Check-weight-constrained quantum codes: Bounds and examples

TL;DR

This paper investigates the limitations imposed by check-weight constraints on quantum low-density parity-check (qLDPC) codes, establishing bounds on their parameters and providing explicit constructions for practical code sizes.

Contribution

It provides the first strong bounds on the parameters of check-weight constrained qLDPC codes and offers explicit constructions near these bounds.

Findings

Checks of weight at most three cannot support nontrivial distance.

Tradeoffs between rate and distance are tight for checks of weight four and two.

Numerical bounds and explicit code examples are identified for practical code sizes.

Abstract

Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.