Improved energy barrier in higher-dimensional hypergraph product codes

TL;DR

This paper demonstrates an improved lower bound on the energy barrier for higher-dimensional hypergraph product codes, showing they can have macroscopic energy barriers even when classical codes do not, advancing quantum error correction understanding.

Contribution

It provides a new, tighter energy barrier lower bound for LDPC HHGP codes based on the classical codes' distance, surpassing previous confinement-based bounds.

Findings

Energy barrier lower bounded by classical code distance

Higher-dimensional codes can have macroscopic barriers without classical counterparts

Bound is tight for codes with size-dependent distances like 3D and 4D toric codes

Abstract

Single-shot error correction outperforms conventional approaches by requiring only one round of stabilizer measurements for decoding, even in the presence of measurement errors. This capability relates to the confinement property of codes, which provides an energy barrier lower bound. Earlier research established a confinement property for higher-dimensional hypergraph product (HHGP) codes (Quintavalle et al. 2021 PRX Quantum), yielding an energy barrier lower bound for these codes. In this work, by analyzing the structure of logical operators, we show an improved energy barrier lower bound for HHGP codes with low-density parity-check (LDPC) property. Our bound exceeds results derived from confinement alone, and unlike standard hypergraph product codes, these higher dimensional variants can possess macroscopic energy barriers even when the underlying classical codes lack this property. Specifically, our analysis shows that the energy barrier of LDPC HHGP codes is lower bounded by the distance of the underlying classical codes. This bound is tight if the underlying classical codes exhibit system size-dependent distances but constant energy barriers, like 3D and 4D toric codes.