Modeling coherent errors in quantum error correction

Abstract

Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ò) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n, in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than  --error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.

Quantum Error Correction Decoheres Noise

“…This critical value is consistent with the value observed in ref. [21] of 1/ǫ 2 , where ǫ is the angle of rotation about the x-axis, and we note that all of our observations hold in their specific case when we replace r in our results with √ ǫ, as that is how the specified noise scales relative to our lemma 1. Because the off diagonal terms and diagonal terms produce the same scaling in a worst-case analysis with coherent noise, the ratio of off diagonal to diagonal errors is independent of the number of rounds of error correction in the worst-case scaling of typical noise.…”

Coherence in logical quantum channels

“…Let us analyze how well this code protects against coherent errors, in which each physical qubit in the code block rotates about the x-axis. Similar calculations were carried out in [13,14]. Understanding this example will prepare us for an analysis of more general stabilizer codes.…”

“…Here we investigate the coherence of the logical channel in the case where the physical noise is fully coherent unitary noise, a problem that has been previously studied [13,14,15]. Our work improves on these past results in that we consider a family of codes with an accuracy threshold (toric codes without boundaries) and prove bounds on the logical coherence which apply in the limit of a large code block.…”

Mitigating coherent noise using Pauli conjugation

“…Since we have only pure Z noise, we only need to look at the symmetry exists in the X stabilisers, leading to additional symmetry in the exchange between qubits (1,2) and between qubits (8,9). Applying on top of the rotational symmetry, we have the following classes of equivalent conjugations:…”