Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes

TL;DR

This paper examines the error-correction capabilities of quantum LDPC codes with diminishing relative distance, demonstrating their potential for reliable quantum error correction in large-scale quantum computers.

Contribution

It analyzes the fault-tolerance of quantum LDPC codes with vanishing relative distance and introduces the quantum hypergraph-product codes as a promising finite-rate alternative to toric codes.

Findings

LDPC codes with distance scaling as a positive power of block length can correct errors at low error rates

Quantum hypergraph-product codes generalize toric codes and offer advantages for large quantum computers

These codes maintain error correction capabilities despite decreasing relative distance

Abstract

We discuss error-correction properties for families of quantum low-density parity check (LDPC) codes with relative distance that tends to zero in the limit of large blocklength. In particular, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^\alpha$, $\alpha>0$, can correct all errors with certainty if the error rate per (qu)bit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Z\'emor. These codes are a finite-rate generalization of the toric codes, and, for sufficiently large quantum computers, offer an advantage over the toric codes.