The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology

TL;DR

This paper investigates how coherent and incoherent errors affect the surface code's logical noise and information measures, revealing exponential suppression of coherence with code distance and limitations of current thresholds.

Contribution

It introduces a phenomenological field theory and novel simulation methods to analyze non-Pauli errors in surface codes, extending understanding beyond traditional Pauli error models.

Findings

Logical noise coherence diminishes exponentially with code distance for any nonzero incoherent component.

Information-theoretic measures require large code distances to detect thresholds under increasing error coherence.

Methods enable large-scale simulation of non-Pauli errors and their maximum-likelihood thresholds.

Abstract

We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X-error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.