Error-correction and noise-decoherence thresholds for coherent errors in planar-graph surface codes

TL;DR

This study investigates how coherent errors affect surface codes on planar graphs, revealing non-universal thresholds and classifying final-state distributions, with implications for quantum error correction resilience.

Contribution

It introduces a Majorana-fermion representation for planar-graph surface codes and extends fermion-linear-optics simulations to analyze logical-state storage.

Findings

Error-correction thresholds vary with graph connectivity.

Three classes of graphs produce distinct final-state distributions.

Some graph classes maintain coherence above the error-correction threshold.

Abstract

We numerically study coherent errors in surface codes on planar graphs, focusing on noise of the form of $Z$- or $X$-rotations of individual qubits. We find that, similarly to the case of incoherent bit- and phase-flips, a trade-off between resilience against coherent $X$- and $Z$-rotations can be made via the connectivity of the graph. However, our results indicate that, unlike in the incoherent case, the error-correction thresholds for the various graphs do not approach a universal bound. We also study the distribution of final states after error correction. We show that graphs fall into three distinct classes, each resulting in qualitatively distinct final-state distributions. In particular, we show that a graph class exists where the logical-level noise exhibits a decoherence threshold slightly above the error-correction threshold. In these classes, therefore, the logical level noise above the error-correction threshold can retain significant amount of coherence even for large-distance codes. To perform our analysis, we develop a Majorana-fermion representation of planar-graph surface codes and describe the characterization of logical-state storage using fermion-linear-optics-based simulations. We thereby generalize the approach introduced for the square lattice by Bravyi \textit{et al}. [npj Quantum Inf. 4, 55 (2018)] to surface codes on general planar graphs.