No More Hooks in the Surface Code: Distance-Preserving Syndrome Extraction for Arbitrary Layouts at Minimum Depth

TL;DR

This paper introduces ZX interleaving syndrome extraction, a method that preserves the full fault distance in surface code layouts at minimal depth, reducing errors without extra overheads, thus enhancing fault-tolerant quantum computing.

Contribution

The paper presents a novel syndrome extraction technique that maintains full fault distance at minimal circuit depth for any regular surface-code layout, avoiding additional overheads.

Findings

Achieves full fault distance $d$ in simulations

Outperforms existing methods with $d-1$ fault distance

Applicable to any regular surface-code tiling

Abstract

Hook errors are a major challenge in implementing logical operations with the surface code, because they can reduce the fault distance below the code distance. This motivates syndrome-extraction circuits that suppress hook-error effects for the stabilizer layouts that appear during logical operations. However, the existing methods either increase circuit depth or require simultaneous execution of measurements and CNOT gates, both of which introduce additional overheads and degrade the threshold. We propose the ZX interleaving syndrome extraction, which preserves the full fault distance $d$ for any surface-code layout with regular stabilizer tiles at minimum depth, i.e., four layers of CNOT gates, without requiring additional circuit depth or simultaneous execution of measurements and CNOT gates. The key idea is to interleave the Z and X stabilizer tiles so that hook-error edges in the decoding graph are shortened and effectively eliminated. Numerical simulations under uniform depolarizing noise for memory and lattice-surgery experiments confirm that the proposed method achieves a full fault distance of $d$, whereas the best existing minimum-depth approach achieves $d-1$. Since the full fault distance is achievable for any regular tiling layout of the surface code, the proposed method may serve as an indispensable technique for practical fault-tolerant quantum computation.