In this work, we develop an efficient decoding method for graph codes, a class of stabilizer quantum error-correcting codes constructed from graph states. While optimal decoding is generally NP-hard, we propose a faster decoder exploiting the structural properties of the underlying graph states. Although distinct error patterns may yield the same syndrome, we demonstrate that the post-measurement state follows a well-defined structure determined by the projective syndrome measurement. Building on this idea, we introduce a hierarchical decoder in which each level can be solved in polynomial time. Additionally, this decoder achieves optimal decoding performance at the lower levels of the hierarchy. This strategy avoids the need for full maximum-likelihood decoding of graph codes. Numerical results illustrate the efficiency and effectiveness of the proposed approach.
Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure
X and pure
Z stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For
m possibly redundant generator rows on
n qubits, the algorithm uses
O(mn2+n4) binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.
Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices.
While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures, and prove via the ZX calculus that this representation is related to stabilizer tableaus by an efficiently computable bijection. This gives a new universal recipe for code construction by way of finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. We construct as examples families of ⟦ n , Θ ( n log n ) , Θ ( log n ) ⟧ and ⟦ n , Ω ( n 4 / 5 ) , Θ ( n 1 / 5 ) ⟧ codes. We use graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. Moreover, code properties such as distance and encoding circuit depth are bounded by simple functions of the graph degree. We prove that key coding algorithms—distance approximation, minimum-weight generator selection, and decoding—are unified as instances of one optimization game on a graph. By studying this game, we construct an efficient greedy decoder and prove that it corrects all recoverable errors for all graphs with cycle lengths no shorter than 13 (reducible to 5 with mild extra constraints); these include the above two families. Our results suggest that graphs are generically useful for the study of stabilizer codes.
We study directionally informed belief propagation (BP) decoding for quantum CSS codes, where anisotropic Tanner-graph structure and biased noise concentrate degeneracy along preferred directions. We formalize this by placing orientation weights on Tanner-graph edges, aggregating them into per-qubit directional weights, and defining a \emph{directional degeneracy enumerator} that summarizes how degeneracy concentrates along those directions. A single bias parameter~
β maps these weights into site-dependent log-likelihood ratios (LLRs), yielding anisotropic priors that plug directly into standard BP
→OSD decoders without changing the code construction. We derive bounds relating directional and Hamming distances, upper bound the number of degenerate error classes per syndrome as a function of distance, rate, and directional bias, and give a MacWilliams-type expression for the directional enumerator. Finite-length simulations under code-capacity noise show significant logical error-rate reductions -- often an order of magnitude at moderate physical error rates -- confirming that modest anisotropy is a simple and effective route to hardware-aware decoding gains.
Abstract Quantum stabilizer codes are essential for protecting quantum information, but systematic searches for high-distance codes become computationally prohibitive as code length increases. Existing code constructions exhibit structural regularities, yet no general framework quantifies how such constraints reduce the stabilizer-code search space or incorporates them into code searches. This project asks whether structural properties of known quantum codes can be formulated as mathematical constraints that quantitatively reduce search spaces while remaining useful for discovering high-distance codes. We develop a unified framework based on
q-Galois matrices and define the search-space retention factor as the asymptotic fraction of logarithmic search space retained after imposing a constraint. We derive retention factors for weight-spectrum, qudit-participation, overlap, support, Pauli-distribution, local non-commutation, graph-theoretic, and symmetry constraints, including combinations. We introduce finite-size correction factors
C and
C′ and find empirically that they decay approximately as
logn/n and
1/n, respectively. The framework is evaluated using 58,902 quantum-code parameter sets, including 34,775 codes with usable matrix data across
q=2,3,4,5,7,8; 1,909 semi-perfect quantum codes are analyzed. We develop a general method to empirically estimate the weight-spectrum and average-weight for unresolved semi-perfect quantum codes, achieving runtime speedups ranging from
108.3608 to
10193.9298 across 4 distinct unresolved codes. Runtime analysis validates that these reductions translate into substantial computational savings. Applying the framework to a
(14,3) qubit search produced a code detecting 91,765 of 91,770 errors of weight at most 4 and achieving 99.9999933% and 99.9733% of the theoretically optimal performance of
[[14,3,5]]2 code at
p=0.01. Thus, the framework converts structural properties into explicit candidate-space reductions that can be imposed during code enumeration, providing a quantitative basis for reducing the computational cost of high-distance stabilizer-code searches.