The realization of fault-tolerant quantum computation relies on Quantum Error-Correcting Codes (QECC). Of particular interest are codes that combine a high encoding rate with sufficient distance. Existing algebraic constructions, such as lifted product (LP) codes, build a large quantum low-density parity-check (qLDPC) code from a classical base matrix. In this work we introduce LiftDiff, a generative approach to the discovery of QECC. A discrete denoising diffusion model is trained to generate base matrices over the group algebra $\mathbb{F}_2[\mathbb{Z}_\ell]$, which are then expanded into LP codes. The reverse diffusion process is guided by an energy function that evaluates quantum code properties, including girth, degree, sparsity and decoding performance, steering the sampler toward matrices that yield LP codes with favorable parameters. Our method discovers codes with parameters such as $[[169,13,11]]$, $[[143,15,9]]$, and $[[260,24,10]]$, including a $[[247,19,14]]$ code that achieves the highest ${kd^2}/{n}$ among the baselines. Moreover, several discovered codes outperform the Gross code across a broad range of the tested physical error rates under both code-capacity and circuit-level noise.
The realization of fault-tolerant quantum computation relies on Quantum Error-Correcting Codes (QECC). Of particular interest are codes that combine a high encoding rate with sufficient distance. Existing algebraic constructions, such as lifted product (LP) codes, build a large quantum low-density parity-check (qLDPC) code from a classical base matrix. In this work we introduce LiftDiff, a generative approach to the discovery of QECC. A discrete denoising diffusion model is trained to generate base matrices over the group algebra F2[Zℓ], which are then expanded into LP codes. The reverse diffusion process is guided by an energy function that evaluates quantum code properties, including girth, degree, sparsity and decoding performance, steering the sampler toward matrices that yield LP codes with favorable parameters. Our method discovers codes with parameters such as [[169,13,11]], [[143,15,9]], and [[260,24,10]], including a [[247,19,14]] code that achieves the highest kd2/n among the baselines. Moreover, several discovered codes outperform the Gross code across a broad range of the tested physical error rates under both code-capacity and circuit-level noise.
Quantum computers could outperform classical machines on important problems, but only if the errors that pervade quantum hardware can be corrected at scale. Quantum low-density parity-check (qLDPC) codes offer a promising route to this goal by combining sparse parity checks with finite encoding rate and growing distance, but their construction remains a challenging discrete design problem. Here we introduce structured concept evolution (SCE), a search framework that pairs a large language model with a structured algebraic mutation grammar to discover lifted-product code families, a class of CSS qLDPC codes. Instead of asking the LLM to design codes from first principles, SCE evolves structured concepts consisting of algebraic specifications paired with executable programs that realize them, using hierarchical mutations that modify the group algebra, protograph geometry, or base space. Running SCE, we discover a diverse set of competitive code families, ranging from abelian constructions to families over non-abelian groups beyond those underlying standard designs such as bivariate-bicycle codes, and characterize them under code-capacity depolarizing noise with BP+OSD decoding. These results are obtained with lightweight models (GPT-5.4-mini and GPT-5.4-nano).
Lifted product (LP) codes form an important class of quantum error correcting codes with favorable code parameters. We introduce a systematic construction of LP code families based on group extensions and graph lifts. The construction increases the code size while preserving the local structure of the Tanner graph, and relates code parameters, logical operators, and fault-tolerant logical-operation gadgets within the families through chain and cochain maps.
As a first application, we obtain LP codes with better code parameters than previously reported ones. We then demonstrate that code-surgery gadgets can be transferred across the selected finite lifts through chain maps and, in several cases, implemented with lower space overhead. We also develop parallel product surgery for lifted clustered cyclic codes. Finally, we propose lifting as a systematic first step toward defining thermodynamic families for algebraically defined qLDPC codes without an underlying Euclidean lattice. For several base codes and selected lifts, coherent information exhibits finite-size crossings, while our results also indicate that additional conditions are needed to determine a unique family.
Quantum error correction (QEC) is a cornerstone of quantum computing, enabling reliable information processing in the presence of noise. Sparse stabilizer codes -- referred to generally as quantum low-density parity-check (QLDPC) codes -- have risen to the forefront of QEC research in recent years. This can be attributed to several key factors. First, classical LDPC codes admit low-complexity belief propagation iterative decoding and near-capacity performance, which contributed to the early interest in QLDPC codes. Then, the result promising constant overhead fault tolerance using QLDPC codes led to the search for code families that go beyond the long-holding n scaling barrier of minimum distance for codelength n. This resulted in recent breakthroughs in the construction of QLDPC codes, which, combined with efficient decoding algorithms and the development of fault-tolerant protocols operating on QLDPC-encoded quantum information, provide a promising pathway to low-overhead, fault-tolerant quantum computation. However, despite their potential, challenges remain, particularly in constructing and decoding finite-length codes that account for, or efficiently leverage, specific characteristics of quantum hardware, such as connectivity, topology, native gate sets, and noise models. This article provides an in-depth examination of QLDPC codes and their iterative decoders, catering to an information theory audience with no or limited background in quantum mechanics. We discuss the theoretical underpinnings, explore unique characteristics of quantum channels, and delineate key code constructions and decoding algorithms, ultimately highlighting the impact and future prospects of QLDPC codes in quantum information science.
The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.
Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width f, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance d and check weight w. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations.
First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by 1 guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an [[1088,128,22]] LP code with f=22 and an [[1024,256,24]] PP code with f≤27, both at w=10. At w=11, our PP constructions attain either d=25 with f≤27 or d=24 with the proved optimum f=25, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group G and two subsets of G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%, approximately matching that of the [[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.