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

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. Here we scrutinize the performance of the topological toric code in 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 multiqubit entanglement gates. By mapping the multiround error detection protocol-from the inception of an imperfectly prepared code state via imperfect stabilizer measurements-to a statistical mechanical model (3D $$ gauge theory coupled with 2D $$ gauge theory), we establish a link between the error threshold and the model's phase transition. Specifically, we find two distinct error thresholds that demarcate varying efficacies in QEC. 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 distance d, it is revealed that maintaining the preparation error rate beneath a crossover scale, proportional to $$ , 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 serious vulnerability of fault tolerant QEC based on 2D toric codes to quantum deviations in code space, highlighting the imperative to address inherent preparation noise.