Correcting coherent errors with surface codes

TL;DR

This paper introduces an efficient simulation algorithm for surface code quantum error correction under coherent noise, revealing that large codes effectively convert coherent errors into Pauli errors, aiding quantum fault tolerance.

Contribution

The authors develop a $O(n^2)$ algorithm to simulate surface code error correction with coherent errors, enabling threshold estimation for large systems and advancing understanding of error behavior.

Findings

Large surface codes convert coherent errors into Pauli errors at the logical level.

The algorithm efficiently simulates systems with over a thousand qubits.

Error thresholds are estimated for models with systematic $Z$-rotations and $SU(2)$ errors.

Abstract

We study how well topological quantum codes can tolerate coherent noise caused by systematic unitary errors such as unwanted $Z$-rotations. Our main result is an efficient algorithm for simulating quantum error correction protocols based on the 2D surface code in the presence of coherent errors. The algorithm has runtime $O(n^2)$, where $n$ is the number of physical qubits. It allows us to simulate systems with more than one thousand qubits and obtain the first error threshold estimates for several toy models of coherent noise. Numerical results are reported for storage of logical states subject to $Z$-rotation errors and for logical state preparation with general $SU(2)$ errors. We observe that for large code distances the effective logical-level noise is well-approximated by random Pauli errors even though the physical-level noise is coherent. Our algorithm works by mapping the surface code to a system of Majorana fermions.