Theory of low-weight quantum codes

TL;DR

This paper investigates the theoretical limits and properties of low-weight quantum stabilizer codes, establishing complexity results, bounds, and practical applications relevant for fault-tolerant quantum computing.

Contribution

It provides the first comprehensive analysis of weight-constrained quantum codes, including NP-hardness results, explicit bounds, and a linear programming scheme for code parameter optimization.

Findings

Calculating optimal code weight is NP-hard.

Stabilizer codes with weight at most 3 have distance 2 and rate ≤ 1/4.

LP scheme yields exact weight bounds for codes with n ≤ 9.

Abstract

Low check weight is practically crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we explore the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that calculating the optimal code weight is an $\mathsf{NP}$-hard problem, demonstrating the necessity of establishing bounds for weight that are analytical or efficiently computable. Then we systematically investigate the feasible code parameters with weight constraints. We provide various explicit analytical lower bounds and in particular completely characterize stabilizer codes with weight at most 3, showing that they have distance 2 and code rate at most 1/4. We also develop a powerful linear programming (LP) scheme for setting code parameter bounds with weight constraints, which yields exact optimal weight values for all code parameters with $n\leq 9$. We further refined this constraint from multiple perspectives by considering the generator weight distribution and overlap. In particular, we consider practical architectures and demonstrate how to apply our methods to e.g.~the IBM 127-qubit chip. Our study brings the weight as a crucial parameter into coding theory and provide guidance for code design and utility in practical scenarios.