Error correction with brickwork Clifford circuits

TL;DR

This paper proves that random 1D Clifford brickwork circuits can serve as effective approximate quantum error correction codes with logarithmic depth, and establishes bounds for exact error correction capabilities.

Contribution

It introduces a novel application of statistical mechanics techniques to analyze error correction properties of 1D Clifford brickwork circuits, providing both approximate and exact bounds.

Findings

Random 1D Clifford brickwork circuits form good approximate quantum error correction codes.

Logarithmic depth suffices for approximate error correction.

Matching bounds are established for the depth needed for exact error correction.

Abstract

We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.