Error thresholds of toric codes with transversal logical gates

• URL: http://arxiv.org/abs/2510.10835v1
• Date: Sun, 12 Oct 2025 22:57:00 GMT
• Title: Error thresholds of toric codes with transversal logical gates
• Authors: Yichen Xu, Yiqing Zhou, James P. Sethna, Eun-Ah Kim,
• Abstract summary: We quantify the impact of two error-spreading mechanisms on the threshold for fault-tolerant quantum computation.<n>We show that the tCNOT gate reduces the optimal bitflip error threshold to $p80$, a $26%$ decrease from the toric code memory threshold.
• Score: 5.203952901790516
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The threshold theorem promises a path to fault-tolerant quantum computation by suppressing logical errors, provided the physical error rate is below a critical threshold. While transversal gates offer an efficient method for implementing logical operations, they risk spreading errors and potentially lowering this threshold compared to a static quantum memory. Available threshold estimates for transversal circuits are empirically obtained and limited to specific, sub-optimal decoders. To establish rigorous bounds on the negative impact of error spreading by the transversal gates, we generalize the statistical mechanical (stat-mech) mapping from quantum memories to logical circuits. We establish a mapping for two toric code blocks that undergo a transversal CNOT (tCNOT) gate. Using this mapping, we quantify the impact of two independent error-spreading mechanisms: the spread of physical bit-flip errors and the spread of syndrome errors. In the former case, the stat-mech model is a 2D random Ashkin-Teller model. We use numerical simulation to show that the tCNOT gate reduces the optimal bit-flip error threshold to $p=0.080$, a $26\%$ decrease from the toric code memory threshold $p=0.109$. The case of syndrome error coexisting with bit-flip errors is mapped to a 3D random 4-body Ising model with a plane defect. There, we obtain a conservative estimate error threshold of $p=0.028$, implying an even more modest reduction due to the spread of the syndrome error compared to the memory threshold $p=0.033$. Our work establishes that an arbitrary transversal Clifford logical circuit can be mapped to a stat-mech model, and a rigorous threshold can be obtained correspondingly.

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