Methods, devices, and systems for decoding syndrome of quantum error correction codes

Abstract

In a method for decoding a syndrome of a QEC code, a syndrome of the QEC code and a plurality of transformation pairs are obtained. Each transformation pair comprises a first transformation and a second transformation. Applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant. For each transformation pair, the first transformation is applied to the syndrome to obtain a transformed syndrome, and an error estimate is determined based on the transformed syndrome and the parity-check matrix. A permuted error estimate is determined for at least one of the error estimates based on the second transformation and the error estimate. An error associated with the syndrome is determined based on the permuted error estimates of the plurality of transformation pairs.

Description

METHODS, DEVICES, AND FOR DECODING SYNDROME OF QUANTUM ERROR CORRECTION CODES Cross-Reference to Related Application

[0001] This application claims priority from US application No.63 / 680560 filed 7 August 2024 and entitled METHODS, DEVICES, AND SYSTEMS FOR DECODING SYNDROME OF QUANTUM ERROR CORRECTION CODES which is hereby incorporated herein by reference for all purposes. For purposes of the United States of America, this application claims the benefit under 35 U.S.C. §119 of US application No.63 / 680560 filed 7 August 2024 and entitled METHODS, DEVICES, AND SYSTEMS FOR DECODING SYNDROME OF QUANTUM ERROR CORRECTION CODES which is hereby incorporated herein by reference for all purposes. Field

[0002] This disclosure generally relates to quantum error correction (QEC), and in particular to methods, devices, and systems for decoding a syndrome of a QEC code. Background

[0003] Classical error correction is a fundamental concept in information theory and computer science that focuses on ensuring that data is transmitted accurately despite the presence of errors. The original data is encoded to produce a codeword that is transmitted over a channel. During transmission, the codeword can be corrupted by errors. These errors can come from various sources, such as noise in the communication channels or interference in the signal processing. The basic premise of classical error correction is to add redundancy to the original data during the encoding process, allowing the system to detect and correct errors without having to retransmit the data. The aim of decoding is to determine the transmitted codeword from the received codeword.

[0004] Automorphism Ensemble (AE) decoders are known to improve decoding performance, i.e., increase the probability of identifying the correct errors. An AE decoder applies a set of different automorphisms to the received, possiblycorrupted, codeword. Each automorphism transforms the received codeword, and a corresponding decoder then attempts decode each transformed codeword. The results of all the decoders are evaluated and the most likely correct codeword is selected.

[0005] For example, the decoder may be an iterative decoder, such as a belief propagation decoder. A permutation of the columns of the received codeword can affect the performance of iterative decoders. For example, a decoder may fail to decode a received codeword, but succeed in decoding a column permutation of the same codeword. AE decoders exploit this phenomenon to increase the chance of successful decoding.

[0006] In quantum information systems, quantum error correction (QEC) can be applied to identify and correct errors. Here, the codeword is associated with the quantum states of the physical qubits. However, due to the nature of quantum mechanics, measuring and copying the codeword would result in the collapse of the quantum state. As a result, classical AE decoder methods cannot be directly applied in QEC. Therefore, there is a need for efficient decoding methods in QEC that offer high-performance and low time complexity. Summary

[0007] The present disclosure has several aspects, including a method, a device, and an information processing system for decoding a syndrome of a QEC code.

[0008] A first aspect of the disclosure provides a method for decoding a syndrome of a QEC code. The syndrome of the QEC code is obtained. A plurality of transformation pairs is obtained. Each transformation pair comprises a first transformation and a second transformation. Applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant. For each transformation pair, the first transformation is applied to the syndrome to obtain a transformed syndrome and an error estimate is determined based on the transformed syndrome and the parity-check matrix. A permuted error estimate is determined for at least one of the error estimates based on the second transformation and the error estimate. An error associated with the syndrome is determined based on the permuted error estimates of the plurality of transformation pairs.

[0009] The disclosure provides a syndrome-based method for decoding a syndrome of a QEC code with increased performance and low time complexity, as will be explained in the following.

[0010] In general, a quantum mechanical operation on one or more qubits is described by a unitary operator. An important class of unitary operators is described by the Pauli group. The single-qubit Pauli group ^^1is the group of Pauli operators that can each act on the qubit. There are 16 Pauli operators. Up to a complex phase, there are 3 distinct non-trivial operators,Pauli ^^,^^, and ^^ operators. The single-qubit Pauli group ^^1is formed by all products of the Pauli ^^,^^, and ^^ operators.

[0011] According to this specification, the term “^^-qubit Pauli group ^^^^” (or simply: “Pauli group”) denotes the group formed by all length ^^ tensor ofelements of the single-qubit Pauli group ^^1, where ^^ is an integer than 1. The Pauli group is a non-Abelian group elements in general do notcommute. The elements of the ^^-qubit group are denoted as Pauli operators.

[0012] According to this specification, the term “QEC code” can be understood as being defined by specifying a code space. Herein, a code space is a vector subspace of a Hilbert space which is spanned by so-called codewords of the QEC code. A codeword is a state which encodes some data. A codeword corresponds to a logical state of the QEC code. A QEC code is used to protect quantum information that is processed in the computation from errors due to quantum noise, e.g., decoherence.

[0013] A QEC code is in general a mapping of ^^ qubits onto ^^ qubits, where ^^ > ^^. The quantum states on ^^ qubits are elements of a Hilbert space of dimension 2^^. The quantum states on ^^ qubits are elements of a Hilbert space of dimension 2^^. Herein, the ^^ qubits are the “logical qubits” or “encoded qubits” that are to be protected from error, e.g., a threshold amount of error. The ^^ qubits are the“physical qubits” implementing the logical qubits. The additional ^^ − ^^ qubits allowthe ^^ logical qubits to be stored in a redundant fashion so that the encoded information is less susceptible to noise or other disturbances.

[0014] A QEC code with block size ^^, encoded qubits ^^, and distance ^^ is denoted as an “[[^^,^^,^^]] QEC code,” where the distance ^^ is the minimum support of anon-identity logical operator in the code. The distance ^^ sets a bound on the number of errors the QEC code can and correct. For example, if there are ^^−1 fewer than ⌊ 2 ⌋ errors, a perfect decoder can identify and correct these errors without the intended quantum state. On the other hand, if there are more ^^−1than ⌊ 2 ⌋ a perfect decoder may be unable to correct all the errors. QEC code can be obtained from the user. In particular, the QEC code can be a stabilizer code ^^. According to this specification, a “stabilizer code” denotes a QEC code defined by a set ^^ of mutually commuting Hermitian Pauli operators that does not contain the negative of the identity operator, −^^.

[0016] According to this specification, the term “Pauli stabilizers,” S, denotes the commuting Hermitian Pauli operators that define the associated stabilizer code ^^. This collection forms a subgroup of the full Pauli group ^^^^. In particular, the composition of two Pauli stabilizers produces another stabilizer. Each Paulistabilizer is an ^^ qubit gate.

[0017] In the case of a Calderbank-Shor-Steane (CSS) stabilizer code (CSS code), the group S of Pauli stabilizers is determined by binary matrices ^^^^and ^^^^: S= {^^^^^^^^  :  ^^ ∈ RowSpan(^^^^),  ^^ ∈ RowSpan(^^^^)}.

[0018] For more general codes, the group S of Pauli stabilizers is determined by asingle matrix ^^ (the parity-check matrix) comprising ^^ − ^^ rows, each of length 2^^:S = {^^^^^^^^  :  (^^|^^) ∈ RowSpan(^^)}.

[0019] More specifically, the codewords ^^ ∈ ^^ of the stabilizer code are the ^^ qubitstates (i.e., vectors in the Hilbert space ℋ of dimension 2^^) that are unchanged when any Pauli stabilizer is applied. The codewords are therefore the fixed points of the Pauli stabilizers.

[0020] The Pauli stabilizers can be used to detect unwanted errors. For a given codeword, a detectable error will transform the codeword to a different vector in the Hilbert space ℋ of dimension 2^^that is no longer a fixed point of the Pauli stabilizers. By determining which Pauli stabilizers still fix the vector, the error can be identified and corrected.

[0021] A QEC code can therefore be defined by its parity-check matrix. According to this specification, the term “parity- matrix” (denoted by ^^) denotes a matrix used in linear error-correcting codes to verify whether a given codeword is valid. For a (^^,^^) linear block code (with ^^ being the length of each codeword and ^^ being the number of information bits in each codeword), the parity check matrix ^^is in general an ^^ × ^^ matrix, where ^^ ≥ ^^ − ^^ and a codeword ^^ is valid, if and onlyif ^^ ⋅ ^^^^ = 0 .

[0022] In other words, codewords are in the kernel of ^^, i.e., ^^ ∈ ker(^^).

[0023] Each Pauli operator can be represented by a binary vector of length 2^^. For example, a Pauli operator ^^ acting on ^^ qubits can be written as: ^^ = ^^^^1^^^^1 ⊗ ^^^^2^^^^2 ⊗ ⋯ ⊗ ^^^^^^^^^^^^ ,where ^^ = (^^1,^^2, …The parity checkmatrix ^^ is then a (^^ − ^^) × 2^^ matrix, where each row corresponds to the binaryrepresentation of one of the ^^ − ^^ stabilizer generators.

[0024] When a quantum state is measured using the stabilizer generators, a syndrome is obtained which indicates whether the state lies within the code space or has been affected by errors.

[0025] According to this specification the term “error associated with the syndrome” denotes a binary vector ^^0, where each entry indicates whether a specific type of error has occurred on a corresponding qubit. For example, bit-flip ^^ errors and / or phase-flip ^^ errors may occur for a respective qubit.

[0026] The syndrome ^^^^is a binary vector obtained by applying the parity check matrix ^^ to the transpose of the error vector ^^0, i.e.: ^^^^ = ^^ ⋅ ^^^^^^.

[0027] Closely connected to parity-check matrices are generator matrices.According to this specification, the term “generator matrix” denotes a matrix ^^associated with a classical or quantum code which is used to encode informationbits (or logical qubits) into codewords (or physical qubits). For a classical (^^, ^^)linear block code, the generator matrix is of dimension ^^ × ^^, where ^^ is thenumber of information bits, ^^ ≥ ^^, and ^^ the length of the codewords. For someinformation vector ^^, the corresponding codeword ^^ is obtained by the following formula: ^^ = ^^ ⋅ ^^ .

[0028] According to the method for decoding a syndrome of a QEC code, copies of the syndrome are transformed several times using a respective first transformation which is a linear transformation. If one of the first transformations is combined with a corresponding second transformation which is a permutation, then the parity check matrix is left invariant. Accordingly, each transformation pair comprising the first transformation and the second transformation is an automorphism of the QEC code. For each transformed syndrome, a respective decoding step is used to determine an error estimate. Finally, the error associated with the syndrome is determined based on the multiple respective error estimates. The method can therefore be performed by an automorphism ensemble (AE) decoder.

[0029] Herein, “automorphism” indicates that the transformation pairs are automorphisms of the code. “Ensemble” means that there are multiple decoding processes for determining respective error estimates, i.e., one decoder for each transmission pair.

[0030] According to this specification, an “automorphism” of a code ^^ is a permutation of the code's symbols that maps codewords to other codewords, preserving the structure of the code ^^. Automorphisms of a QEC code can be considered as a way of presenting the same QEC code in a different way. Essentially, an automorphism is a symmetry of the QEC code, i.e., a mapping to an equivalent representation. More specifically, the transformation pairs are automorphisms because a combined application of the first transformation and the second transformation of a given transformation pair leaves the parity check matrix invariant.

[0031] The automorphisms of the code ^^ forms a group, which is denoted by Aut(^^). The order of the automorphism group Aut(^^) is denoted by |Aut(^^)|. The order corresponds to the number of its elements and therefore relates to the size ofthe automorphism group. The size of the automorphism group depends on the structure of the parity check matrix of code.

[0032] According to this specification, an “ensemble” of decoders denotes a collection of decoders, each utilizing different automorphisms of the code i.e., one decoder for each transmission pair. An ensemble therefore relates to multiple decoding processes for determining respective error estimates.

[0033] In classical error correction, an AE decoder can be used based on permutations of the codeword. After the individual decoders estimate respective codewords, the initial transformation can be compensated by applying the respective inverse permutation. Herein, a column permutation and its inverse form an automorphism of the classical code, i.e., they merely change the representation of the code, rather than changing the code itself. In QEC correction, this approach is not directly applicable because the codewords correspond to the quantum states of physical qubits. This means the codeword cannot be measured without collapsing any superposition.

[0034] The method according to the specification is therefore syndrome-based instead of codeword-based. However, the classical concept of AE decoders based on permutations is still not directly applicable because permuting the columns of the syndrome before decoding and then performing the inverse of the column permutation on the error estimates is not equivalent to applying an automorphic transformation on the QEC code. Further, it is in general not possible to apply an inverse of the column permutation on the error estimates because the syndrome and the error estimates will have different numbers of columns.

[0035] It is an underlying idea of the specification that automorphisms of the code can be constructed from transformation pairs comprising a first transformation and a second transformation. The first transformation is an invertible linear transformation operating on the rows of a parity check matrix which transforms each row of the parity check matrix of the QEC code into a linear combination of rows of the parity check matrix. Specifically, the first transformation is therefore a linear transformation on the row-space of the parity check matrix. The first transformation basically replaces the initial column permutation used in classical AE decoders. The second transformation is a permutation which permutes columns of the parity-check matrix.

[0036] An advantage of using multiple decoders, i.e., an AE-type decoder, is that sometimes a decoder may fail to find a from a given input i.e., the syndrome but the solution can often be found in a different representation of the input. An underlying reason is that the numerical treatment used in decoding method sometimes fails to converge to the true solution, for example if the method gets trapped near a local minimum which is not the true global minimum. In other situations, the method may not converge at all, e.g., if there are cycles in a graph of an iterative algorithm. In general, a maximum number of iterations can be set and the decoding algorithm terminates if no converging solution is found after the maximum number of iterations has been reached. In a different representation of the code, the behavior of the method can be different and the method may converge to the true solution.

[0037] According to an embodiment of the method for decoding a syndrome of a QEC code, the transformation pairs are pairwise different, i.e., for any two transformation pairs, the corresponding first transformation and / or second transformation are different.

[0038] According to an embodiment of the method for decoding a syndrome of a QEC code, the transformation pairs comprise the trivial transformation pair comprising the first transformation being the identity operation and the second transformation being the identity operation. That is, the untransformed syndrome is also decoded. Computing the transformed syndrome is not necessary in this case because the explicit application of an identity matrix can simply be omitted.

[0039] According to an embodiment of the method for decoding a syndrome of a QEC code, for the ithtransformation pair of the plurality of transformation pairs, the first transformation is represented by a linear transformation matrix ^^^^and the second transformation is represented by a permutation matrix ^^^^which satisfy: ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ ,wherein ^^ is the parity-check matrix of the QEC code.

[0040] According to an embodiment of the method for decoding a syndrome of a QEC code, for the ithtransformation pair, applying the first transformation to the syndrome ^^^^to obtain the transformed syndrome ^^^^comprises computing the transformed syndrome ^^^^according to:^^^^ = ^^^^ ⋅ ^^^^.

[0041] The invertible linear ^^^^can therefore be applied to the syndrome before decoding.

[0042] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the permuted error estimate for at least one of the error estimates comprises determining whether the error estimate satisfies a predefined condition. For each error estimate that satisfies the predefined condition, a permuted error estimate is determined based on the second transformation and the error estimate. The predefined condition can be that the error estimate converges. An error estimate converges if the error estimate corresponds to the transformed syndrome. Error estimates which do not correspond to the transformed syndrome can be identified and ignored in the following method steps.

[0043] According to an embodiment of the method for decoding a syndrome of a QEC code, for the ithtransformation pair, the predefined condition is given by: ^^ ⋅ ^^ ^^^^ = ^^^^ .

[0044] Herein, ^^^^^^is a transpose of the error estimate.

[0045] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the permuted error estimate ^^′^^based on the second transformation and the error estimate comprises computing the permuted error estimate ^^′^^according to: ^^′^^ = ^^^^ ⋅ ^^^−^^^, wherein ^^−^^^^is an inverse of the permutation matrix ^^^^.

[0046] That is, the inverse of the permutation matrix ^^^^can be applied to the output of the individual decoders. After transformation, the permuted error estimate ^^′^^will correspond to an estimate of the original error corresponding to the syndrome associated with the quantum computation. This will be explained in the following.

[0047] The original syndrome ^^^^measured for a set of physical qubits encoded according to the QEC code with parity check matrix ^^ is associated with an error ^^0on the physical qubits according to the following equation: ^^^^ = ^^ ⋅ ^^^^ ^^

[0048] The relation ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ between the permutation matrix ^^^^, the lineartransformation ^^^^and the parity check ^^ can be rewritten as: ^^^^ ⋅ ^^ = ^^ ⋅ ^^^−^^^,

[0049] Applying both sides of the equation to the transpose of the error ^^0results in the following relation: ^^^^ ⋅ ^^ ⋅ ^^ ^^0 = ^^ ⋅ ^^−^^^^ ⋅ ^^^0^.

[0050] Substituting the expression ^^^^ = ^^ ⋅ ^^^^^^gives the relation: ^^^^ ⋅ ^^ −^^^^ = ^^ ⋅ (^^^^ ⋅ ^^^0^) .

[0051] This illustrates that decoding the syndrome ^^^^that has been transformed according to the linear transformation ^^^^results in an error estimate ^^^^given by: ^^^^^^ = ^^−^^1^^^^^^.

[0052] The permutation matrix ^^^^corresponds to a permutation and therefore satisfies: ^^^^^^ = ^^−^^1.

[0053] Therefore: ^^ = (^^ ^^ ⋅ ^^ ^^^)^^ = (^^ ^^ ^^ ^^ ^^^^ ^^ ^ ^^) ⋅ (^^^^ ) = ^^^^ ⋅ ^^^^ .

[0054] The permuted error estimate ^^′^^ = ^^^^ ⋅ ^^^−^^^is therefore given by: ^^′^^ = ^^^^ ⋅ ^^−^^ −^^^^ = ^^^^ ⋅ ^^^^ ⋅ ^^^^ = ^^^^ .

[0055] Therefore, the permuted error estimate ^^′^^is an error estimate of the original error ^^^^.

[0056] Accordingly, it follows basically from the condition ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ that thepermuted error estimate ^^′^^corresponds to the original error ^^^^. This condition can be understood also in the following way. By applying the inverse ^^−^^^^of the permutation matrix ^^^^on both sides of the condition, the following equation is equivalent to the above condition: ^^^^ ⋅ ^^ = ^^ ⋅ ^^−^^^^ = ^^ ⋅ ^^ ^^ .

[0057] Here, ^^^^is another permutation matrix because the inverse of a permutation matrix is also a permutation matrix. Each transformation pair can be considered tocomprise a first transformation represented by a linear transformation ^^^^and another second transformation by a permutation matrix ^^^^. The aboveequation then means that the first transformation ^^^^ of the parity-check matrix ^^ isequivalent to the other second transformation ^^^^of the parity check matrix ^^, i.e., they give the same transformed parity check matrix ^^ : ^^ = ^^^^ ⋅ ^^ = ^^ ⋅ ^^ ^^ .

[0058] Yet another way to understand the transformation pairs is as follows. If ^^ denotes a codeword, i.e., a vector of quantum states of physical qubits, then therelation ^^^^ ⋅ ^^ = ^^ ⋅ ^^ ^^ applied to ^^ gives:(^^^^ ⋅ ^^) ⋅ ^^ = ^^ ⋅ (^^ ^^ ⋅ ^^) .

[0059] On the left-hand side, applying the first transformation ^^^^to the parity checkmatrix ^^ corresponds to the transformed parity check matrix ^^ given by:^^ = ^^^^ ⋅ ^^ .

[0060] On the right-hand side, applying the other second transformation ^^^^to thecodeword ^^ corresponds to a transformed codeword ^̃^ (a transformed vector ofquantum states of physical qubits) given by: ^̃^ = ^^ ^^ ⋅ ^^ .

[0061] Then, the condition ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ implies that an action of the transformedparity check matrix ^^ on the codeword ^^ is identical to an action of the original parity check matrix ^^ of the QEC code on the transformed codeword ^̃^.

[0062] According to an embodiment of the method for decoding a syndrome of a QEC code, physical qubits can be re-ordered via permutations (by the other second transformation ^^^^) and the syndrome can be measured after each permutation.

[0063] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the error estimate comprises applying a syndrome-based decoding method to the transformed syndrome and the parity-check matrix of the QEC code in order to determine the error estimate. Several QEC codes can be decoded efficiently with syndrome-based methods.

[0064] According to an embodiment of the method for decoding a syndrome of a QEC code, the syndrome-based an iterative decoding method. Iterative decoding methods typically comprise an initialization phase, where the decoder is initialized with the received data and initial error estimates. The error estimates are updated through a series of iterations, where each iteration involves exchanging information between multiple decoders or nodes (e.g., variable nodes and check nodes in quantum low-density parity check codes). The iterative updates are repeated until the error estimate converges to a stable solution (i.e., the convergence condition is satisfied) or until a maximum number of iterations is reached.

[0065] According to an embodiment of the method for decoding a syndrome of a QEC code, the iterative decoding method is a belief propagation decoding method. The belief propagator comprises parameters such as the maximum number of iterations. Once the maximum number of iterations is reached and a converging solution is not found, the belief propagation decoding method terminates. In this case, no solution has been obtained and the specific transformation pair is ignored in the following determination of the error associated with the syndrome.

[0066] According to an embodiment of the method for decoding a syndrome of a QEC code, the belief propagation decoding method is a scaled minimum sum algorithm, scaled-MSA, belief propagation decoding method. The scaled-MSA belief propagation decoding method is an effective decoder for quantum low- density parity check codes.

[0067] According to an embodiment of the method for decoding a syndrome of a QEC code, differing iterative decoding methods are used in determining the error rates for at least some of the transformation pairs. Using different decoding methods introduces additional variety which increases the probability that at least one decoder finds the actual error.

[0068] According to an embodiment of the method for decoding a syndrome of a QEC code, the differing iterative decoding methods differ in a type of the iterative decoding method. For example, one of decoding methods can be a belief propagation decoding method and another decoding method is not a belief propagation decoding method (e.g., a successive cancellation decoding method if the QEC code is a polar code).

[0069] According to an embodiment of the method for decoding a syndrome of a QEC code, the differing iterative methods differ in at least one parameter of the iterative decoding method. By varying parameters of the iterative decoding method, some variety is introduced which increases the probability that at least some decoders find a converging error estimate.

[0070] According to an embodiment of the method for decoding a syndrome of a QEC code, the differing iterative decoding methods differ in an a-priori probability of the iterative decoding method. Herein, the a-priori probability is an input parameter of the iterative decoding method and corresponds to the error rate of the quantum information system that is to be expected. A variation can be included by using different a-priori probabilities for different transformation pairs, e.g., by varying within a predetermined range around a predetermined value. This helps to find converging solutions, e.g., in the presence of loops in a graph of a belief propagation decoding method.

[0071] According to an embodiment of the method for decoding a syndrome of a QEC code, for each transformation pair, the a-priori probability of the iterative decoding method is selected from a predetermined distribution of a-priori probabilities.

[0072] According to an embodiment of the method for decoding a syndrome of a QEC code, the distribution of a-priori probabilities is a uniform distribution of a- priori probability.

[0073] According to an embodiment of the method for decoding a syndrome of a QEC code, a scaled-MSA belief propagation decoding method is used which has a parameter ^^. According to the scaled-MSA belief propagation decoding method, the values of log-likelihood ratios computed by check nodes are scaled by the parameter ^^ which is a scaling factor. The scaling factor ^^ can be a constant or it can depend on the iteration number. For example, the scaling factor may be given by the formula ^^ = 1 − 2−^^^^ ,where ^^ is the iteration number and ^^ is a constant. For different transformation pairs, different decoder instances can be used with different values for the scalingfactor ^^ and / or the constant ^^. The performance of the method can increase by varying parameters of decoding different transformation pairs.

[0074] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the error associated with the syndrome comprises determining the error associated with the syndrome based on a subset of the permuted error estimates of the plurality of transformation pairs. Sometimes, it may not be necessary to include all of the permuted error estimates in the step of determining the error. This can speed up the computation.

[0075] According to an embodiment of the method for decoding a syndrome of a QEC code, the error associated with the syndrome is determined based on a subset of the permuted error estimates having a predefined size. The final calculation of the error can be based on a fixed number of permuted error estimates even if more converging error estimates have been determined, i.e., the number of converging solutions during decoding exceeds the predefined number of elements of the subset. The computational requirements for determining the error estimate can therefore be bounded. In some embodiments, if there are more permuted error estimates than the predetermined size of the subset, permuted error estimates for the subset can be selected randomly.

[0076] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the error associated with the syndrome comprises applying a maximum likelihood method to a set of the permuted error estimates of the plurality of transformation pairs in order to obtain the error associated with the syndrome. The maximum likelihood method may use a likelihood of the respective permuted error estimates. These likelihoods may be provided by the decoding method. For example, a belief propagation decoding method provides a likelihood together with the error estimate.

[0077] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the error associated with the syndrome comprises applying a majority vote method to a set of the permuted error estimates of the plurality of transformation pairs in order to obtain the error associated with the syndrome. It is assumed that the permuted error estimate which appears most often corresponds to the actual error.

[0078] According to an embodiment of the method for decoding a syndrome of a QEC code, determining the error with the syndrome comprises determining the error associated with the syndrome to be the permuted error estimate of the plurality of transformation pairs having the lowest number of single physical qubit errors. In general, the physical qubit is affected by an error with a certain probability p which is a small number close to zero. The probability of having an additional error therefore decreases with the single error probability p. Accordingly, it is much more likely that a smaller number of errors has occurred as compared to a larger number of errors. Therefore, the permuted converging error with the lowest number of single physical qubit errors is most likely the actual error.

[0079] According to an embodiment of the method for decoding a syndrome of a QEC code, the error estimates are determined in parallel and if a predetermined number of converging error estimates has been found, determination of the error estimates for the other transformation pairs is stopped. That is, if enough converging error estimates have been found, the error can be determined based on these error estimates and no further solutions are required. This may accelerate the process of decoding the syndrome because it is not necessary to wait until all error estimates are found.

[0080] According to an embodiment of the method for decoding a syndrome of a QEC code, the QEC code is a stabilizer QEC code and determining the syndrome associated with the quantum computation comprises performing a stabilizer measurement on the stabilizers of the QEC code.

[0081] According to an embodiment of the method for decoding a syndrome of a QEC code, the QEC code is a quantum low-density parity-check (QLDPC) code. QLDPC codes are a class of QEC codes that are inspired by classical low-density parity-check (LDPC) codes. QLDPC codes are defined by sparse parity-check matrices, meaning that each row and column of the matrix contains only a small number of non-zero entries. This sparsity leads to efficient encoding and decoding algorithms. The parity-check matrix can be represented as a bipartite Tanner graph, where one set of nodes represents qubits and the other set represents parity checks. The edges indicate which qubits participate in each parity check. The QLDPC code can be implemented using the stabilizer formalism. The sparse nature of QLDPC codes results in lower overhead in terms of the number ofphysical qubits required for encoding logical qubits, compared to other types of QEC codes.

[0082] For QLDPC codes, iterative decoders such as belief propagation decoders are known to be computationally efficient and can be implemented in a scalable manner. However, the error correction performance of QLDPC code under a belief propagation decoder can be impaired by many factors, such as the existence of short cycles, the existence of trapping sets, and degeneracy errors. Ordered Statistics (OS) decoder can be used in conjunction with belief propagation decoder for correcting all the non-converging errors at the output of belief propagation decoder. However, the time complexity induced by the OS decoder is at least in the order of ^^(^^3), where N is the number of physical qubits.

[0083] According to the specification, an AE-type decoder can be used having competitive error correction performance while inducing a lower time complexity. The AE-type decoder can be used with iterative decoders, in particular belief propagation decoders. Accordingly, in some embodiments, a respective iterative decoder is used to determine the error estimate based on the transformed syndrome and the parity-check matrix of the QEC code. In particular, the iterative decoder may be a belief propagation decoder.

[0084] According to an embodiment of the method for decoding a syndrome of a QEC code, the QEC code is a Reed-Muller code. Reed-Muller codes denote a class of linear error-correcting codes which can be used to define stabilizer codes.

[0085] According to an embodiment of the method for decoding a syndrome of a QEC code, the QEC code is a polar code. The polar code can be decoded using a belief propagation decoding method or by successive cancellation decoding (SCD).

[0086] According to an embodiment of the method for decoding a syndrome of a QEC code, the determined error associated with the syndrome is corrected. For example, gates corresponding to Pauli ^^ and ^^ operators can be applied to physical qubits in order to correct bit-flip (^^) errors and phase-flip (^^) errors affecting the physical qubits, respectively.

[0087] A second aspect of the disclosure provides a computer program product comprising executable program code configured to, when executed by acomputing device, perform the method for decoding a syndrome of a QEC code according to the first aspect.

[0088] A third aspect of the disclosure provides a non-transitory, computer- readable storage medium comprising executable program code configured to, when executed by a computing device, perform the method for decoding a syndrome of a QEC code according to the first aspect.

[0089] A fourth aspect of the disclosure provides a device for decoding a syndrome of a QEC code. The device comprises at least one processor and at least one tangible computer-readable storage device communicatively coupled to the at least one processor and which stores processor-executable instructions. When executed by the at least one processor, the processor-executable instructions cause the at least one processor to obtain the syndrome of the QEC code. The processor obtains a plurality of transformation pairs, each transformation pair comprising a first transformation and a second transformation, wherein applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant. For each transformation pair, the processor applies the first transformation to the syndrome to obtain a transformed syndrome and determines an error estimate based on the transformed syndrome and the parity-check matrix. The processor determines a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate. The at least one processor determines an error associated with the syndrome based on the permuted error estimates of the plurality of transformation pairs.

[0090] A fifth aspect of the disclosure provides an information processing system for decoding a syndrome of a QEC code. The information processing system comprises a quantum hardware system, at least one processor and at least one tangible computer-readable storage device communicatively coupled to the at least one processor and which stores processor-executable instructions. When executed by the at least one processor, the processor-executable instructions cause the at least one processor to obtain the syndrome of the QEC code. The processor obtains a plurality of transformation pairs, each transformation pair comprising a first transformation and a second transformation, wherein applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant. For each transformation pair, the processor applies the firsttransformation to the syndrome to obtain a transformed syndrome and determines an error estimate based on the syndrome and the parity-check matrix. The processor determines a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate. The at least one processor determines an error associated with the syndrome based on the permuted error estimates of the plurality of transformation pairs. The at least one processor controls the quantum hardware system to correct the determined error associated with the syndrome.

[0091] The disclosure relates to all combinations of the above features, even if these are recited in different aspects or different claims. In particular, the device for decoding a syndrome of a QEC code may be configured to execute each or a combination of the described embodiments of the method for decoding a syndrome of a QEC code Brief description of the drawings

[0092] In the following, further aspects and exemplary embodiments will be described with reference to the accompanying drawings. However, the present disclosure is not limited to the described exemplary embodiments and may be modified in various different ways. Consequently, the drawings and description are intended to be illustrative in nature and not limiting. Fig.1 schematically shows a block diagram illustrating an information processing system according to an embodiment of the disclosure; Fig.2 schematically shows a module of a quantum hardware system of the information processing system of Fig.1 according to an example embodiment of the disclosure; Fig.3 shows a flow diagram illustrating a method for decoding a syndrome of a QEC code according to an embodiment of the disclosure; and Fig.4 shows a flow diagram illustrating a method for decoding a syndrome of a QEC code according to an embodiment of the disclosure. Detailed description

[0093] Figure 1 schematically shows a block diagram illustrating an information processing system 100.

[0094] The information processing system 100 comprises a quantum hardware system 400 for performing quantum The quantum hardware system 400 can optionally be a modular system and comprise a plurality of interconnected chips or modules 401. The modules 401 can be coupled, e.g., by optical links or one or more optical networks, to entangle physical qubits on different modules 401. In other embodiments, the quantum hardware system 400 has only a single chip or module 401.

[0095] The quantum hardware system 400 is a physical device or machine which can be used to implement quantum algorithms (e.g., a quantum computer). The quantum hardware system 400 operates the multiple modules 401 when executing a quantum process and can perform “intra-modular” and “inter-modular” operations. The term “intra-modular operation” relates to operations (e.g., gates) between physical qubits of the same module 401. The term “inter-modular operation” relates to operations (e.g., gates) between physical qubits of different modules 401. In certain applications, it can be advantageous to link multiple modules 401 instead of building ever-larger monolithic quantum supercomputers (i.e., having only a single chip or module). Herein, the modules 401 are separate components and different modules 401 can be spatially separated.

[0096] The quantum hardware system 400 can be a non-locally connected quantum hardware system, as described in Simmons, “Scalable Fault-Tolerant Quantum Technologies with Silicon Colour Centres”, arXiv:2311.04858, 2023, and in Afzal et al., “Distributed Quantum Computing in Silicon”, arXiv: 2406.01704, which are hereby incorporated by reference in their entirety.

[0097] Figure 2 schematically shows module 401 of quantum hardware system 400 of information processing system 100 according to an example embodiment. In this embodiment, module 401 comprises physical qubits 401-1 that can be optically initialized into at least first and second quantum states that can be used to represent quantum information and which can exist in a quantum superposition.

[0098] Module 401 may include an excitation source 401-2 (e.g., a laser) for optically exciting physical qubits 401-1. Module 401 may include one or more antennas 401-5 for providing optical and microwave signals for controlling / manipulating the quantum state of physical qubits 401-1.

[0099] Module 401 may include one or more optical links 401-3 that facilitate optical coupling between different qubits 401-1. For example, optical links 401-3 may facilitate optical coupling between different physical qubits 401-1 via an optical switch 401-6 and / or a detector 401-7.

[0100] Optical switch 401-6 may be controlled to select specific optical links 401-3 for connecting a physical qubit of module 401 to at least one other physical qubit of module 401, either of the same module 401 or of another module 401. In some embodiments, by controlling the optical switches 401-6 in a suitable manner, any pair of physical qubits of the modules 401 may be connected to each other.

[0101] Detector 401-7 may be used in heralding entanglement between the physical qubits of the modules 401 based on a photon detection pattern of photon states associated with said physical qubits of the modules 401. The entanglement may be generated according to an entanglement protocol, such as the Barrett-Kok protocol.

[0102] The physical qubits 401-1 of module 401 can be matter qubits. Examples of matter qubits include luminescent defects, trapped ions, trapped atoms, neutral atoms, superconducting qubits, quantum dots, quantum wells, nuclear spins within dissolved molecules, trapped atoms coupled to high-finesse cavities, Bose- Einstein condensates, and the like.

[0103] In some embodiments, physical qubit 401-1 is a spin-photon interface that combines a long-lived solid-state spin and a photonic degree of freedom. Spins generally have long coherence times, so information can be stored for long times. For example, the quantum hardware system 400 may comprise a semiconductor body with luminescent defects which form the physical qubits of the modules 401. The semiconductor body may comprise silicon or similar semiconductor materials. For example, the semiconductor material may include natural silicon, silicon carbide, silicon germanium, isotopically purified paramagnetic silicon, a so-called silicon vacuum or combinations thereof. The semiconductor body may be processed to remove a large fraction of non-paramagnetic isotopes (e.g., silicon- 29). The semiconductor body may comprise enriched or purified silicon that has been processed to remove some to nearly all non-zero-nuclear spin isotopes, such as silicon-29. Purified silicon includes material enriched to various levels of silicon- 28, such as, 99%, 99.9%, and 99.99%. Purified silicon includes material enrichedwith silicon-28. Purified silicon includes silicon where spectroscopic linewidths are at least ten to hundred times sharper in natural silicon. The semiconductor body may also comprise an epilayer of isotopically purified silicon, grown on top of a natural silicon wafer.

[0104] The luminescent defects may comprise luminescence centres or colour centres. The luminescent defects may comprise radiation damage centres. The luminescent defects may comprise T centres, as described in any of US 2022 / 0366290 A1, US 2022 / 0327416 A1, and US 2024 / 0012749 A1, which are hereby incorporated by reference in their entirety.

[0105] In embodiments where physical qubits 401-1 of module 401 are associated with T centres, pairs of T centres may be optically connected by means of a telecom photonic interface (i.e., operating in the telecom frequency band, such as the telecom O-band) of the T centre in the silicon substrate. In some embodiments, each T centre can be optically connected to any other T centre, i.e., all-to-all optical connection between the T centres is possible. The optical connection (e.g., optical links) between a pair of T centres may comprise at least one photonic waveguide integrated in the silicon substrate (“on-chip”). In addition to or alternatively, the optical connection can also comprise optical fibres or other components which can be external to the silicon substrate (“off-chip”). In particular, the modules 401 can be arranged in different chips and coupling devices can comprise at least some off-chip hardware components for coupling physical qubits on different modules 401.

[0106] The optical connection may be configured to facilitate entanglement between T centres. The entanglement can be generated and / or distributed by photons which are transmitted over the optical connections connecting the T centres. The quantum hardware system 400 can be configured to prepare maximally entangled Bell pairs, using the T centres.

[0107] The quantum hardware system 400 may be used for any practical application in quantum sensing, quantum computing or quantum communication.

[0108] Quantum sensing comprises measurements which utilize quantum effects such as entanglement, interference, or quantum state squeezing.

[0109] Quantum computing comprises any processing of information based on quantum effects, such as physical qubits of the modules 401 and (de-)coherence or entanglement of physical qubits of the modules 401.

[0110] Quantum communication comprises the transmission of classical information or of quantum states between different devices, e.g., between the quantum hardware system 400 and another quantum hardware system based on quantum effects as described above.

[0111] Quantum hardware system 400 is operable to execute a quantum algorithm. The quantum algorithm may be represented as a logical quantum circuit. Logical quantum circuits can be implemented on the quantum hardware system 400 by physical quantum circuits. Logical quantum circuits comprise a plurality of logical operators, such as gates.

[0112] According to this specification, the term “logical quantum circuit” denotes a quantum computation. The logical quantum circuit generally comprises a sequence of operations, such as quantum gates, measurements, and initializations of logical qubits to predefined values. Herein, logical quantum gates are basic blocks that operate on a small number of logical qubits.

[0113] In order to implement the logical quantum circuit, the logical qubits are implemented by (a generally larger number of) physical qubits of the quantum hardware system 400.

[0114] According to the specification, a “physical operator implementation” of a logical operator denotes a physical gate or more generally a sequence of physical gates that implement the logical operator. The physical gates are not necessarily already gates acting on hardware physical qubits of the quantum hardware system 400. Rather, the physical gates can act on “physical” (or virtual) qubits of a virtual circuit which is implemented on a physical circuit (i.e., the quantum hardware system 400) at a subsequent stage.

[0115] Returning now to Fig.1, the information processing system 100 comprises a syndrome determining device 600 for determining a syndrome of a quantum error correcting (QEC) code. The syndrome determining device 600 may comprise at least one measurement device for performing measurements on the quantumhardware system 400. The measurements may include measurement of the stabilizers of a QEC code.

[0116] The syndrome determining device 600 may determine the syndrome to detect errors in a quantum state of at least one physical qubit without collapsing the quantum state itself. In some embodiments, auxiliary qubits are prepared which interact with the physical qubits holding encoded quantum information of a quantum operation. Using quantum gates, the auxiliary qubits are entangled with the physical qubits holding the encoded quantum information. The type of the quantum gates can depend on stabilizers of the QEC code. The quantum gates may comprise controlled-NOT (CNOT) gates.

[0117] Herein, the term “CNOT gate” denotes a two-qubit gate in quantum computing that performs a NOT operation (bit-flip) on the second qubit (target qubit) only if the first qubit (control qubit) is in the state |1^. In matrix form, acting on the states|00^, |01^, |10^, and |11^, the CNOT gate is represented as: 10 0 00 1 0 0.

[0118] The syndromethe auxiliary qubits in the computational basis. The outcome of the measurement provides the syndrome, which is a binary vector indicating which stabilizers are violated. In some embodiments, the syndrome is derived from measuring all stabilizers of the QEC code. In some embodiments, the syndrome is derived from measuring a subset of stabilizers of the QEC code.

[0119] The information processing system 100 further comprises a device 200 for decoding a syndrome of a QEC code. The device 200 for decoding a syndrome of a QEC code comprises at least one processor 203, and at least one memory 204 (i.e., a tangible computer-readable, or processor-readable storage device) communicatively coupled to the at least one processor 203.

[0120] The processor 203 can be a logic processing unit and can comprise a central processing unit (CPU), a graphics processing unit (GPU), a microcontroller (μC), an integrated circuit (IC), an application-specific integrated circuit (ASIC), adigital signal processor (DSP), a field programmable gate array (FPGA), a program logic unit (PLU), a network processor or a combination thereof.

[0121] The memory 204 can comprise at least one of a magnetic hard disk, an optical disc (e.g., compact disc, digital video disc, Blu-ray disc), a solid-state disc (SSD), a magneto-optical memory or a hard disc drive (HDD). For example, the memory 204 can comprise a volatile semiconductor or solid-state memory, e.g., a random-access memory (RAM), dynamic RAM (DRAM), or static RAM (SRAM). The memory 204 can comprise a non-volatile semiconductor or solid-state memory, e.g., a read only memory (ROM), programmable ROM (PROM), erasable PROM (EPROM), or the like.

[0122] The memory 204 stores processor-executable instructions and / or processor-readable data associated with the operation of the device 200 for decoding a syndrome of a QEC code. Specifically, when executed by processor 203, the instructions cause the processor 203 to decode a syndrome of a QEC code. The processor-executable instructions and / or processor-readable data can comprise an operating system, peripheral drivers, server instructions, application instructions, calibration instructions, or communication channel instructions.

[0123] The device 200 for decoding a syndrome of a QEC code further comprises a user interface 205, having at least one of a display, a keyboard, a touch screen, a mouse, buttons, a microphone, loudspeakers, and the like. A user may provide or receive information regarding the operation of the information processing system 100 via the user interface 205.

[0124] Although the user interface 205 is illustrated as a component of the device 200 for decoding a syndrome of a QEC code, the user interface 205 can also be an external device of the information processing system 100. For example, the device 200 for decoding a syndrome of a QEC code can be implemented as a remote server which a user can access via the user interface 205.

[0125] An interface 202 is provided for connecting the processor 203 with the quantum hardware system 400, the syndrome determining device 600, a cooling device 500 and an actuator device 300. The interface 202 can be any port or link or interface capable of communicating information to another system, e.g., a wiredconnection or a wireless connection (e.g., wireless LAN, Bluetooth®, ethernet, or the like).

[0126] All of the components of the device 200 for decoding a syndrome of a QEC code described above can be controlled and / or can communicate over at least one bus 201. The processor 203 may be configured to control the interface 202 and user interface 205 of the device 200 for decoding a syndrome of a QEC code.

[0127] The information processing system 100 comprises the cooling device 500. The cooling device 500 may maintain the quantum hardware system 400 at a predefined operating temperature of the quantum hardware system 400. The operating temperature may be a cryogenic temperature, e.g., in a range from about 1 mK to 77 K, or more particularly in a range from about 1.5 K to 4 K. In some embodiments, the cooling device 500 may be omitted. The quantum hardware system 400 may, optionally, also be kept at constant air pressure, e.g., a stable vacuum.

[0128] The information processing system 100 comprises the actuator device 300. The actuator device 300 can comprise a plurality of actuators. For example, the actuator device 300 can comprise an electromagnet to apply a time-invariant electric field, a time-varying electric field, or a pulsed electric field to the quantum hardware system 400.

[0129] The quantum hardware system 400 can be operated in order to perform quantum operations, such as quantum communication or quantum computation. A logical quantum circuit can be provided together with a QEC code. The logical quantum circuit and / or the QEC code can be predetermined or obtained, e.g., from a user via the user interface 205 or from a remote device (e.g., a remote server). In some embodiments, a user can provide the logical quantum circuit to a job handler of a Quantum as a Service (QaaS) platform.

[0130] A physical implementation of the logical quantum circuit may be determined, for example, based on the QEC code, to determine a virtual quantum circuit comprising physical qubits. The physical qubits are mapped to hardware physical qubits of the quantum hardware system 400 in order to obtain a physical quantum circuit which is executed on the quantum hardware system 400 to perform the quantum operations.

[0131] During the operation of the quantum hardware system 400, the syndrome determining device 600 determines the of a QEC code and provides the syndrome to the device 200 for decoding a syndrome of a QEC code which decodes the syndrome.

[0132] Decoding the syndrome is performed using a plurality of transformation pairs associated with a parity check matrix of the QEC code. Each transformation pair comprises a first transformation and a second transformation. The first transformation transforms each row of the parity check matrix of the QEC code into a linear combination of rows of the parity check matrix. The second transformation permutes columns of the parity-check matrix. Each transformation pair is such that, when applied to the parity check matrix, the parity check matrix remains unchanged (i.e., invariant).

[0133] The processor 203 carries out a decoding process for each transformation pair or at least some of the transformation pairs of the plurality of transformation pairs. In some embodiments, the decoding processes for the transformation pairs are performed in parallel, i.e., using parallel computing. For each pair, the processor 203 applies the first transformation to the syndrome of the QEC code to obtain a transformed syndrome. The processor 203 then determines an error estimate based on the transformed syndrome and the parity-check matrix of the QEC code. Next, the processor 203 determines a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate.

[0134] Using the permuted error estimates, the processor 203 determines an error associated with the syndrome. The error can correspond to at least one bit-flip error and / or at least one phase-flip error that affects at least one physical qubit.

[0135] The device 200 for decoding a syndrome of a QEC code may cause the quantum hardware system to correct the error associated with the syndrome. For example, device 200 for decoding a syndrome of a QEC code may provide a control command to the quantum hardware system 400 that causes the quantum hardware system 400 to correct the error associated with the syndrome. In some embodiments, the device 200 for decoding a syndrome of a QEC code may control the quantum hardware system 400 to apply one or more quantum gates in order to correct the determined error. For example, gates corresponding to Pauli ^^ and ^^operators can be applied to physical qubits in order to correct bit-flip (^^) errors and phase-flip (^^) errors affecting the respectively.

[0136] Figure 3 shows a flow diagram illustrating a method for decoding a syndrome of a QEC code. The method can be performed using the device 200 for decoding a syndrome of a QEC code of Figure 1. The method may be carried out by processor 203 when executing instructions processor-executable instructions stored by memory 204.

[0137] The QEC code can be a predetermined QEC code or can be obtained from a user. For example, the user may provide, via a user interface (e.g., user interface 205), a parity check matrix of the QEC code. Further, a logical quantum circuit can be obtained from the user or can be predetermined. Based on the logical quantum circuit and the parity check matrix of the QEC code, a physical quantum circuit is determined and executed on a quantum hardware system 400.

[0138] In some embodiments, the QEC code is a stabilizer code, e.g., a CSS code. The QEC code can be a quantum Reed-Muller code, or a quantum polar code. The QEC code can in some embodiments be a QLDPC code.

[0139] In step S101, a syndrome ^^^^of the QEC code is obtained. The syndrome ^^^^is a binary vector obtained by multiplying the transpose of an unknown error vector ^^0by the parity check matrix ^^, i.e., ^^^^ = ^^ ⋅ ^^^^^^.

[0140] The syndrome can be obtained by measuring stabilizers of the QEC code using the syndrome determining device 600. For example, auxiliary qubits can become entangled with the physical qubits holding the encoded quantum information. The auxiliary qubits can be measured in the computational basis in order to determine the syndrome. In some embodiments, the syndrome is obtained from a user via a user interface (e.g., user interface 205).

[0141] In step S102, a plurality of transformation pairs associated with the parity check matrix of the QEC code is obtained. The plurality of transformation pair may, for example, be provided by a user, via a user interface (e.g., user interface 205). In some embodiments, the plurality of transformation is generated by processor 203 based on the parity check matrix of the QEC code. In some embodiments, the parity check is retrieved from memory 204.

[0142] The size (i.e., number of transformation pairs) of the plurality of transformation pairs may, for example, based on the structure (e.g., symmetry) of the parity check matrix.

[0143] Each transformation pair in the plurality of transformation pairs comprises a first transformation and a second transformation. The first transformation transforms each row of a parity check matrix of the QEC code into a linear combination of rows of the parity check matrix. The second transformation permutes columns of the parity-check matrix. Applying each transformation pair (i.e., both the first and second transformations) to the parity check matrix leaves the parity check matrix invariant.

[0144] In some embodiments, each transformation pair is an automorphism of the QEC code, and the plurality of transformation pairs forms an automorphism group of the QEC code.

[0145] In step 103, a transformation pair is selected from the plurality transformation pairs to be used in a decoding process. The selected transformation pair is the ithtransformation pair for some integer ^^ between 1 and ^^, where ^^ denotes the total number of transformation pairs that are taken into account for decoding the syndrome.

[0146] In step S104, the first transformation of the selected transformation pair is applied to the syndrome to obtain (e.g., determine) a transformed syndrome. The transformed syndrome ^^^^of the selected transformation pair can be computed according to: ^^^^ = ^^^^ ⋅ ^^^^ ,where ^^^^is a linear transformation matrix corresponding to the first transformation of the selected transformation pair. In some embodiments, the first transformation is an invertible linear transformation.

[0147] Step S105 corresponds to a decoding step, i.e., an error estimate is determined based on the transformed syndrome ^^^^and the parity-check matrix ^^ of the QEC code, using a decoding method. The decoding method that is used for determining the error estimate can depend on the type of the QEC code. For example, the QEC code can be a QLDPC code and the decoding method can be a syndrome-based decoding method. In some embodiments, the syndrome-baseddecoding method is an iterative decoding method, e.g., a belief propagation method.

[0148] In step S106, it is checked whether the error estimate determined in step S105 satisfies a predefined convergence condition (i.e., whether the predefined convergence condition is met). For example, the error estimate ^^^^can be classified as a converging error estimate if the error estimate is an error in compliance with the transformed syndrome ^^^^. This can comprise determining whether the error estimate satisfies the following equation: ^^ ⋅ ^^ ^^^^ = ^^^^ .

[0149] Herein, ^^^^^^is the transpose of the error estimate and ^^ is the parity check matrix of the QEC code.

[0150] If the error estimate does not satisfy the convergence condition (i.e., does not satisfy the above equation), the error estimate of the corresponding transformation pair is discarded, S107. For example, the error estimate may not satisfy the above condition after a predetermined maximum number of iterations. In this case, it can be determined that the convergence condition is not satisfied, in which case said error estimate is discarded.

[0151] If the convergence condition is satisfied, a permuted error estimate ^^′^^is computed (e.g., determined) by applying the second transformation to the error estimate in step S108. The permuted error estimate ^^′^^can be computed according to: ^^′^^ = ^^^^ ⋅ ^^^−^^^, wherein ^^−^^^^is an inverse of the permutation matrix ^^^^which corresponds to the second transformation of the selected transformation pair. The linear transformation matrix ^^^^and the permutation matrix ^^^^satisfy the following relation: ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ .

[0152] In step S109, the permuted error estimate is stored. For example, the permuted error estimate may be stored in memory 204.

[0153] In step S110, the processor determines if there are any transformation pairs remaining. If there is no transformation pair remaining, the error associated with the syndrome is determined using the permuted error estimate(s), step S111. Forexample, the error can be determined by applying a majority vote method to the permuted error estimates stored in 204 at step S109. In other words, the most frequently occurring converging error estimate is identified as the error associated with the syndrome.

[0154] In some embodiments, the error associated with the syndrome is determined based on the permuted error estimate having the lowest number of single physical qubit errors. The probability of an error is relatively small. Accordingly, the probability of having a large number of errors is much smaller than the probability of having a small number of errors. Therefore, the error associated with the syndrome is determined by the permuted error estimate with the lowest number of single physical qubit errors.

[0155] In other embodiments, a maximum likelihood method is applied to the permuted error estimates in order to obtain the error associated with the syndrome. In some cases, the decoding method provides a likelihood of the respective permuted error estimate. The likelihood can be considered in the maximum likelihood method in order to determine the error based on the permuted error estimates.

[0156] In some embodiments, only a subset of the permuted error estimates of the plurality of transformation pairs is used to determine the error.

[0157] The determined error can be used for correcting errors. For example, gates corresponding to Pauli ^^ and ^^ operators can be applied to physical qubits in order to correct bit-flip (^^) errors and phase-flip (^^) errors affecting the physical qubits, respectively.

[0158] The decoding method used for determining the error estimate can be the same for all transformation pairs. In other embodiments, a plurality of decoding methods can be used. In some cases, a separate decoding method is used for each transformation pair. For example, different types of decoding method can be used. In other embodiments, different parameters can be used for different transformation pairs. For example, iterative decoding methods are used which differ in an a-priori probability of the iterative decoding method. The a-priori probability corresponds to what is expected to be the actual error rate of the quantum hardware system 400. A variation can be included by using different a-priori probabilities for different transformation pairs. For example, if the actual error rate of the quantum information system expected to be a predetermined value (e.g., 0.1), then the a-priori probability may be selected from a predetermined range around this predetermined value, e.g., ±10% around this value.

[0159] In some embodiments, the error estimate is determined using a scaled-MSA belief propagation method. According to this method, the values of the log- likelihood ratios computed by check nodes are scaled by a scaling factor, ^^. The scaling factor ^^ can be a constant (taking values between 0 and 1) or it can depend on the iteration number. For example, the scaling factor may be given by the formula ^^ = 1 − 2−^^^^ ,where ^^ is the iteration number and ^^ is a constant. For different transformation pairs, different decoder instances can be used with different values for the scaling factor ^^ and / or the constant ^^. For example, the value of ^^ may be sampled from a uniform distribution in a predetermined range. For example, the range can be given by [(1 − ^^) ⋅ ^^0 , (1 + ^^) ⋅ ^^0],for some value ^^0 (e.g., ^^0 = 0.5) and some value ^^ (e.g., ^^ = 0.1).

[0160] The decoding processes for the respective transformation pairs can be serially computed, i.e., one decoding process after another. In other embodiments, all decoding processes are performed in parallel, i.e., steps S103 to S109 are performed in parallel for all transformation pairs.

[0161] Figure 4 shows a flow diagram illustrating a method for decoding a syndrome of a QEC code. A plurality of parallel decoders is used to decode the syndrome. In some embodiments, each decoder in the plurality of decoders performs a respective syndrome-based decoding method. Each decoder is associated with a respective transformation pair of a plurality of transformation pairs that is determined based on the parity check matrix of the QEC code as described below. Each transformation pair comprises a first transformation which is represented by a linear transformation matrix ^^^^and a second transformationwhich is represented by a permutation matrix ^^^^ for ^^ = 1, … , ^^, where ^^ is thetotal number of transformation pairs.

[0162] The first transformation transforms each row of a parity check matrix ^^ of the QEC code into a linear combination rows of the parity check matrix ^^. The second transformation permutes columns of the parity-check matrix ^^. Applying both the transformation pair (i.e., both the first transformation and the second transformation) to the parity check matrix ^^ leaves the parity check matrix invariant ^^, i.e., the linear transformation matrix ^^^^and the permutation matrix ^^^^satisfy the condition: ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ .

[0163] In step S201, a syndrome of the QEC code is transformed using the first transformation, i.e., a transformed syndrome is computed according to: ^^^^ = ^^^^ ⋅ ^^^^for all transformation pairs with a corresponding linear transformation matrix ^^^^.

[0164] In step S202, the transformed syndrome is decoded and a corresponding error estimate ^^^^is determined: ^^^^ = Decode(^^^^, ^^) .

[0165] Parallel decoding is used for decoding all transformed syndromes. Decoding can be based on a master decoder which comprises multiple copies of a syndrome-based decoder, for example. The decoding method can be an iterative decoding method, e.g., a belief propagation method.

[0166] In step S203, convergence elimination is performed, i.e., it is checked whether the determined error estimates ^^^^satisfy a convergence condition. An error estimates ^^^^is converging if the following condition holds: ^^ ⋅ ^^ ^^^^ = ^^^^ ,where, ^^^^^^is the transpose of the error estimate.

[0167] In step S204, a decoding error is returned if the convergence condition is not satisfied and the respective error estimates ^^^^is discarded and ignored in the remaining steps in the method.

[0168] In step S205, if the convergence condition is satisfied, the error estimates ^^^^is permuted. The permuted error estimate ^^′^^is calculated according to: ^^′^^ = ^^^^ ⋅ ^^^−^^^,where ^^−^^^^is an inverse of the permutation matrix ^^^^which corresponds to the second transformation of the pair.

[0169] In some embodiments, a respective decoder performs steps S201 to S205 using a corresponding transformation pair. The corresponding permuted error estimate from each decoder may be stored locally or provided to a master decoder that determines an error associated with the syndrome based on the permuted error estimates.

[0170] In step S206, the error ^^0associated with the syndrome is determined based on a likelihood of each successfully decoded solution, i.e., each permuted error estimate ^^′^^: ^^0 = arg max^^(^^′^^) . ^^

[0171] Herein, ^^(^^′^^) denotes the likelihood of the permuted error estimates ^^′^^. The value of ^^(^^′^^) can be an output of the iterative method, e.g., belief propagation.

[0172] In other embodiments, the error ^^0associated with the syndrome is determined based on a majority vote among the permuted error estimates ^^′^^. In some embodiments, the error ^^0associated with the syndrome is determined bases on the error estimate ^^′^^with the lowest number of single physical qubit errors.

[0173] As will appreciated, the error associated with the syndrome is determined based on the permuted error estimate of at least one of the decoders.

[0174] In summary, the technology provides methods, devices, and systems for decoding a syndrome of a QEC code with high performance and low time complexity. The technology may be implemented using a plurality of individual decoders. Each decoder uses a corresponding transformation pair comprising a linear transformation matrix and a permutation matrix in determining respective permuted error estimate. An error associated with the syndrome is determined based on the permuted error estimates.

[0175] The devices, apparatuses and systems described in the present disclosure may comprise electronic components and circuits known to those skilled in the art. Therefore, details of the circuitry and its components have not be explained in anygreater extent than that considered necessary for the understanding and appreciation of the underlying concepts the present disclosure.

[0176] Where reference is made to a component, such as a device, component, software module or the like, the reference to that component is intended to include as equivalents any component being functionally equivalent, i.e., performing the same function, even though the component is not necessarily structurally equivalent to the component that performs in the exemplary embodiments.

[0177] In the above description, embodiments have been described with reference to specific details, e.g., parts of a method, components, materials, and the like. A person skilled in the art will understand that embodiments may be implemented without one or more of these specific details.

[0178] All of the US patents, US patent application publications, US patent applications, foreign patents, foreign patent applications, and non-patent publications referred to in this specification, or referred to on any application data sheet, are incorporated by reference in their entireties for all purposes herein.

[0179] A person skilled in the art may understand that certain method steps may be described or depicted in a particular order of occurrence while such specificity with respect to sequence is not actually required.

[0180] Phrases like “an embodiment” and “another embodiment” are used in the sense that particular features described in connection with the embodiment are included in at least one embodiment. Those phrases do not necessarily all refer to the same embodiment. Terms such as "first", "second", “third”, and so on, are used to distinguish between the elements described by these terms. These terms do not necessarily imply any temporal or other prioritization of such elements.

[0181] As used herein, the singular forms "a," "one," and "the " are also intended to encompass the plural forms unless the context indicates otherwise. In addition, it is understood that the expressions "includes" and / or "including" when used in this specification relates to the presence of features, numbers, steps, operations, elements, and / or components, but does not exclude the presence or addition of one or more features, numbers, steps, operations, elements, and / or combinations thereof. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed elements.

[0182] Terms such as “horizontal”, “vertical”, “upper”, “lower”, “above”, “below”, “forward” and “backward” refer to orientations of components and / or events in time and / or space. The skilled person understands that may therefore depend on the specific orientation and may change if the components and / or events are oriented differently.

[0183] In this specification, the present disclosure has been described with reference to the accompanying drawings, in which exemplary embodiments are shown. However, the present disclosure is not limited to the described exemplary embodiments described and may be modified in various different ways. Consequently, the drawings and description are intended to be illustrative in nature and not limiting. Identical reference numbers denote identical elements in the specification.

[0184] As used herein, the terms "about," "approximately," or "substantially" refer to a value, amount, or property that is close to the specified value, amount, or property. The value, amount, or property is such that a desired function or result is still achieved. According to an example, an amount may be less than 10%, 5%, 1%, or 0.1% of the specified amount, respectively.

[0185] Even if the disclosure has been described and illustrated with reference to illustrative embodiments, various modifications may be made without departing from the scope of the present disclosure as defined in the claims. Such modifications may comprise replacement of features, components and / or method steps with equivalent features, components and / or method steps; mixing of features, components and / or method steps from different embodiments; and / or omitting or combining features, components and / or method steps from described embodiments. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.

Claims

WHAT IS CLAIMED IS:

1. A method for decoding a syndrome of a quantum error correction, QEC, code, the method comprising the steps: obtaining the syndrome of the QEC code; obtaining a plurality of transformation pairs, each transformation pair comprising a first transformation and a second transformation, wherein applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant; for each transformation pair, applying the first transformation to the syndrome to obtain a transformed syndrome, and determining an error estimate based on the transformed syndrome and the parity-check matrix; determining a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate; and determining an error associated with the syndrome based on the permuted error estimates.

2. The method of claim 1, wherein, for the ithtransformation pair of the plurality of transformation pairs, the first transformation is represented by a linear transformation matrix ^^^^and the second transformation is represented by a permutation matrix ^^^^which satisfy: ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ ,wherein ^^ is the parity-check matrix of the QEC code.

3. The method of claim 2, wherein, for the ithtransformation pair, applying the first transformation to the syndrome ^^^^to obtain the transformed syndrome ^^^^comprises computing the transformed syndrome ^^^^according to: ^^^^ = ^^^^ ⋅ ^^^^.

4. The method of claim 2 or 3, wherein determining the permuted error estimate for at least one of the error estimates comprises: determining whether each error estimate satisfies a predefined condition; anddetermining, for each error estimate that satisfies the predefined condition, the permuted error based on the second transformation and the error estimate.

5. The method of claim 4, wherein the predefined condition is given by: ^^ ⋅ ^^ ^^^^ = ^^^^where ^^^^^^is a transpose of the error estimate, and ^^^^is the transformed syndrome.

6. The method of any of claims 4 or 5, wherein determining the permuted error estimate ^^′^^based on the second transformation and the error estimate comprises computing the permuted error estimate ^^′^^according to: ^^′^^ = ^^^^ ⋅ ^^−^^^^, wherein ^^−^^^^is an inverse of the permutation matrix ^^^^.

7. The method of any of the preceding claims, wherein determining the error estimate comprises applying a syndrome-based decoding method to the transformed syndrome and the parity-check matrix of the QEC code in order to determine the error estimate.

8. The method of claim 7, wherein the syndrome-based decoding method is an iterative decoding method.

9. The method of claim 8, wherein the iterative decoding method is a belief propagation decoding method.

10. The method of claim 9, wherein the belief propagation decoding method is a scaled minimum sum algorithm, scaled-MSA, belief propagation decoding method.

11. The method of any of claims 8 to 10, wherein differing iterative decoding method are used in determining the error rates for at least some of the transformation pairs.

12. The method of claim 11, wherein the differing iterative decoding methods differ in a type of the iterative method.

13. The method of claim 11 or 12, wherein the differing iterative decoding methods differ in at least one parameter of the iterative decoding method.

14. The method of any of claims 11 to 13, wherein the differing iterative decoding methods differ in an a-priori probability of the iterative decoding method.

15. The method of claim 14, wherein, for each transformation pair, the a-priori probability of the iterative decoding method is selected from a predetermined distribution of a-priori probabilities.

16. The method of claim 15, wherein the distribution of a-priori probabilities is a uniform distribution of a-priori probabilities.

17. The method of any of the preceding claims, wherein determining the error associated with the syndrome comprises: determining the error associated with the syndrome based on a subset of the permuted error estimates.

18. The method of claim 17, wherein the error associated with the syndrome is determined based on the subset of the permuted error estimates having a predefined size.

19. The method of any of the preceding claims, wherein determining the error associated with the syndrome comprises: applying a maximum likelihood method to a set of the permuted error estimates of the plurality of transformation pairs in order to obtain the error associated with the syndrome.

20. The method of any of claims 1 to 18, wherein determining the error associated with the syndrome comprises:applying a majority vote method to a set of the permuted error estimates of the plurality of pairs in order to obtain the error associated with the syndrome.

21. The method of any of claims 1 to 18, wherein the error associated with the syndrome corresponds to the permuted error estimate having the least number of single physical qubit errors.

22. The method of any of the preceding claims, wherein the QEC code is a stabilizer QEC code and wherein obtaining the syndrome of the QEC code comprises performing a stabilizer measurement on stabilizers of the QEC code.

23. The method of any of the preceding claims, wherein the QEC code is a quantum low-density parity-check, QLDPC, code.

24. The method of any of claims 1 to 22, wherein the QEC code is a quantum Reed-Muller code.

25. The method of any of claims 1 to 22, wherein the QEC code is a quantum polar code.

26. The method of any of the preceding claims, further comprising the step of correcting the determined error associated with the syndrome.

27. A computer program product comprising executable program code configured to, when executed by a computing device, perform the method according to any of claims 1 to 26.

28. A non-transitory, computer-readable storage medium comprising executable program code configured to, when executed by a computing device, perform the method according to any of claims 1 to 26.

29. A device for decoding a syndrome of a quantum error correction, QEC, code, comprising: at least one processor; and at least one tangible computer-readable storage device communicatively coupled to the at least one processor and which stores processor-executable instructions which, when executed by the at least one processor, cause the at least one processor to: obtain the syndrome of the QEC code; obtain a plurality of transformation pairs, each transformation pair comprising a first transformation and a second transformation, wherein applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant for each transformation pair, apply the first transformation to the syndrome to obtain a transformed syndrome, and determine an error estimate based on the transformed syndrome and the parity-check matrix; determine a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate; and determine an error associated with the syndrome based on the permuted error estimates.

30. The device of claim 29, wherein, for the ithtransformation pair of the plurality of transformation pairs, the first transformation is represented by a linear transformation matrix ^^^^and the second transformation is represented by a permutation matrix ^^^^which satisfy: ^^^^ ⋅ ^^ ⋅ ^^^^ = ^^ ,wherein ^^ is the parity-check matrix of the QEC code.

31. The device of claim 30, wherein, for the ithtransformation pair, the processor is configured to apply the first transformation to the syndrome ^^^^to obtain the transformed syndrome ^^^^by computing the transformed syndrome ^^^^according to: ^^^^ = ^^^^ ⋅ ^^^^.

32. The device of claim 30 or 31, wherein the processor is configured to determine the permuted error for at least one of the error estimates by: determining whether each error estimate satisfies a predefined condition; and determining, for each error estimate that satisfies the predefined condition, the permuted error estimate based on the second transformation and the error estimate.

33. The device of claim 32, wherein the predefined condition is given by: ^^ ⋅ ^^ ^^^^ = ^^^^where ^^^^^^is a transpose of the error estimate, and ^^^^is the transformed syndrome.

34. The device of claim 32 or 33, wherein the processor is configured to determine the permuted error estimate ^^′^^based on the second transformation and the error estimate by computing the permuted error estimate ^^′^^according to: ^^′^^ = ^^^^ ⋅ ^^−^^^^, wherein ^^−^^^^is an inverse of the permutation matrix ^^^^.

35. The device of any of claims 29 to 34, wherein the processor is configured to determine the error estimate by applying a syndrome-based decoding method to the transformed syndrome and the parity-check matrix of the QEC code in order to determine the error estimate.

36. The device of claim 35, wherein the syndrome-based decoding method is an iterative decoding method.

37. The device of claim 36, wherein the iterative decoding method is a belief propagation decoding method.

38. The device of claim 37, wherein the belief propagation decoding method is a scaled minimum sum MSA, belief propagation decoding method.

39. The device of any of claims 36 to 38, wherein the processor is configured to use differing iterative decoding methods in determining the error rates for at least some of the transformation pairs.

40. The device of claim 39, wherein the differing iterative decoding methods differ in a type of the iterative decoding method.

41. The device of claim 39 or 40, wherein the differing iterative decoding methods differ in at least one parameter of the iterative decoding method.

42. The device of any of claims 39 to 41, wherein the differing iterative decoding methods differ in an a-priori probability of the iterative decoding method.

43. The device of claim 42, wherein, for each transformation pair, the processor is configured to select the a-priori probability of the iterative decoding method from a predetermined distribution of a-priori probabilities.

44. The device of claim 43, wherein the distribution of a-priori probabilities is a uniform distribution of a-priori probabilities.

45. The device of any of claims 29 to 44, wherein the processor is configured to determine the error associated with the syndrome by: determining the error associated with the syndrome based on a subset of the permuted error estimates.

46. The device of claim 45, wherein the processor is configured to determine the error associated with the syndrome based on the subset of the permuted error estimates having a predefined size.

47. The device of any of claims 29 to 46, wherein the processor is configured to determine the error associated the syndrome by: applying a maximum likelihood method to a set of the permuted error estimates of the plurality of transformation pairs in order to obtain the error associated with the syndrome.

48. The device of any of claims 29 to 47, wherein the processor is configured to determine the error associated with the syndrome by: applying a majority vote method to a set of the permuted error estimates of the plurality of transformation pairs in order to obtain the error associated with the syndrome.

49. The device of any of claims 29 to 48, wherein the error associated with the syndrome corresponds to the permuted error estimate having the least number of single physical qubit errors.

50. The device of any of claims 29 to 49, wherein the QEC code is a stabilizer QEC code and wherein the processor is configured to obtain the syndrome of the QEC code based on a stabilizer measurement on stabilizers of the QEC code.

51. The device of any of claims 29 to 50, wherein the QEC code is a quantum low-density parity-check, QLDPC, code.

52. The device of any of claims 29 to 50, wherein the QEC code is a Reed- quantum Muller code.

53. The device of any of claims 29 to 50, wherein the QEC code is a quantum polar code.

54. The device of any of claims 29 to 53, wherein the processor is configured to control a quantum hardware system to correct the determined error associated with the syndrome.

55. An information processing system for decoding a syndrome of a quantum error correction, QEC, code, a quantum hardware system; at least one processor communicatively coupled to the quantum hardware system; and at least one tangible computer-readable storage device communicatively coupled to the at least one processor and which stores processor-executable instructions which, when executed by the at least one processor, cause the at least one processor to: obtain a plurality of transformation pairs, each transformation pair comprising a first transformation and a second transformation, wherein applying each transformation pair to a parity check matrix of the QEC code leaves the parity check matrix invariant; obtain the syndrome of the QEC code associated by executing a quantum operation on the quantum hardware system; for each transformation pair, apply the first transformation to the syndrome to obtain a transformed syndrome, and determine an error estimate based on the transformed syndrome and the parity-check matrix; determine a permuted error estimate for at least one of the error estimates based on the second transformation and the error estimate; determine an error associated with the syndrome based on the permuted error estimates; and control the quantum hardware system to correct the determined error associated with the syndrome.

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