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A Framework for Efficient Search of High-Distance Quantum Stabilizer Codes

Utsav Karia

Published Aug 22, 2026• 0 citations
Publisher

Abstract

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 $\log n/n$ 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 $10^{8.3608}$ to $10^{193.9298}$ 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.

DOI
10.5281/zenodo.22899978

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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 qqq-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 CCC and C′C'C′ and find empirically that they decay approximately as log⁡n/n\log n/nlogn/n and 1/n1/n1/n, respectively. The framework is evaluated using 58,902 quantum-code parameter sets, including 34,775 codes with usable matrix data across q=2,3,4,5,7,8q=2,3,4,5,7,8q=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 108.360810^{8.3608}108.3608 to 10193.929810^{193.9298}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)(14,3)(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[[14,3,5]]_2[[14,3,5]]2​ code at p=0.01p=0.01p=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.
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Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024
The codeword-stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [see A. Cross et al., e-print arXiv:0708.1021], but only for binary states. Here we generalize the CWS framework to the nonbinary case (of both prime and nonprime dimensions) and map the search for nonbinary quantum codes to a corresponding search problem for classical nonbinary codes with specific error patterns. We show that while the additivity properties of nonbinary CWS codes are similar to the binary case, the structural properties of the nonbinary codes differ substantially from the binary case, even for prime dimensions. In particular, we identify specific structure patterns of stabilizer groups, based on which efficient constructions might be possible for codes that encode more dimensions than any stabilizer codes of the same length and distance; similar methods cannot be applied in the binary case. Understanding of these structural properties can help prune the search space and facilitate the identification of good nonbinary CWS codes.
Reducing the qubit overhead of quantum error correction is a central challenge for scalable fault-tolerant quantum computing. Recent ultra-high-rate quantum codes offer a promising route toward this goal, with some constructions requiring as few as two physical data qubits per logical qubit. However, systematic principles for navigating the tradeoffs among encoding rate, distance, check weight, and blocklength remain lacking. Here, we develop and analyze principles for exploring this design space, and use them to design compact code constructions with improved performance. We develop code templates based on a pair-partition construction and a halving transformation that further reduces blocklength. Motivated by ensemble analysis of the degree distributions, we identify column weight as a key design parameter: increasing the column weight enables larger distances at compact blocklengths, at the cost of heavier checks. We find that at physical error rates of 0.1%, the benefits of increased distance often outweigh the penalty associated with heavier checks. Applying this framework, we identify numerous compact codes with favorable parameters, including [[90,21,11]], [[140,31,15]], and [[200,43,20]] non-CSS codes with check weight 10. Moreover, we develop symmetry-informed strategies for identifying low-weight logical bases. These results provide systematic strategies for designing ultra-high-rate quantum codes and navigating their Pareto frontier.
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