Coherence in logical quantum channels

TL;DR

This paper investigates how quantum error correction, specifically the toric code, can suppress coherence in logical channels under coherent noise, suggesting fault-tolerant quantum computing can be effective against such errors.

Contribution

It demonstrates that the logical channel becomes increasingly incoherent with larger code size under certain conditions, providing evidence that quantum codes can mitigate coherent noise effects.

Findings

Logical channels become more incoherent as code size increases.

Error correction effectiveness improves when noise strength decreases inversely with code distance.

Results suggest fault-tolerance can handle coherent errors effectively.

Abstract

We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.