Charge-Informed Quantum Error Correction

TL;DR

This paper explores the physics of quantum error correction in charge-conserving topological memories, revealing phase transitions and universality classes, and demonstrating improved decoding strategies over charge-agnostic methods.

Contribution

It introduces a charge-informed decoding framework, analyzes its phase transition behavior, and compares its performance to charge-agnostic decoders in topological quantum memories.

Findings

Optimal decoder exhibits Berezinskii-Kosterlitz-Thouless universality.

Disorder-dominated loop-glass phase identified at low temperatures.

Charge-informed decoders outperform charge-agnostic counterparts.

Abstract

We investigate the statistical physics of quantum error correction in ${\rm U}(1)$ symmetry-enriched topological quantum memories. Starting from a phenomenological error model of charge-conserving noise, we study the optimal decoder assuming the local charges of each anyon can be measured. The error threshold of the optimal decoder corresponds to a continuous phase transition in a disordered two-dimensional integer loop model on the Nishimori line. Using an effective replica field theory analysis and Monte Carlo numerics, we show that the optimal decoding transition exhibits Berezinskii-Kosterlitz-Thouless universality with a modified universal jump in winding number variance. We further generalize the model beyond the Nishimori line, which defines a large class of suboptimal decoders. At low nonzero temperatures and strong disorder, we find numerical evidence of a disorder-dominated loop-glass phase which corresponds to a "confidently incorrect" decoder. The zero-temperature limit defines the minimum-cost flow decoder, which serves as the ${\rm U}(1)$ analog of minimum-weight perfect matching in $\mathbb{Z}_2$ topological codes. Both the optimal and minimum-cost flow decoders are shown to dramatically outperform the charge-agnostic optimal decoder in symmetry-enriched topological codes.