Decoding coherent errors in toric codes on honeycomb and square lattices: duality to Majorana monitored dynamics and symmetry classes

TL;DR

This paper investigates how coherent errors affect the decodability of toric codes on honeycomb and square lattices, revealing a duality with Majorana fermion dynamics and classifying phase transitions by symmetry classes.

Contribution

It establishes a duality between decoding problems under coherent errors and monitored Majorana fermion dynamics, linking universal phase transition behavior to symmetry classes.

Findings

Decodability transitions are governed by Altland-Zirnbauer symmetry classes.

Honeycomb and square lattice codes map to class-DIII and class-D dynamics.

Spatially varying errors make surface codes more vulnerable to decoding failure.

Abstract

Topological stabilizer codes, such as the toric and surface codes, are leading candidates for fault-tolerant quantum computation. While their decodability under stochastic noise has been extensively studied, the effects of coherent errors, which involve quantum interference, remain less explored. In this work, we study the decodability of toric codes on honeycomb and square lattices subject to $X$- and $Z$-type coherent errors generated by the $X$- and $Z$-rotations on each qubit. We establish a duality between these decoding problems and 1+1D monitored dynamics of non-interacting Majorana fermions. This duality shows that the Altland-Zirnbauer symmetry class of the dual Majorana dynamics governs the universal structure of the decodability phase diagram. We show that the honeycomb-lattice toric code (hTC) with $X$-type error is dual to class-DIII dynamics, while the hTC with $Z$-type error and the square-lattice toric code (sTC) with both error types are dual to class-D dynamics. The key distinction arises from time-reversal symmetry. In class DIII, the generic transition out of the decodable phase is dual to a measurement-induced transition between dynamical phases with area-law and logarithmic entanglement scaling. In contrast, in class D, the generic decodability transition corresponds to a transition between two topologically distinct area-law phases. To explore these transitions in microscopic models, we consider hTC and sTC with $X$-type errors as representatives and introduce a minimal two-parameter coherent error model with spatially varying rotation angles. Using analytical and numerical methods, we map out the decodability phase diagrams and characterize the universal behavior of the transitions. We find that the decodability of sTC is more vulnerable to spatially varying coherent errors than uniform ones.