Vulnerability of fault-tolerant topological quantum error correction to quantum deviations in code space

Abstract

Quantum computers face significant challenges from quantum deviations or coherent noise, particularly during gate operations, which pose a complex threat to the efficacy of quantum error correction (QEC) protocols. In this study, we scrutinize the performance of the topological toric code in 2 dimension (2D) under the dual influence of stochastic noise and quantum deviations, especially during the critical phases of initial state preparation and error detection facilitated by multi-qubit entanglement gates. By mapping the protocol for multi-round error detection--from the inception of an imperfectly prepared code state via imperfect stabilizer measurements--to a statistical mechanical model characterized by a 3-dimensional $\mathbb{Z}_2$ gauge theory coupled with a 2-dimensional $\mathbb{Z}_2$ gauge theory, we establish a novel link between the error threshold and the model's phase transition point. We find two distinct error thresholds that demarcate varying efficacies in error correction. The empirical threshold that signifies the operational success of QEC aligns with the theoretical ideal of flawless state preparation operations. Contrarily, below another finite theoretical threshold, a phenomenon absent in purely stochastic error models emerges: unidentifiable measurement errors precipitate QEC failure in scenarios with large code distances. For codes of finite or modest distance $d$, it is revealed that maintaining the preparation error rate beneath a crossover scale, proportional to $1/\log d$, allows for the suppression of logical errors. Considering that fault-tolerant quantum computation is valuable only in systems with large scale and exceptionally low logical error rates, this investigation explicitly demonstrates the vulnerability of 2D toric codes to quantum deviations in code space.