Single-shot error correction of three-dimensional homological product codes

TL;DR

This paper introduces a new framework called confinement for quantum codes, proves its sufficiency for single-shot error correction, and demonstrates high thresholds in 3D homological product codes through simulations.

Contribution

It establishes confinement as a key concept for single-shot decoding and applies it to 3D homological product codes, achieving the highest known single-shot thresholds.

Findings

3D homological product codes exhibit confinement in their X-components.

Monte Carlo simulations show thresholds of approximately 3.08% and 2.90%.

Numerical evidence supports single-shot protection for local stochastic phase-flip noise.

Abstract

Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their $X$-components and are therefore single-shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of $3.08(4)\%$ and $2.90(2)\%$ for 3D surface and toric codes respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed 3D homological product code.