Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry

TL;DR

This paper calculates optimal error thresholds for fracton-based quantum error correcting codes by mapping error processes to statistical models, revealing higher thresholds than traditional topological codes, indicating fracton phases' potential for quantum memory.

Contribution

It introduces a novel mapping of error correction in fracton codes to statistical models with subsystem symmetry, enabling threshold calculation and comparison with existing codes.

Findings

X-cube fracton code has a minimum error threshold of 7.5%.

Thresholds are higher than those of 3D topological codes like toric and color codes.

No glass order at the Nishimori line suggests stability of fracton codes as quantum memories.

Abstract

Fracton models provide examples of novel gapped quantum phases of matter that host intrinsically immobile excitations and therefore lie beyond the conventional notion of topological order. Here, we calculate optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multi-body couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We perform large-scale parallel tempering Monte Carlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold ($7.5\%$) that is much higher than 3D topological codes such as the toric code ($3.3\%$), or the color code ($1.9\%$). This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms.