Low-density parity-check codes as stable phases of quantum matter

TL;DR

This paper establishes conditions under which quantum error correcting codes, specifically low-density parity-check codes, define stable quantum phases of matter that are resistant to perturbations, linking quantum information theory with condensed matter physics.

Contribution

It proves that LDPC codes with logarithmic code distance and check soundness define stable quantum phases, expanding understanding of quantum code stability and phase robustness.

Findings

LDPC codes with logarithmic code distance define stable phases

Quantum toric code phases are robust to nonlocal perturbations

Classical codes' phases are stable against symmetric perturbations

Abstract

Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error correcting code, when does it define a stable gapped quantum phase of matter, whose ground state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits "check soundness". Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.