Finite-rate sparse quantum codes aplenty

TL;DR

This paper presents a new method for generating sparse quantum stabilizer codes using constraint satisfaction problems, demonstrating the existence of a satisfiability threshold and codes that approach channel capacity.

Contribution

It introduces a flexible CSP-based framework for creating finite-rate sparse quantum codes with customizable properties, advancing quantum code design.

Findings

Existence of a satisfiability threshold for the codes.

Sparse codes in the satisfiable phase nearly achieve channel capacity.

Finding intermediate-size finite-rate sparse quantum codes is computationally easy.

Abstract

We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, $X/Z$ balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery.