Decoding of Quantum Data-Syndrome Codes via Belief Propagation
DDecoding of Quantum Data-Syndrome Codes viaBelief Propagation
Kao-Yueh Kuo ∗‡ , I-Chun Chern †§ , and Ching-Yi Lai ∗†‡§∗ Institute of Communications Engineering and † Department of Electrical and Computer Engineering,National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan ‡ Institute of Communications Engineering and § Department of Electrical and Computer Engineering,National Chiao Tung University, Hsinchu 30010, Taiwan, {kykuo, ethanc.eed06, cylai}@nctu.edu.tw
Abstract —Quantum error correction is necessary to protectlogical quantum states and operations. However, no meaningfuldata protection can be made when the syndrome extractionis erroneous due to faulty measurement gates. Quantum data-syndrome (DS) codes are designed to protect the data qubits andsyndrome bits concurrently. In this paper, we propose an efficientdecoding algorithm for quantum DS codes with sparse checkmatrices. Based on a refined belief propagation (BP) decodingfor stabilizer codes, we propose a DS-BP algorithm to handlethe quaternary quantum data errors and binary syndrome biterrors. Moreover, a sparse quantum code may inherently beable to handle minor syndrome errors so that fewer redundantsyndrome measurements are necessary. We demonstrate this withsimulations on a quantum hypergraph-product code.
I. I
NTRODUCTION
Quantum error-correction codes are indispensable in fault-tolerant quantum computation (FTQC) and quantum commu-nication [1]–[4]. Quantum stabilizer codes are an importantclass of quantum codes [5]–[7] since they allow simple encod-ing and decoding processes similar to classical error-correctingcodes. This decoding process is analogous to the classicalsyndrome-based decoding. However, quantum operations areinevitably faulty, making robust syndrome extraction difficult.Using imperfect syndrome measurements, the outcome willbe inaccurate and it may also corrupt the data qubits. As aconsequence, we will be led into using a wrong syndrome,which will in turn provide us with an incorrect decoding resultfor error recovery.Conventionally, an error recovery operation is chosen byrepeated syndrome measurements, followed by a certain deci-sion strategy [1]. For topological codes, the minimum-weightperfect-matching decoder is able to locate a likely error withhigh complexity [8]. For the ease of analysis, a simple errormodel is considered, where each qubit independently suffersa Pauli error and each syndrome bit independently suffersa bit-flip error. It has been shown that quantum stabilizercodes could be capable of correcting data errors and syndromeerrors simultaneously, known as quantum data-syndrome (DS)codes [9]–[12]. Recently, there has been a proposal put for-ward to construct quantum convolutional DS codes with a gen-eralized Viterbi decoding algorithm [13]. Following Bombin’sseminal work on the single-shot fault-tolerant quantum error-correction to handle measurement errors, other approacheshave also been proposed for topological codes [14]–[16]. We would like to develop a decoding algorithm with lowcomplexity for FTQC by first studying the simple error model.In this paper we study the decoding of sparse-graph quantumcodes [17]–[19] with independent data and syndrome errorsusing belief propagation (BP). BP has been shown to bepowerful and efficient for classical decoding [20], [21], ar-tificial intelligence decision [22], and quantum decoding [18],[23], [24]. Since the Pauli errors { I, X, Y, Z } are quaternarybut a syndrome bit-flip error is binary, a hybrid quaternary-binary nature of the DS codes is created, which makes theirdecoding complicated. Previously BP for quantum codes (BP )has been refined to pass only scalar messages because ofthe fact that the error syndromes for a stabilizer code arebinary [24]. Accordingly, we propose a decoding algorithm(called DS-BP ) for quantum DS codes, showing that thequaternary data error information and binary syndrome bit-flipinformation can be handled by scalar message passing.A quantum DS code has additional redundant stabilizersbeing measured and usually a two-stage sequential decodingprocess is applied [9], [13]. In reality, syndrome measurementstake a longer time than simple logical operations. One wouldlike to have redundant measurements as few as possible. Wedemonstrate that DS-BP on a quantum hypergraph-product(HP) code without any redundant syndrome bits may have per-formance close to the case of perfect syndrome measurements(where the loss in block error rate is less than an order forthe [[129 , HP code). On the other hand, we assume thatthe error rate is higher if repeated syndrome measurements areconducted since this usually takes a longer time. Then DS-BP on the HP code without any redundant syndrome bits performsbetter than the two-stage sequential decoding.The paper is organized as follows. In Sec. II, we introducethe basics of stabilizer codes and quantum DS codes. InSec. III, a DS-BP decoding algorithm for quantum DScodes is proposed along with the simulation results. Then weconclude in Sec. IV.II. Q UANTUM D ATA -S YNDROME C ODES
A. Quantum Stabilizer Codes
We consider binary stabilizer codes [5], [7]. Suppose that S is a subgroup of the N -fold Pauli group G N and generated by N − K independent N -fold Pauli operators S , S , . . . , S N − K such that S m S m (cid:48) = S m (cid:48) S m and − I ⊗ N / ∈ S . An [[ N, K ]] a r X i v : . [ qu a n t - ph ] F e b tabilizer code C ( S ) encodes K logical qubits into N physicalqubits and its code space is the joint (+1) eigenspace of theelements in S . The elements in S are called stabilizers . Ifa Pauli error anticommutes with some stabilizers, it can bedetected by measuring the eigenvalues of the stabilizers. Thusthe measurement outcomes, called error syndrome , are used todetermine a correction operation. Sometimes additional redun-dant stabilizers { S m } Mm = N − K +1 are measured for enhancingBP in decoding [25] or handling syndrome errors [11], [12].Without loss of generality, S m is of the form S m = S m ⊗ S m ⊗ · · · ⊗ S mN , where S mn ∈ { I = [
10 01 ] , X = [
01 10 ] , Z =[
10 0 − ] , Y = iXZ } for n = 1 , , . . . , N . We will ignore thenotation ⊗ without confusion. Then S = [ S mn ] ∈ { I, X, Y, Z } M × N is called the check matrix of the stabilizer code. Two Paulioperators of the same dimension, E and F , either commuteor anticommute with each other. We define (cid:104) E, F (cid:105) G = (cid:40) , if EF = F E ;1 , otherwise . (1)For an error E = E E · · · E N ∈ G N , its binary errorsyndrome is given by z = ( z , z , . . . , z M ) ∈ { , } M , where z m = (cid:104) E, S m (cid:105) G = N (cid:88) n =1 (cid:104) E n , S mn (cid:105) G mod 2 . Let wt G ( E ) denote the number of non-identity entries in E ∈G N . The minimum distance of C ( S ) is defined as d = min { wt G ( E ) | E ∈ { I, X, Y, Z } N \S , (cid:104) E, S m (cid:105) = 0 ∀ m } . A stabilizer code with minimum distance d can correct anyerrors E with wt G ( E ) ≤ t , where t = (cid:98) d − (cid:99) . B. Quantum Data-Syndrome (DS) Codes
In addition to a Pauli error E ∈ G N on the data qubits,each syndrome bit z m suffers an independent bit-flip error e m ∈ { , } . Now the syndrome bit relation becomes z m = (cid:104) E, S m (cid:105) G + e m mod 2 . (2)Consequently the check matrix for the DS code is defined by ˜ S = [ S I M ] , (3)where I M is an M × M binary identity matrix. The m -th rowof ˜ S is denoted by ˜ S m = ( S m , ( I M ) m ) ∈ { I, X, Y, Z } N ×{ , } M . Define the product of e, f ∈ { , } M by (cid:104) e, f (cid:105) b = (cid:80) Mj =1 e j f j mod 2 . Then the product of ( E, e ) , ( F, f ) ∈G N × { , } M is defined by (cid:104) ( E, e ) , ( F, f ) (cid:105) = (cid:104) E, F (cid:105) G + (cid:104) e, f (cid:105) b mod 2 . (4)For e ∈ { , } M , let wt b ( e ) denote the number of its nonzeroentries. Then the weight of ( E, e ) is defined as wt( E, e ) = wt G ( E ) + wt b ( e ) . Definition 1.
Let S ∈ { I, X, Y, Z } M × N be a check matrixof an [[ N, K ]] stabilizer code, where M ≥ N − K . We saythat S induces an [[ N, K | M ]] quantum DS code ˜ C with a DScheck matrix ˜ S = [ S I M ] , where ˜ C = { ( F, f ) ∈ { I, X, Y, Z } N ×{ , } M | (cid:104) ( F, f ) , ˜ S m (cid:105) = 0 ∀ m } . The (DS) minimum distance of ˜ C is defined as ˜ d = min { wt( F, f ) | ( F, f ) ∈ ˜ C \ ˜ S} , where ˜ S (cid:44) { ( F, | F ∈ S} . Theorem 1. [12] An [[ N, K | M ]] quantum DS code withDS minimum distance ˜ d can correct any error ( E, e ) ∈ G N ×{ , } M with wt( E, e ) ≤ ˜ t , where ˜ t = (cid:98) ˜ d − (cid:99) . III. B
ELIEF P ROPAGATION FOR Q UANTUM
DS C
ODES
The minimum distance d of a stabilizer code is an upperbound on its induced DS minimum distance ˜ d . The DS mini-mum distance usually achieves this upper bound by additionalredundant measurements [11]–[13].In some conditions, we can have the syndrome protectedwithout redundant measurements [10] and this is a desiredproperty especially when the syndrome measurements areexpensive: the syndrome error rate could be higher thanthe data error rate since a syndrome measurement involvesmany two-qubit operations and single qubit measurements;also more syndrome measurements take more processing time,which in turn incurs a higher data error rate.In the next section, we will simulate a case of CSScodes [26], [27] since they are commonly used in FTQC. Thestabilizer group of a CSS code can be chosen to be products ofonly X operators or only Z operators. Thus we have a binarycheck matrix H = [ H X O | OH Z ] , where H X ∈ { , } M × N , H Z ∈ { , } M × N , M = M + M , and H X H TZ = O , where O is the all-zero matrix whose dimension can be inferred fromthe context. The induced quantum DS code has a (binary)check matrix: ˜ H = [ H X O I M O | OH Z OI M ] . (5)It is not hard to obtain the following theorem. Theorem 2.
Let ˜ H be defined as in (5), where H X and H Z arethe parity-check matrices of two classical codes, respectively,each with minimum distance at least 3. If every column of H X and H Z is of weight at least 2, then the induced quantumDS code has minimum distance at least 3. The conditions can be generalized for higher-weight errorsbut it is complicated. We remark that the minimum distanceof a code provides only a reference for its error performance;we care more about the following practical decoding problem:
The quantum DS decoding problem : Given a DS checkmatrix ˜ S = [ S I M ] , where S ∈ { I, X, Y, Z } M × N , a binarysyndrome z ∈ { , } M of ( E, e ) ∈ G N × { , } M , and certaincharacteristics of the error model, the decoder has to infer ( ˆ E, ˆ e ) , where ˆ E ∈ { I, X, Y, Z } N and ˆ e ∈ { , } M such that (cid:104) ( ˆ E, ˆ e ) , ˜ S m (cid:105) = z m for all m = 1 , , . . . , M and ˆ E ∈ E S . . The DS-BP Algorithm We would like to have a successful decoding with probabil-ity as high as possible (according to a specific error model). Ingeneral, achieving an optimum decoding is extremely difficultfor conventional stabilizer codes [28], [29] and this is also thecase for DS codes. However, if S is a sparse matrix, then ˜ S is also sparse. Using BP to decode DS codes is also efficientand possible to have good performance like stabilizer codes[18], [23], [24].The DS check matrix ˜ S = [ S I M ] corresponds to aTanner graph consisting of N data(-variable) nodes, M syndrome(-variable) nodes, and M check nodes. There is anedge connecting data node n and check node m if S mn (cid:54) = I .There is always an edge connecting check node m andsyndrome node m . For example, if S = [ XZ YZ IY ] , then ˜ S = [ S I M ] = [ XZ YZ IY
10 01 ] and its corresponding Tanner graphis shown in Fig. 1. E E E e e ( ˜ S ) : (cid:104) E , X (cid:105) G + (cid:104) E , Y (cid:105) G + e = z ( ˜ S ) : (cid:104) E , Z (cid:105) G + (cid:104) E , Z (cid:105) G + (cid:104) E , Y (cid:105) G + e = z XYZ
Fig. 1. The Tanner graph of ˜ S = [ XZ YZ IY
10 01 ] . There are three types ofedges connecting a data node and a check node. Suppose that the data nodes and syndrome nodes are num-bered from to N + M . Let M ( n ) be the neighboring checknodes of a variable node n , and N ( m ) be the neighboringvariable nodes of a check node m . Using a similar derivationin [24, Algorithm 3] (or more simply, in [30]), it is not sodifficult to generalize the quaternary BP (BP ) algorithm forstabilizer codes [24, Algorithm 3] to DS-BP for quantum DScodes as in Algorithm 1 that handles only scalar messages.The initial probabilities of data and syndrome errors canbe assigned in DS-BP . Suppose that each qubit suffers amemoryless depolarizing channel of depolarizing rate (cid:15) D andeach syndrome bit suffers a memoryless binary symmetricchannel (BSC) of crossover rate (cid:15) S . Then ( p In , p Xn , p Yn , p Zn ) areinitialized to (1 − (cid:15) D , (cid:15) D , (cid:15) D , (cid:15) D ) for each data-variable node n ∈ { , . . . , N } , and ( p (0) n , p (1) n ) are initialized to (1 − (cid:15) S , (cid:15) S ) for each syndrome-variable node n ∈ { N + 1 , . . . , N + M } .Algorithm 1 performs the update according to a parallelschedule [24] and is referred to as parallel DS-BP . Wealso consider an update order according to a serial schedulerunning along the check nodes as in Algorithm 2, which isreferred to as serial DS-BP . Consider the example in Fig. 1again. Its message update order using parallel DS-BP (resp.serial DS-BP ) is illustrated in Fig. 2 (resp. Fig. 3). Themessage update order plays an important role in BP, especiallywhen the Tanner graph has numerous short cycles [24], [31]. Algorithm 1 : Quaternary DS-BP decoding for quantum DScodes with a parallel schedule (parallel DS-BP ) Input : ˜ S = [ S I M ] , S ∈ { I, X, Y, Z } M × N , z ∈ { , } M ,and initial { ( p In , p Xn , p Yn , p Zn ) } Nn =1 , { ( p (1) n , p (0) n ) } N + Mn = N +1 . Initialization.
For all n = 1 to N + M and all m ∈ M ( n ) : • If n ≤ N , let q Wmn = p Wn for W ∈ { I, X, Y, Z } ,and let q (0) mn = q Imn + q S mn mn and q (1) mn = 1 − q (0) mn . • If n > N , let q (0) mn = p (0) n and q (1) mn = p (1) n . • Calculate d n → m = q (0) mn − q (1) mn . (6) Horizontal Step.
For all m = 1 to M and all n ∈ N ( m ) : • Compute δ m → n = ( − z m (cid:89) n (cid:48) ∈N ( m ) \ n d mn (cid:48) . (7) Vertical Step.
For all n = 1 to N + M and all m ∈ M ( n ) : • Let r (0) mn = (1 + δ mn ) / and r (1) mn = (1 − δ mn ) / . • If n ≤ N , compute q Wmn = p Wn (cid:89) m (cid:48) ∈M ( n ) \ m r ( (cid:104) W,S m (cid:48) n (cid:105) ) m (cid:48) n , W ∈ { I, X, Y, Z } , (8) q (0) mn = a mn ( q Imn + q S mn mn ) ,q (1) mn = a mn ( (cid:80) W (cid:48) ∈{ X,Y,Z }\ S mn q W (cid:48) mn ) . • If n > N , compute q ( b ) mn = a mn p ( b ) n (cid:89) m (cid:48) ∈M ( n ) \ m r ( b ) m (cid:48) n , b ∈ { , } . (9) • Each a mn is a chosen scalar such that q (0) mn + q (1) mn = 1 . • Update: d n → m = q (0) mn − q (1) mn . Hard Decision.
For all n = 1 to N + M : • If n ≤ N , compute q Wn = p Wn (cid:89) m ∈M ( n ) r ( (cid:104) W,S mn (cid:105) ) mn , W ∈ { I, X, Y, Z } , and let ˆ E n = arg max W ∈{ I,X,Y,Z } q Wn . • If n > N , compute q ( b ) n = p ( b ) n (cid:89) m ∈M ( n ) r ( b ) mn , b ∈ { , } , and let ˆ e n = 0 , if q (0) n > q (1) n , and ˆ e n = 1 , otherwise. • Let ˆ E = ˆ E ˆ E · · · ˆ E N and ˆ e = (ˆ e N +1 , ˆ e N +2 , . . . , ˆ e N + M ) . – If (cid:104) ( ˆ E, ˆ e ) , ˜ S m (cid:105) = z m for all m = 1 to M , halt andreturn “CONVERGED”. – Otherwise, if a maximum number of iterations isreached, halt and return “FAIL”. – Otherwise, repeat from the horizontal step. lgorithm 2 : Quaternary DS-BP decoding with a serialschedule along the check nodes (serial DS-BP ) Input : The same as in Algorithm 1.
Initialization.
For all m = 1 to M and all n ∈ N ( m ) : • Let δ m → n = 0 . Serial Update.
For each check node m = 1 to M : • For every n ∈ N ( m ) , do the same as in the five bulletpoints around (8) and (9), with the order specified here. • For every n ∈ N ( m ) , compute δ m → n = ( − z m (cid:89) n (cid:48) ∈N ( m ) \ n d mn (cid:48) . (10) Hard Decision. • Do as in Algorithm 1, except that “repeat from thehorizontal step" must be replaced by “repeat from theserial update step". E E E e e (a) E E E e e (b) Fig. 2. The message update order of parallel DS-BP for the example inFig. 1. (a) The initialization step, as well as the vertical step. (b) The horizontalstep. The message update order will be iterated between (a) and (b). E E E e e (a) E E E e e (b) E E E e e (c) E E E e e (d) Fig. 3. The message update order of serial DS-BP (along check nodes) forthe example in Fig. 1. (a) and (b): serial update for check node 1. (c) and(d): serial update for check node 2. The message update order will be iteratedfrom (a) to (d). B. Simulation Results
We construct a [[129 , CSS-type HP code [19] based onthe [7 , , and [15 , , BCH codes as in [24], [32]. Thiscode has minimum distance d = 3 and can be a candidatefor Theorem 2. Its raw check matrix has some columns ofweight one. After row multiplications by adjacent rows, wecan obtain a check matrix, each column of which is of weightat least 2. Then by Theorem 2, we have a [[129 , | quantum HP DS code with minimum distance ˜ d = 3 .We first explain the serial schedule that will be conductedin the following simulations. In [24], it is demonstrated thatbased on the raw check matrix of the [[129 , HP code, theparallel BP decoding does not perform well due to decodingoscillation (which is caused by the numerous short cyclesand symmetric sub-graphs in the Tanner graph [24], [31]); on Since the raw check matrix has a cyclic-like structure, the multiplicationof two adjacent rows will not have high weight and the locality is slightlyaffected. the other hand, the serial BP along variable nodes performsquite well by using the raw matrix [24]. We have createda check matrix so that each of its column has weight ≥ for Theorem 2; however, the serial update along variablenodes is too aggressive at certain variable node for the newcheck matrix (when computing the hard-decision and outgoingmessages). This causes for some weight-one errors to bedecoded as weight-three errors, and the syndrome is falselymatched. Fortunately, this can be improved by using a serialupdate along the check nodes (which also breaks the symmetryin the short cycles and sub-graphs; however, it provides a moregradual update for each coordinate n at each iteration).In this subsection, the simulation of BP allows a maximumnumber of 12 iterations, and each data point is based oncollecting at least 100 blocks of logical errors.We begin with the simulation without syndrome errors( (cid:15) S = 0 ) and compare the decoding results of parallel DS-BP (Algorithm 1) and serial DS-BP (Algorithm 2). In this case,DS-BP is equivalent to the usual BP in [24].We use bounded-distance decoding (BDD) as a benchmarkfor error performance. In general, BDD with radius t can cor-rect any error of weight no larger than t . We consider a moregeneral BDD as follows. Let t ≥ and γ = ( γ , γ , . . . , γ t ) ,where γ j denotes the percentage of weight- j errors assumedto be corrected. Then the generalized BDD with respect to t and γ has a logical error rate at (cid:15) ( = (cid:15) D here) as follows P e,BDD ( N, t, γ ) = 1 − (cid:16)(cid:80) tj =0 γ j (cid:0) Nj (cid:1) (cid:15) j (1 − (cid:15) ) ( N − j ) (cid:17) . (11)The [[129 , code can correct any error of weight one.If we make a lookup table for decoding by assigning eachsyndrome to low-weight errors, then this code can correctabout . of the weight-2 errors. Thus, with t = 2 ,this lookup-table decoding provides a generalized BDD tohave γ = 1 , γ = 1 , and γ ≈ . . This codeis not degenerate; its stabilizers have weight larger than 3.Consequently these three γ j values are fixed whether thedegeneracy is considered or not, and they dominate the errorperformance.Figure 4 shows the simulations of Algorithm 1 and Algo-rithm 2 at (cid:15) S = 0 , together with several BDD reference curves.Note that γ j = 1 for j ≤ t if not specified. Serial DS-BP achieves a performance quite close to the lookup-table decoder P e,BDD ( N, , γ = 98 . . This matches Gallager’s expecta-tion that the performance of BP can be as close as to twotimes of the BDD performance with radius t = d − [20].Next, we assume that each syndrome bit is flipped with rate (cid:15) S (cid:54) = 0 . For simplicity, assume (cid:15) S = (cid:15) D , which allows us touse (11) as a benchmark. We focus on the serial schedulesince it provides a better performance. The serial DS-BP performance is plotted in Fig. 5, which has a performance lossof less than an order compared to the case of no syndromeerror (cid:15) S = 0 . It can be seen that serial DS-BP improves serialBP ( (cid:15) S = (cid:15) D ) if no repeated measurements are conducted asexpected. We also provide a curve for serial BP ( (cid:15) S = (cid:15) D )with r = 3 repeated measurements, and it performs quite well ig. 4. Decoding performance of the [[129 , HP code (with syndromeerror rate (cid:15) S = 0 ). The serial schedule is along the check nodes. SpecificBDD reference performance curves, per (11), are plotted.Fig. 5. Decoding performance of the [[129 , | HP DS code. Allthe serial-schedule results are based on Algorithm 2. When it is labeled with“Serial BP ”, it means that Algorithm 2 is run with (cid:15) S = 0 regardless of theactual (cid:15) S value. If there are r repeated measurements (meas.), a majority votewill be run to decide the syndrome before running the decoding algorithm. as seen in Fig. 5; however, this only reveals the importance ofeliminating the effect caused by noisy measurements.Performing repeated measurements would require additionaltime (and gates), so these practical issues should be consideredin comparison. As described in [33], [34], the fidelity ofa physical qubit decays exponentially over the operationaltime τ . Assume that the fidelity is − (cid:15) = e − λτ (12)for some decay factor λ . Suppose that a round of syndromemeasurement takes 740 ns [35]. We further assume thatthe measurement time dominates the overall error-correctiontime, since the decoder should run much faster in a clas- Fig. 6. Comparing the different decoding strategies in Fig. 5, at a certainfidelity decay factor λ as in (12), in which we assume that (cid:15) = (cid:15) D and τ = r × ns (where there is only one case with r = 3 , as labeled, andthe other cases have r = 1 ). sical hardware [36]. So now one round of measurementswith one round of serial DS-BP takes about ns, and r = 3 rounds of measurements with one round of serialBP takes about × ns. Given (cid:15) ( = (cid:15) D in Fig. 5) and τ ( = 740 or × ns), a corresponding λ in (12) canbe derived. Figure 6 provides the rescaled curves of Fig. 5.The results show that using the DS-BP approach can take theadvantage of less measurement time to outperform a decodingstrategy with repeated measurements.IV. C ONCLUSION & F
UTURE W ORKS
Faulty syndrome measurement is an issue that cannot beneglected in fault-tolerant quantum error-correction. A poten-tial solution is to apply quantum DS codes. By generalizingthe refined BP [24], we proposed a DS-BP decoding al-gorithm and have demonstrated that it can efficiently achievesatisfactory results. The DS-BP approach has proved to besuitable for low-weight stabilizers and can be used with less(or even without) redundant measurements. This decreases themeasurement time and increases the probability of a successfuldecoding.We simulated the [[129 , | HP DS code with min-imum distance 3. For codes with higher minimum distance,we may use additional syndrome measurements to compensatethe effects of syndrome errors. For example, the surface codesare the current state-of-the-art candidate for FTQC. It is stillunknown whether BP works for codes with strong degeneracythat their stabilizers may have weight much lower than theminimum distance. Also, the error model considered in thispaper is too ideal. To apply the DS-BP approach in a morepractical model, like the faulty circuit model [8], [13], is ourongoing research.
CKNOWLEDGMENT
CYL was financially supported from the Young Scholar Fel-lowship Program by the Ministry of Science and Technology(MOST) in Taiwan, under Grant MOST109-2636-E-009-004.R
EFERENCES[1] P. W. Shor, “Fault-tolerant quantum computation,” in
Proc. 37th Annu.Conf. Found. Comput. Sci. (FOCS) , pp. 56–65, IEEE, 1996.[2] E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,”
Phys. Rev. A , vol. 55, p. 900, 1997.[3] A. M. Steane, “A tutorial on quantum error correction,” in
Proc. Int.School Phys. Enrico Fermi , vol. 162, p. 1, IOS Press; Ohmsha; 1999,2007.[4] D. A. Lidar and T. A. Brun,
Quantum error correction . CambridgeUniversity press, 2013.[5] D. Gottesman,
Stabilizer codes and quantum error correction . PhDthesis, California Institute of Technology, 1997.[6] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,“Quantum error correction via codes over GF(4),”
IEEE Trans. Inf.Theory , vol. 44, pp. 1369–1387, 1998.[7] M. A. Nielsen and I. Chuang, “Quantum computation and quantuminformation,” 2000.[8] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surfacecodes: Towards practical large-scale quantum computation,”
Phys. Rev.A , vol. 86, p. 032324, 2012.[9] A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Robust quantum errorsyndrome extraction by classical coding,” in
Proc. IEEE Int. Symp. Inf.Theory (ISIT) , pp. 546–550, IEEE, 2014.[10] Y. Fujiwara, “Ability of stabilizer quantum error correction to protectitself from its own imperfection,” vol. 90, p. 062304, Dec 2014.[11] A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Correction of data andsyndrome errors by stabilizer codes,” in
Proc. IEEE Int. Symp. Inf.Theory (ISIT) , pp. 2274–2278, IEEE, 2016.[12] A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum data-syndromecodes,”
IEEE J. Sel. Areas Commun. , vol. 38, pp. 449–462, 2020.[13] W. Zeng, A. Ashikhmin, M. Woolls, and L. P. Pryadko, “Quantumconvolutional data-syndrome codes,” in
Proc. IEEE Int. Workshop SignalProcess. Adv Wireless Commun. (SPAWC) , pp. 1–5, 2019.[14] H. Bombín, “Single-shot fault-tolerant quantum error correction,”
Phys.Rev. X , vol. 5, p. 031043, 2015.[15] B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant errorcorrection with the gauge color code,”
Nat. Commun. , vol. 7, pp. 1–8,2016.[16] N. P. Breuckmann, K. Duivenvoorden, D. Michels, and B. M. Terhal,“Local decoders for the 2D and 4D toric code,”
Quantum Inf. Comput. ,vol. 17, p. 181–208, 2016.[17] A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,”
Ann.Phys. , vol. 303, pp. 2–30, 2003.[18] D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-graphcodes for quantum error correction,”
IEEE Trans. Inf. Theory , vol. 50,pp. 2315–2330, 2004.[19] J.-P. Tillich and G. Zémor, “Quantum LDPC codes with positive rate andminimum distance proportional to the square root of the blocklength,”
IEEE Trans. Inf. Theory , vol. 60, pp. 1193–1202, 2014.[20] R. G. Gallager,
Low-Density Parity-Check Codes . no. 21 in ResearchMonograph Series, Cambridge, MA: MIT Press, 1963.[21] D. J. C. MacKay, “Good error-correcting codes based on very sparsematrices,”
IEEE Trans. Inf. Theory , vol. 45, pp. 399–431, 1999.[22] J. Pearl,
Probabilistic reasoning in intelligent systems: networks ofplausible inference . Morgan Kaufmann, 1988.[23] D. Poulin and Y. Chung, “On the iterative decoding of sparse quantumcodes,”
Quant. Inf. Comput. , vol. 8, pp. 987–1000, 2008.[24] K.-Y. Kuo and C.-Y. Lai, “Refined belief propagation decoding ofsparse-graph quantum codes,”
IEEE J. Sel. Areas Inf. Theory , vol. 1,pp. 487–498, 2020.[25] A. Rigby, J. C. Olivier, and P. Jarvis, “Modified belief propagationdecoders for quantum low-density parity-check codes,”
Phys. Rev. A ,vol. 100, p. 012330, 2019.[26] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codesexist,”
Phys. Rev. A , vol. 54, p. 1098, 1996.[27] A. M. Steane, “Error correcting codes in quantum theory,”
Phys. Rev.Lett. , vol. 77, p. 793, 1996. [28] K.-Y. Kuo and C.-C. Lu, “On the hardnesses of several quantumdecoding problems,”
Quant. Inf. Process. , vol. 19, pp. 1–17, 2020.[29] P. Iyer and D. Poulin, “Hardness of decoding quantum stabilizer codes,”
IEEE Trans. Inf. Theory , vol. 61, pp. 5209–5223, 2015.[30] K.-Y. Kuo and C.-Y. Lai, “Refined belief-propagation decoding ofquantum codes with scalar messages,” in
Proc. IEEE Global Commun.Conf. (GLOBECOM), to be appeared , 2020.[31] N. Raveendran and B. Vasi´c, “Trapping sets of quantum LDPC codes,” e-print arXiv:2012.15297 , 2020.[32] Y.-H. Liu and D. Poulin, “Neural belief-propagation decoders forquantum error-correcting codes,”
Phys. Rev. Lett. , vol. 122, p. 200501,2019.[33] S. Muralidharan, C.-L. Zou, L. Li, and L. Jiang, “One-way quantumrepeaters with quantum Reed-Solomon codes,”
Phys. Rev. A , vol. 97,p. 052316, 2018.[34] J. M. Nichol, L. A. Orona, S. P. Harvey, S. Fallahi, G. C. Gardner,M. J. Manfra, and A. Yacoby, “High-fidelity entangling gate for double-quantum-dot spin qubits,” npj Quantum Inf. , vol. 3, pp. 1–5, 2017.[35] R. Versluis, S. Poletto, N. Khammassi, B. Tarasinski, N. Haider, D. J.Michalak, A. Bruno, K. Bertels, and L. DiCarlo, “Scalable quantumcircuit and control for a superconducting surface code,”
Phys. Rev. Appl. ,vol. 8, p. 034021, 2017.[36] S. Varsamopoulos, B. Criger, and K. Bertels, “Decoding small surfacecodes with feedforward neural networks,”