Quantum error correction (QEC) is essential for operating quantum computers in the presence of noise. Here, we accurately decode arbitrary Calderbank-Shor-Steane (CSS) codes via the maximum satisfiability (MaxSAT) problem. We show how to map quantum maximum likelihood problem of CSS codes of arbitrary geometry and parity check weight into MaxSAT problems. We incorporate the syndrome measurements as hard clauses, while qubit and measurement error probabilities, including biased and non-uniform, are encoded as soft MaxSAT clauses. For the code capacity of color codes on a hexagonal lattice, our decoder has a higher threshold and superior scaling in noise suppression compared to belief propagation with ordered statistics post-processing (BP-OSD), while showing similar scaling in computational cost. Further, we decode surface codes and recently proposed bivariate quantum low-density parity check (QLDPC) codes where we find lower error rates than BP-OSD. Finally, we connect the complexity of MaxSAT decoding to a computational phase transition controlled by the clause density of the MaxSAT problem, where we show that our mapping is always in the computationally ''easy`` phase. Our MaxSAT decoder can be further parallelised or implemented on ASICs and FPGAs, promising potential further speedups of several orders of magnitude. Our work provides a flexible platform towards practical applications on quantum computers.
Quantum error correction (QEC) is essential for operating quantum computers in the presence of noise. Here, we accurately decode arbitrary Calderbank-Shor-Steane (CSS) codes via the maximum satisfiability (MaxSAT) problem. We show how to map quantum maximum likelihood problem of CSS codes of arbitrary geometry and parity check weight into MaxSAT problems. We incorporate the syndrome measurements as hard clauses, while qubit and measurement error probabilities, including biased and non-uniform, are encoded as soft MaxSAT clauses. For the code capacity of color codes on a hexagonal lattice, our decoder has a higher threshold and superior scaling in noise suppression compared to belief propagation with ordered statistics post-processing (BP-OSD), while showing similar scaling in computational cost. Further, we decode surface codes and recently proposed bivariate quantum low-density parity check (QLDPC) codes where we find lower error rates than BP-OSD. Finally, we connect the complexity of MaxSAT decoding to a computational phase transition controlled by the clause density of the MaxSAT problem, where we show that our mapping is always in the computationally ''easy`` phase. Our MaxSAT decoder can be further parallelised or implemented on ASICs and FPGAs, promising potential further speedups of several orders of magnitude. Our work provides a flexible platform towards practical applications on quantum computers.
Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3 -fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Quantum low-density parity check (QLDPC) codes have rapidly developed in the field of quantum error correction in recent years. A variety of quantum code families have emerged from numerous research activities. In this paper, we employ concatenated Calderbank-Shor-Steane (CSS) codes, hypergraph product codes, and Bacon-Shor codes to train belief-propagation (BP) and neural BP (NBP) decoders. At the same time, we employ a concatenation scheme in BP and NBP, the amplified construction of CSS codes is realized, and the training of longer length parameter quantum codes is realized. The numerical results show that the concatenated CSS codes demonstrate better error correction performance compared to both hypergraph product codes and Bacon-Shor codes. In addition, the concatenated CSS codes are trained using the concatenation scheme to improve the error correction performance of the decoder.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.
Abstract Quantum error correction (QEC) is the core mechanism enabling scalable and fault-tolerant quantum computation. Among various QEC schemes, Calderbank–Shor–Steane (CSS) codes play a central role. Efficiently verifying whether a quantum state remains within the target CSS code subspace is therefore essential both for understanding the theoretical performance of these codes and for assessing the error-correcting capability of practical quantum platforms. In this work, we develop an efficient framework for verifying whether a quantum system remains within the encoded subspace of a CSS code. We first show that the stabilizer structure of any CSS code can be mapped to a two-colorable graph, enabling the design of a hybrid verification strategy based solely on local Pauli measurements. We further analytically optimize the measurement probabilities and derive the resulting sample complexities for representative CSS codes. Specifically, for the Toric code, the efficiency of our hybrid strategy increases with lattice size and asymptotically approaches near-optimal performance. These results establish a unified and scalable approach to CSS code subspace verification, providing a foundation for certifiable and fault-tolerant quantum information processing.