# Efficient simulation of quantum error correction under coherent error based on non-unitary free-fermionic formalism

## TL;DR

This paper introduces an efficient method using non-unitary free-fermionic formalism to accurately simulate quantum error correction under coherent noise, revealing the impact of noise coherence on error thresholds.

## Contribution

It develops a novel, efficient simulation scheme for quantum error correction under non-Clifford, coherent noise, enabling precise threshold evaluation without approximation.

## Key findings

- Error threshold drops to one third with fully coherent noise.
- Method applicable to 1D repetition and surface codes.
- Threshold dependence on coherence explained by leading-order analysis.

## Abstract

In order to realize fault-tolerant quantum computation, tight evaluation of error threshold under practical noise models is essential. While non-Clifford noise is ubiquitous in experiments, the error threshold under non-Clifford noise cannot be efficiently treated with known approaches. We construct an efficient scheme for estimating the error threshold of one-dimensional quantum repetition code under non-Clifford noise. To this end, we employ non-unitary free-fermionic formalism for efficient simulation of the one-dimensional repetition code under coherent noise. This allows us to evaluate the effect of coherence in noise on the error threshold without any approximation. The result shows that the error threshold becomes one third when noise is fully coherent. Our scheme is also applicable to the surface code undergoing a specific coherent noise model. The dependence of the error threshold on noise coherence can be explained with a leading-order analysis with respect to coherence terms in the noise map. We expect that this analysis is also valid for the surface code since it is a two-dimensional extension of the one-dimensional repetition code. Moreover, since the obtained threshold is accurate, our results can be used as a benchmark for approximation or heuristic schemes for non-Clifford noise.