Gauge-model analysis of the XZZX surface code with biased data noise and measurement errors

Abstract

Surface codes are a leading approach to fault-tolerant quantum computation, and the XZZX variant is designed to exploit biased noise, in which one Pauli error channel dominates. The decoding threshold of such a code can be studied by mapping the decoding problem to a statistical-mechanics model with quenched disorder, whose order–disorder transition coincides with the threshold. In this work we extend this mapping to the XZZX code subject to dephasing-biased data noise and phenomenological measurement errors, modeled as independent bit-flips on the stabilizer outcomes. Building on the random coupled-plaquette gauge model previously applied to the CSS surface code, we show that the mixed stabilizers of the XZZX code cause measurement errors to couple the X- and Z-error sectors, resulting in a three-dimensional gauge model for any Pauli noise model. The model is sampled using Metropolis–Hastings with parallel tempering, and a modified Polyakov line is used to probe the deconfinement transition associated with decoding. Simulations are performed at a single code distance d = 19 for noise biases η ∈ {1, 10, 100, 1000}. For weak bias we observe a smooth crossover consistent with an order–disorder transition, but as the bias increases the crossover fades and the system does not disorder throughout the sampled range, preventing threshold extraction. Proposed explanations include finite-size effects, open boundaries, and the limitations of the Polyakov line on a finite lattice. The natural next steps are finite-size scaling and the consideration of a wider range of order-disorder metrics.