Theory of low-weight quantum codes

TL;DR

This work studies stabilizer quantum codes under explicit weight constraints, introducing the optimal-weight quantity $W_{\mathrm{opt}}(n,k,d)$ and proving that computing the stabilizer-generator weight $W(G)$ is $\mathsf{NP}$-hard. It develops linear-programming bounds based on weight enumerators and the quantum MacWilliams identities, augmented with overlap, parity, and architecture-aware constraints to yield tight finite-size bounds (exact for $n\le 9$) and practical certificates. A complete characterization shows that any code with generator weight at most $3$ must have distance $d=2$ and rate $k/n\le 1/4$, implying higher weights are necessary for better performance. The authors also provide explicit weight-bounded constructions and demonstrate hardware-aware bounds (e.g., for IBM's Eagle connectivity), offering concrete guidance for designing low-weight, fault-tolerant quantum codes with real-world applicability.

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.

Figures (2)

• Figure 1: Qubit layout of the IBM Eagle chip. With checks centered at the 27 qubits colored in orange, our LP lower bound implies that radius $r\ge5$ is required to achieve a $[\![127,100,6]\!]$ code. The three shaded areas show the radius-5 neighborhoods of qubits 0, 62, and 83.
• Figure 2: Constructions are weight 6 BB codes taken from rmy6-9n89. Bounds for $(k,d)$ of weight 6 codes are drawn in gray and are obtained by Algorithm \ref{['algo:Weight_LB_Table']}.

Theorems & Definitions (40)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Definition 2
• Proposition 3
• Theorem 4
• Theorem 5
• Corollary 6
• Theorem 7: No-go criteria from LP family (informal)