Coherent error induced phase transition

TL;DR

This paper studies how coherent errors cause a phase transition in quantum stabilizer codes, affecting the stability of logical information and the effectiveness of error correction.

Contribution

It introduces the concept of a coherent error induced phase transition and analyzes its impact on quantum error correction in toric and non-local stabilizer codes.

Findings

Below a critical error threshold, logical information remains stable.

Above the threshold, the syndrome state shifts, impairing error correction.

The phase transition involves qualitative changes in syndrome distribution features.

Abstract

We investigate the stability of logical information in quantum stabilizer codes subject to coherent unitary errors. Beginning with a logical state, we apply a random unitary error channel and subsequently measure stabilizer checks, resulting in a syndrome-dependent post-measurement state. By examining both this syndrome state and the associated syndrome distribution, we identify a phase transition in the behavior of the logical state. Below a critical error threshold pc, the syndrome state remains in the same logical state, enabling successful recovery of the code's logical information via suitable error-correction protocols. Above pc, however, the syndrome state shifts to a different logical state, signaling the breakdown of efficient error correction. Notably, this process can often induce an effective unitary rotation within the logical space. This transition is accompanied by qualitative changes in both the global and local features of the syndrome distribution. We refer to this phenomenon as a coherent error induced phase transition. To illustrate this transition, we present two classes of quantum error correcting code models the toric code and non-local random stabilizer codes thereby shedding light on the design and performance limits of quantum error correction under coherent errors.