A $\boldsymbol{2d \times d \times d}$ Spacetime Volume Implementation of a Logical S Gate in the Surface Code

Abstract

The logical S gate implemented via twist defect braiding in the surface code is one of the major sources of overhead in fault-tolerant quantum computing, since an S-gate correction is required in every logical T-gate teleportation. Existing logical S-gate implementations require spacetime volumes of \(2d \times 2d \times d\) or \(2d \times 1.5d \times d\), where $d$ is the code distance of the surface code. To the best of our knowledge, their circuit-level implementations have not yet been shown, hindering quantitative comparisons of fault distances and logical error rates. In this work, we provide these missing circuit-level implementations. Additionally, we propose a novel twist defect braiding protocol that reduces the spacetime volume to \(2d \times d \times d\). First, we construct an implementation of the proposed method using constant-length non-local gates, and then refine it to utilize only nearest-neighbor two-qubit gates on a square grid, without requiring additional two-qubit gate depth beyond that of standard syndrome extraction circuits. Through numerical simulations, we evaluate the fault distances and logical error rates for both existing and proposed methods. Our results show that, although the proposed method reduces the fault distance by one or three, its logical error rates remain comparable to those of existing methods at large code distances (\(d \ge 5\)) and at physical error rates near \(p = 10^{-3}\). This demonstrates that the proposed method is promising for near-term fault-tolerant quantum computing.