fur-ther decomposed using the identity − M Q = − K X i | w i ih w i | = K Y i ( − | w i ih w i | ) . (26)We use the graph state encoding unitary U G fromEq. (17) and the following decomposition [20] of thestandard-form basis of the CWS code in terms of thegraph state | s i and the classical states | c i i : | w i i = Z c i | s i = Z c i U G | i n = U G X c i | i n = U G | c i i . The measurement operator M Q is rewritten as a product M Q = − U G M C U †G , M C ≡ K Y i =1 M c i (27)where the measurement operator M c i ≡ − | c i ih c i | = X c i (cid:0) − | i n h | n (cid:1) X c i , (28)stabilizes the orthogonal complement of the classicalstate | c i i . The components of the binary vector c i arethe respective complements of c i , c i,j = 1 ⊕ c i,j . Theoperator in the parentheses in Eq. (28) is the ( n − Z gate (the Z -operator is applied to the n -thqubit only if all the remaining qubits are in state | i ).It can also be represented as n -qubit controlled phasegate C θn ≡ exp( iθP | i n ) with θ = π , where the opera-tor P | i n projects onto the state | i n . This can be fur-ther decomposed as a product of two Hadamard gates and an ( n − n ≥ n −
4) three-qubit Toffoli gates [39] and therefore haslinear complexity in n . With no ancillas, the complexityof the ( n − O ( n ) [39]. C π ≡ •• Z = •• H (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) H FIG. 3: Decomposition of n -qubit controlled- Z gate C Zn interms of ( n − n -qubit Toffoli gate)for n = 3. The corresponding ancilla-based measurement for M c i can be constructed with the help of two Hadamard gates[Fig. 1] by adding an extra control to each C πn gate. In-deed, this correlates the state | i of the ancilla with M c i acting on the n qubits, and the state | i of the ancillawith
1. When constructing the measurement for theproduct of the operators M c i , it is sufficient to use onlyone ancilla, since for each basis state | w i i only one ofthese operators acts non-trivially. The classical part ofthe overall measurement circuit without the graph stateencoding U G is shown in Fig. 4.The complexity of measuring M Q [Eq. (27)] then be-comes K times the complexity of ( n + 1)-qubit Toffoligate for measuring each M c i , plus the complexity of theencoding circuit U G and its inverse U †G , which is at most n . Overall, for large n , the measurement complexity isno larger than n + K O ( n ), or (1 + K ) O ( n ) for a circuitwithout additional ancillas.We would like to emphasize that so far we have onlyconstructed the measurement for error detection . Actual error correction for a non-additive code in this schemeinvolves constructing measurements M E ≡ EM Q E † forall corrupted subspaces corresponding to different degen-eracy classes given by different Cl G ( E ). This relies on theorthogonality of the corrupted subspaces, see AppendixA. For a general t -error correcting code, the number ofthese measurements can reach the same exponential or-der B ( n, t ) as the number of correctable errors in Eq. (1).For non-degenerate codes, we cannot do better using thismethod.Note that the measurement circuit derived in this sec-tion first decodes the quantum information, then per-forms the measurement for the classical code, and finallyre-encodes the quantum state. IV. MEASUREMENTS FOR UST CODES
In this section we construct a quantum circuit for themeasurement operator M Q of a USt code (( n, K k , d )).To this end, we define the logical combinations of non-Pauli measurements in agreement with analogous combi-nations defined in Ref. [30] for the projection operators, | i H Z Z · · ·
Z H | i P M C | ψ i + | i Q M C | ψ i| ψ i / n X c • X c X c • X c · · · X c K • X c K | {z } | {z } | {z } C M C M C M K FIG. 4: Generic measurement of the classical part M C of the CWS code stabilizer M Q [Eq. (27)] uses K ( n +1)-qubit controlled- Z gates. Here X c i indicates that a single-qubit gate X c i,j is applied at the j th qubit, j = 1 , . . . n , and C M i is the controlled- M c i gate [see Eq. (28)]. This can be further simplified by combining the neighboring X ci,j -gates and replacing the controlled- Z gatesby ( n + 1)-qubit Toffoli gates as in Fig. 3. | i H • H X • X H • H | i| i H • H X • X H • H | i| i (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) | f i| ψ i / n M | a i M | b i | c i M M FIG. 5: Measurement M ∧ M by performing logical AND operation on two ancillas. We use the notations P i ≡ P M i , Q i ≡ − P i , i = 0 , P P = P P . The circuit returns | f i = | i P P | ψ i + | i (1 − P P ) | ψ i . Intermediateresults are: | a i = | i P | ψ i + | i Q | ψ i , | b i = | i P P | ψ i + | i P Q | ψ i + | i Q P | ψ i + | i Q Q | ψ i , | c i = | i P P | ψ i + | i P Q | ψ i + | i Q P | ψ i + | i Q Q | ψ i ; the last two groups of gates return the first two ancillas tothe state | i . | i H • H X • • •
X H • H | i| i H • H (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) | i P P | ψ i + | i ( − P P ) | ψ i| ψ i / n M | a i M | b i M FIG. 6: Simplified measurement for M ∧ M ; notations as in Fig. 5. Intermediate results (cf. Fig. 1) are: | a i = | i P | ψ i + | i Q | ψ i , | b i = | i ( | i P + | i Q ) P | ψ i + | i Q | ψ i . The effect of the last block is to select the component with the firstancilla in the state | i ; this requires that the projectors P and P commute. The result is equivalent to | i P M ∧ M | ψ i + | i (1 − P M ∧ M ) | ψ i . and construct the circuits for logical combinations AND[Figs. 5, 6] and XOR [Figs. 7, 8]. We use these circuits toconstruct the measurement for M Q with complexity notexceeding 2 K ( n + 1)( n − k ). A. Algebra of measurements
Logical AND : Given two commuting measurement op-erators M and M , let M ∧ M denote the measurementoperator that stabilizes all states in the subspace P ( M ∧ M ) ≡ P ( M ) ∩ P ( M ) . (29)The output of the measurement M ∧ M is identical tothe logical AND operation performed on the output ofmeasurements M and M . This measurement can be im-plemented by the circuit in Fig. 5. Here the first two an-cillas are entangled with the two measurement outcomes;the third ancilla is flipped only if both ancillas are in the | i state, which gives the combination | i P M ∧ M | ψ i .The projector onto the positive eigenspace of M ∧ M satisfies the identity P M ∧ M = P M P M . (30)This identity can be used to obtain a simplified circuitwhich only uses two ancillas, see Fig. 6, with the price oftwo additional controlled-Hadamard gates [Fig. 9].The circuits in Figs. 5 and 6 can be generalized toperform the measurement corresponding to the logicalAND of ℓ > M ∧ M ∧ M = M ∧ ( M ∧ M ). The generalization of the simplified circuitin Fig. 6 requires only two ancillas for any ℓ >
1. Thecorresponding complexity is 2 ℓ − M gate, plus 2 ℓ − M i are n -qubitPauli operators, the overall complexity with two ancillasis (2 ℓ − n + 1). Logical XOR : In analogy to the logical “exclusive OR”,we define the symmetric difference A △ B △ C . . . of vectorspaces A , B , C , . . . as the vector space formed by thebasis vectors that belong to an odd number of the original | i H • H • H • H | i| i H • H (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) | f i| ψ i / n M M M FIG. 7: Measurement M ⊞ M by performing logical XOR gate on the two ancillas and subsequent recovery of the first ancilla.Notations as in Fig. 5. The final result is | f i = | i ( P Q + P Q ) | ψ i + | i ( P P + Q Q ) | ψ i . | i H • • H | f i| ψ i / n M M FIG. 8: Simplified measurement for M ⊞ M . Notations asin Fig. 5. The result (cf. Fig. 1) | f i = | i ( Q P + P Q ) | ψ i + | i ( P P + Q Q ) | ψ i is equivalent to | i P M ⊞ M | ψ i + | i (1 − P M ⊞ M ) | ψ i . • H = • R − π/ y Z R π/ y FIG. 9: Implementation of the controlled- H gate based onthe identity H = exp( − iY π/ Z exp( iY π/ vector spaces. For two vector spaces A △ B ≡ ( A ∩ B ⊥ ) ⊕ ( B ∩ A ⊥ ) . (31)This operation is obviously associative, A △ ( B △ C ) =( A △ B ) △ C . For two commuting measurement operators M , M , let M ⊞ M be the measurement operator thatstabilizes the subspace P ( M ) △ P ( M ). Explicitly, P ( M ⊞ M ) ≡ P ( M ) △ P ( M )= (cid:2) P ( M ) ∩ P ⊥ ( M ) (cid:3) ⊕ (cid:2) P ( M ) ∩ P ⊥ ( M ) (cid:3) . (32)The output of measuring M ⊞ M is identical to thelogical XOR operation performed on the outputs of mea-surements M and M . The corresponding measurementcan be implemented by combining the two ancillas witha CNOT gate [Fig. 7].To simplify this measurement, we show that M ⊞ M = − M M . Indeed, Eq. (32) implies that for theprojection operators P i ≡ P M i , i = 0 , P M ⊞ M = P ( − P ) + P ( − P ) . The corresponding measurement operator is factorizedwith the help of the projector identities (3), M ⊞ M = 2[ P ( − P ) + P ( − P )] − − (2 P − P −
1) = − M M . This implies that P ( M ⊞ M ) = P ⊥ ( M M ). In otherwords, the measurement of M ⊞ M can be implementedsimply as an (inverted) concatenation of two measure-ments, see Fig. 8. The same circuit can also be obtained from that in Fig. 7 by a sequence of circuit simplifications(not shown).The circuit in Fig. 8 is immediately generalized to acombination of more than two measurements, M ⊞ . . . ⊞ M ℓ = ( − l − M . . . M ℓ . The corresponding complexityfor computing the XOR of ℓ measurements is simply thesum of the individual complexities, implying that thisconcatenation has no overhead. B. Error detection for union stabilizer codes
A USt code Q = (( n, K k , d )) is a direct sum (15) of K mutually orthogonal subspaces obtained by translat-ing the originating stabilizer code Q = [[ n, k, d ]]. Formutually orthogonal subspaces A ⊥ B , we have A ⊂ B ⊥ and B ⊂ A ⊥ , and the direct sum is the same as thesymmetric difference (31), A ⊕ B = A △ B .In turn, the stabilizer code Q is an intersection of thesubspaces stabilized by the generators G i of the stabi-lizer, see Eq. (7). The translated subspaces t j ( Q ) arestabilized by the Pauli operators M i,j = t j G i t † j . We cantherefore decompose the USt code Q as Q = K M j =1 (cid:20) n − k \ i =1 P ( M i,j ) (cid:21) = K △ j =1 (cid:20) n − k \ i =1 P ( M i,j ) (cid:21) . (33)This gives the decomposition of the measurement opera-tor M Q whose positive eigenspace is the code Q as M Q = K ⊞ j =1 " n − k ^ i =1 M i,j . (34)Recall that the complexity of each of the K logicalAND operations is [2( n − k ) − n + 1). No additionaloverhead is required to form the logical XOR of the re-sults. Thus, we obtain the following Theorem 1
Error detection for a USt code of length n and dimension K k , formed by a translation set of size K > , has complexity at most K ( n − k )( n + 1) . Note that in the special case of a CWS code ( k = 0),the prefactor of K is quadratic in n whereas the corre-sponding prefactor obtained in Sec. III is linear in n . Thereason is that in Eq. (27) the graph encoding circuit Q G with complexity O ( n ) is used only twice, and the pro-jections onto the classical states have linear complexity.In Eq. (34) we are using K projections onto basis statesof the quantum code. The advantage of the more com-plex measurement constructed in this section is that itdoes not involve having unprotected decoded qubits forthe entire duration of the measurement. V. STRUCTURED MEASUREMENT FOR CWSCODESA. Grouping correctable errors
Recall from Section II C that for stabilizer codes therepresentatives of the error degeneracy classes form anAbelian group whose generators are in one-to-one corre-spondence with the generators of the stabilizer. Measur-ing the n − k generators of the stabilizer of a stabilizercode [[ n, k, d ]] uniquely identifies the degeneracy class ofthe error.In this section we establish a similar structure for CWScodes. First, for any subset D ⊂ E of correctable errorsof a quantum code Q , we define the set E D of unrelated errors which do not fall in the same degeneracy class withany error from D . The formal definition E D ≡ { E : E ∈ E ∧ C E,E = 0 , ∀ E ∈ D} (35)is based on the general error correction condition (5) andthe orthogonality of corrupted spaces, see Appendix A.When errors in E are non-degenerate, the definition (35)is equivalent to the set difference, E \ D . In the generalcase, since we do not distinguish between mutually de-generate errors, E D can be thought of as the differencebetween the sets of degeneracy classes in E and in D .Definition (35) implies that the subspaces D ( Q ) ≡ M E ∈D E ( Q )and E D ( Q ), defined analogously, are mutually orthog-onal. Moreover, if the elements of the set D form agroup D ≡ D , the subspace D ( Q ) is also orthogo-nal to E D [ D ( Q )] [see Eq. (40) below]. In other words, Q D ≡ D ( Q ) can be viewed as a quantum code whichdetects errors from E D .This observation, together with the error-detectionmeasurement for USt codes constructed in the previoussection, forms the basis of our error grouping technique.We prove the following Theorem 2
For a CWS code Q = ( G , C ) in standardform and a group D formed by graph images of somecorrectable errors in E , the code Q D ≡ D ( Q ) is a UStcode which detects all errors in E D . Proof . First, we show that the subspace D ( Q ) is a UStcode. The corresponding set of basis vectors is D ( {| w i , . . . , | w K i} ) ≡ [ e α ∈ D K [ i =1 { e α | w i i} . (36) These vectors are mutually orthogonal, h w i | e † β e α | w j i = 0 , ∀ i, j ≤ K, e α , e β ∈ D , e α = e β , (37)since every element e α = Z Cl G ( E α ) of the group D is arepresentative of a separate error degeneracy class. Fur-ther, the group D is Abelian, and its elements commutewith the codeword generators W i = Z c i , c i ∈ C . There-fore, using Eq. (20), we can rearrange the set (36) as D ( {| w i , . . . , | w K i} ) = K [ i =1 W i (cid:0) { e α | s i} e α ∈ D (cid:1) . (38)The set in the parentheses on the right hand side is a basisof the additive CWS code Q D formed by the group D acting on the graph state | s i . Then, we can write thesubspace D ( Q ) explicitly as a USt code [cf. Eq. (15)] D ( Q ) = K M i =1 W i ( Q D ) , (39)where the translations are given by the set of codewordoperators W ≡ { W i } Ki =1 of the original code Q . Orthog-onality condition (14) is ensured by Eq. (37).Second, we check the error-detection condition (4) forthe code (39). Explicitly, for an error E ∈ E D , and forthe orthogonal basis states e α W i | s i , h s | W i † e † α Ee β W j | s i = ± h w i | Ee † α e β | w j i = 0 (40)for all α, β, i, j , according to Eqs. (5), (35) and the groupproperty of D . (cid:3) Now, to correct errors in groups, we just have to find asuitable decomposition of the graph images of the originalerror set into a collection of groups, Cl G ( E ) = S j D j , andperform individual error-detection measurements for theauxiliary codes Q D j until the group containing the erroris identified.To find an error within a group D ≡ h g , . . . , g m i with m generators, we can try all m subgroups of D with onegenerator missing. More specifically, for a generator g l weconsider the subgroup D ( l ) = h g , . . . , g l − , g l +1 , . . . , g m i and perform error detection for the code Q ( l ) D ≡ D ( l ) ( Q ).After completing m measurements, we obtain a repre-sentative of the actual error class. This is the product ofall generators g l for which the corresponding code Q ( l ) D detected an error. B. Complexity of a combined measurement
To actually carry out the discussed program, we needto construct the n − m generators G i of the stabilizer ofthe code Q D . The generators have to commute with the m generators g α in the group D .This can be done with the Gram-Schmidt (GS) orthog-onalization [33] of the graph-state generators S i [Eq. (16)]0with respect to the generators g α . As a result, we obtainthe orthogonalized set of independent generators S ′ i suchthat g α S ′ i = ( − δ iα S ′ i g α . We can take the last n − m of the obtained generators as the generators of the stabi-lizer, G i = S ′ i + m , i = 1 , . . . , n − m .The orthogonalization procedure is guaranteed to pro-duce exactly m generators S ′ α anti-commuting with thecorresponding errors g α , α = 1 , . . . , m . Indeed, the GSorthogonalization procedure can be viewed as a sequenceof row operations applied to the original n × m binarymatrix B with the elements b iα which define the originalcommutation relation, S i g α = ( − b iα g α S i . (41)The generator g α anti-commutes with at least one op-erator in S if and only if the α -th column of B is notan all-zero column. Then all m generators are indepen-dent (no generator can be expressed as a product of someothers) if and only if B has full column rank.By this explicit construction, the generators G i of thestabilizer of the auxiliary code Q D [Eq. (39)] are Pauli op-erators in the original graph-state basis. The complexityof each error-detection measurement M ( Q D ) is thereforegiven by Theorem 1. C. Additive CWS codes
The procedure described above appears to be ex-tremely tedious, much more complicated than the syn-drome measurement for a stabilizer code. However, itturns out that for stabilizer codes this is no more diffi-cult than the regular syndrome-based error correction.Indeed, for a stabilizer code Q = [[ n, k, d ]], the degen-eracy classes for all correctable errors form a group ofall translations of the code, D ≡ T = h g , . . . , g n − k i ,with n − k generators. To locate the error, we justhave to go over all n − k USt codes D ( l ) ( Q ) gener-ated by the subgroups of D with the generator g l miss-ing. Since the originating code Q is a stabilizer code,the USt codes D ( l ) ( Q ) are actually stabilizer codes, en-coding ˜ k = k + ( n − k −
1) = n − E D ( l ) = h g l i . The corresponding sta-bilizers S ( l ) have only one generator each. The neces-sary measurements are just independent measurementsof n − k Pauli operators, the same as needed to measurethe syndrome. Moreover, if the error representatives g l , l = 1 , . . . , n − k , are chosen to satisfy the orthogonalitycondition G i g l = ( − δ il g l G i as in Example 1, the op-erators to be measured are the original generators G i ofthe stabilizer, and the corresponding measurement is justthe syndrome measurement. Example 5.
Consider the additive code ((5 , , , , T = h Z , Z , Z , Z i . This group contains all error degener-acy classes, D = T . With the addition of the logicaloperator X ≡ ZZZZZ , these can generate the entire5-qubit Hilbert space H ⊗ from the graph state | s i ; wehave T ( Q ) = H ⊗ .Indeed, if we form a measurement as for a genericCWS code, we first obtain the stabilizer of the auxil-iary code Q D [Eq. (39)] which in this case has only onegenerator, S = h S i , where S = ZIIZX , see Exam-ple 4. Translating this code with the set (in this case,group) W = { I, X } of codeword operators, we get theauxiliary USt code W ( Q D ) as the union of the positiveeigenspaces of the operators [see Eq. (33)], M , = S , M , = XS X † = − S , which is the entire Hilbert space, W ( Q D ) = D ( Q ) = H ⊗ , as expected.To locate the error within the group D with m = 4 gen-erators, we form a set of smaller codes Q ( l ) D ≡ D ( l ) ( Q ), l = 1 , . . . ,
4, where the group D ( l ) is obtained from D by removing the l -th generator. The corresponding sta-bilizers are S (1) = h S , S i , S (2) = h S , S i , etc. Thematrices M ( l ) i,j of conjugated generators have the form,e.g., M (1) i,j = (cid:18) S , − S S , − S (cid:19) , M (2) i,j = (cid:18) S , − S S , − S (cid:19) , . . . The code Q ( l ) D is formed as the union of the commonpositive eigenspaces of the operators in the columns of thematrix M ( l ) i,j . Clearly, these codes can be more compactlyintroduced as positive eigenspaces of the operators ˜ G l = S l S , l = 1 , . . . ,
4. Such a simplification only happenswhen the original code Q is additive. While the operators˜ G i are different from the stabilizer generators in Eq. (11),they generate the same stabilizer S = h ˜ G , . . . , ˜ G i ofthe original code Q . It is also easy to check that the sameprocedure gives the original generators G i [Eq. (11)] if westart with the error representatives (13). (cid:3) D. Generic CWS codes
Now consider the case of a generic CWS code Q =(( n, K, d )). Without analyzing the graph structure, itis impossible to tell whether there is any set of clas-sical images of correctable errors that forms a largegroup. However, since we know its minimum distance,we know that the code can correct errors located on t = ⌊ ( d − / ⌋ qubits. All errors located on a givenset of qubits form a group. Therefore, by taking an in-dex set A ⊂ { , . . . , n } of s ≤ t different qubit positions,we can ensure that the corresponding correctable errors { E j } s j =1 form a group with 2 s independent generators.The corresponding graph images Cl G ( E j ) ∈ D A obey thesame multiplication table, but they are not necessarily in-dependent. As a result, the Abelian group D A generally1has m ≤ s generators. Since all group elements corre-spond to correctable errors, the conditions of Theorem 2are satisfied.Overall, to locate an error of weight t or less, we needto iterate over each (but the last one) of the (cid:0) nt (cid:1) index setsof size t and perform the error-detecting measurementsin the corresponding USt codes E A ( Q ) until the indexset with the error is found. This requires up to (cid:0) nt (cid:1) − m ≤ t measurements. Thiscan be summarized as the following Theorem 3
A CWS code of distance d can correct errorsof weight up to t = ⌊ ( d − / ⌋ by performing at most N ( n, t ) ≡ (cid:18) nt (cid:19) + 2 t − measurements. For any length n ≥
3, this scheme reduces the total num-ber (1) of error patterns by a factor B ( n, t ) N ( n, t ) ≥ n + 1 n + 1 , if t = 1 , t , if t > . (43) Example 6.
Consider the ((5 , , d = 2is too small to correct arbitrary errors, we can correctan error located at a given qubit. Assume that an er-ror may have happened on the second qubit. Then weonly need to check the index set A = { } . The errors { , X , Y , Z } located in A form a group with genera-tors { X , Z } ; the corresponding group of classical er-ror patterns induced by the ring graph in Fig. 2(b) is D A = h Z Z , Z i . The three generators G i of the sta-bilizer of the originating USt code Q D A can be chosenas, e.g., G = S S = XIXZZ , G = S = IIZXZ , G = S = ZIIZX . Using the classical codewords (25)for the translation operators t j = Z c j , we obtain theconjugated generators M i,j = t j G i t † j M i,j = G , − G , G , G , G , − G G , G , − G , − G , G , − G G , − G , G , − G , − G , G . (44)According to Eq. (33), the auxiliary code Q D A is a directsum of the common positive eigenspaces of the operatorsin the six columns of the matrix (44).To locate the actual error in this 24-dimensional space,we consider the two subgroups D (1) = h ZIZII i and D (2) = h IZIII i of D A . The stabilizers of the correspond-ing auxiliary codes Q (1) D A and Q (2) D A can be obtained byadding G (1)4 = S = ZXZII and G (2)4 = S = IZXZI ,respectively; this adds one of the rows M (1)4 ,j = ( S , − S , S , − S , S , − S ) , (45) M (2)4 ,j = ( S , − S , − S , S , − S , S ) (46) to the matrix (44). The original code Q is the inter-section of the codes Q (1) D A and Q (2) D A ; the corrupted space X ( Q ) is located in Q (1) D A , but not in Q (2) D A , while, e.g.,the corrupted space Y ( Q ) is located in Q D A , but not in Q (1) D A or Q (2) D A . (cid:3) E. General USt codes
A similar procedure can be carried over for a generalUSt code (( n, K k , d )), with the only difference that thedefinitions of the groups D and the auxiliary codes Q D [Eq. (39)] should also include the k generators of the orig-inating stabilizer code Q [Sec. II D]. Overall, the com-plexity of error recovery for a generic USt code can besummarized by the following Theorem 4
Consider any t -error correcting USt code oflength n and dimension K k , with the translation set ofsize K . Then this code can correct errors using (cid:0) nt (cid:1) +2 t − or fewer measurements, each of which has complexity K ( n + 1)( n − k − or less. F. Error correction beyond t For additive quantum codes, the syndrome measure-ment locates all error equivalence classes, not only thosewith “coset leaders” of weight s < d/
2. The same couldbe achieved with a series of clustered measurements, byfirst going over all clusters of weight s = t , then s = t + 1,etc. This ensures that the first located error has thesmallest weight. In contrast, such a procedure will likelyfail for a non-additive code where the corrupted spaces E ( Q ) and E ( Q ) can partially overlap if either E or E is non-correctable. For instance, the measurement inExample 6 may destroy the coherent superposition if theactual error (e.g., Z i , i = 2) was not on the second qubit.Therefore, if no error was detected after (cid:0) nt (cid:1) − t indexset. With a non-additive CWS code, generally we haveto do a separate measurement for each additional cor-rectable error of weight s > t . VI. CONCLUSIONS
For generic CWS and USt codes, we constructed a structured recovery algorithm which uses a single non-Pauli measurement to check for groups of errors locatedon clusters of t qubits. Unfortunately, for a generic CWScode with large K and large distance, both the number ofmeasurements and the corresponding complexity are ex-ponentially large, in spite of the exponential accelerationalready achieved by the combined measurement.To be deployed, error-correction must be comple-mented with some fault-tolerant scheme for elementary2gates. It is an important open question whether a fault-tolerant version of our measurement circuits can be con-structed for non-additive CWS codes. It is clear, how-ever, that such a procedure would not help for any CWScode that needs an exponential number of gates for recov-ery. Therefore, the most important question is whetherthis design can be simplified further.We first note that the group-based recovery [see The-orem 2] is likely as efficient as it can possibly be, illus-trated by the example of additive codes in Sec. V C wherethis procedure is shown to be equivalent to syndrome-based recovery. Also, while it is possible that for fixed K the complexity estimate of Theorem 1 can be reducedin terms of n (e.g., by reusing ancillas with measuredstabilizer values), we think that for a generic code thecomplexity is linear in K .However, specific families of CWS codes might be rep-resented as unions of just a few stabilizer codes whichmight be mutually equivalent as in Eq. (15), or non-equivalent [28]. The corresponding measurement com-plexity for error detection would then be dramatically re-duced. Examples are given by the quantum codes derivedfrom the classical non-linear Goethals and Preparatacodes [23, 24].Another possibility is that for particular codes, largersets of correctable errors may form groups. Indeed, wesaw that for an additive code (( n, k , d )), all error de-generacy classes form a large group of size 2 n − k whichmay include some errors of weight well beyond t . Sucha group also exists for a CWS code which is a subcodeof an additive code. There could be interesting familiesof non-additive CWS codes which admit groups of cor-rectable errors of size beyond 2 t . For such a code, thenumber of measurements required for recovery could beadditionally reduced. Acknowledgment
This research was supported in part by the NSF grantNo. 0622242. Centre for Quantum Technologies is a Re-search Centre of Excellence funded by Ministry of Edu-cation and National Research Foundation of Singapore.The authors are grateful to Bei Zeng for the detailed ex-planation of the CWS graph construction.
Appendix A: Orthogonality of corrupted spaces
As discussed in Sec. II B, for a general non-additivequantum code Q and two linearly independent cor-rectable errors, the corrupted spaces E ( Q ) and E ( Q )may be neither identical nor orthogonal [17]. However,for CWS and USt codes it is almost self-evident thatwhen E ( Q ) and E ( Q ) do not coinside, they are mutu-ally orthogonal. This orthogonality is inherited from theoriginating stabilizer code Q . In particular, in someprevious publications (e.g., Ref. [20]) orthogonality is implied in the discussion of degenerate errors for CWScodes. However, to our knowledge, it was never explic-itly discussed for CWS or USt codes. Since our recoveryalgorithms for CWS and USt codes rely heavily on thisorthogonality, we give here an explicit proof.First, consider a stabilizer code Q . For any Paulioperator E ≡ E † E , there are three possibilities: ( i ) E is proportional to a member of the stabilizer group, E ≡ γS , where S ∈ S and γ = i m , m = 0 , . . . ,
3, ( ii ) E isin the code normalizer N but is linearly independent ofany member of the stabilizer group, and ( iii ) E is outsideof the normalizer, E N .Case ( i ) implies that the space E ( Q ) is identical to thecode Q ; the errors E and E are mutually degenerate.Indeed, for any basis vector | i i , the action of the error E | i i = γS | i i = γ | i i just introduces a common phase γ ;any vector | ψ i ∈ Q is mapped to γ | ψ i and hence norecovery is needed.In case ( ii ) the operator E also maps Q to itself, butno longer identically. Therefore, at least one of the twoerrors E , E is not correctable. Indeed, in this case wecan decompose E (see Sec. II C) as the product of anelement S ∈ S in the stabilizer and logical operators,i.e., E ≡ i m SX a Z b , where m = 0 , . . . , S ∈ S acts trivially on the code,the logical operator specified by the binary-vectors a , b is non-trivial, wgt( a ) + wgt( b ) = 0. Using the explicitbasis (9), it is easy to check that the error-correctioncondition (5) is not satisfied for the operators E , E .Finally, in case ( iii ) the spaces E ( Q ) and Q are mu-tually orthogonal. Indeed, since E is outside of the codenormalizer N , there is an element of the stabilizer group S ∈ S that does not commute with E . Therefore, forany two states in the code, | ϕ i , | ψ i ∈ Q , we can write h ϕ | E | ψ i = h ϕ | ES | ψ i = − h ϕ | SE | ψ i = − h ϕ | E | ψ i , (A1)which gives h ϕ | E | ψ i = 0, and the spaces Q and E ( Q )[also, E ( Q ) and E ( Q )] are mutually orthogonal.Now, consider the same three cases for a USt code (15)derived from Q . In case ( i ) the code is mapped to itself, E ( Q ) = Q . The operator E acts trivially on the code(and the errors E , E are mutually degenerate) if E either commutes (A2) or anti-commutes (A3) with theentire set of translations generating the code:( Et j = t j E, j = 1 , . . . , K ) (A2)or ( Et j = − t j E, j = 1 , . . . , K ) . (A3)If neither of these conditions is satisfied, the error-correction condition (5) is violated. This is easily checkedusing the basis | j, i i ≡ t j | i i .Similarly, in case ( ii ), the code is mapped to itself, E ( Q ) = Q , but the error-correction condition (5) cannotbe satisfied.Finally, in case ( iii ), the space E ( Q ) is either orthogo-nal to Q , or the error correction condition is not satisfied.The latter is true if E is proportional to an element in3one of the cosets t † j t j ′ S , where j = j ′ , 1 ≤ j, j ′ ≤ K .Then the inner product h j, i | E | j ′ , i i 6 = 0, i = 1 , . . . , k ,which contradicts the error-correction condition (5). Inthe other case, namely, when E is linearly independent ofany operator of the form t † j t j ′ S , j, j ′ = 1 , . . . , K , S ∈ S , E must be a member of a different coset t α S of thestabilizer S of the code Q in P n . This implies orthog-onality: h j, i | E | j ′ , i ′ i ≡ h i | t † j E t j ′ | i ′ i = i m h i | t † j t j ′ t α | i ′ i = 0 , where m = 0 , . . . , Q and the Pauli operators E , E , the spaces E ( Q ) and E ( Q ) either coincide, or areorthogonal. Since CWS codes can be regarded as UStcodes originating from a one-dimensional stabilizer code Q , [Sec. II F], the same is also true for any CWS code. [1] P. W. Shor, Algorithms for quantum computation: dis-crete logarithms and factoring, in Proc. 35th IEEEFound. Comp. Sci. , pp. 124–134, 1994.[2] M. A. Nielsen and I. L. Chuang,
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