Fault-tolerant syndrome extraction in [[n,1,3]] non-CSS code family generated using measurements on graph states

TL;DR

This paper introduces a family of fault-tolerant quantum error-correcting codes that are resilient to noisy syndrome measurements, using graph states and a custom decoder, with improved performance under depolarizing noise.

Contribution

It systematically constructs and analyzes a new family of [[n,1,3]] non-CSS QECCs that are fault-tolerant against hook errors during syndrome extraction, using graph codes and simulation.

Findings

Demonstrates fault tolerance against hook errors using the bare-ancilla method.

Provides simulation results showing trade-offs between code rate and performance.

Identifies optimized codes under anisotropic and circuit-level depolarizing noise.

Abstract

The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. In this work, we construct a family of $[[n,1,3]]$ non-CSS QECCs, which are fault-tolerant (FT) against noisy syndrome measurements. We employ the bare-ancilla method of Muyuan Li \emph{et al.} to demonstrate fault tolerance against hook errors during syndrome extraction. We present a systematic protocol for generating these QECCs using graph codes and propose a family of $[[n,1,3]]$ codes that preserve the fault-tolerant properties of the bare ancilla codes. We use a custom lookup-table decoder and simulate the code's performance under both anisotropic and circuit-level depolarizing noise. Our results reveal a trade-off in performance with respect to the code rate and identify optimized codes under these noise models. We benchmark our results against the flag-qubit method of Chao \emph{et al}. Notably, we report a new bare ancilla code with improved code rate while maintaining the same distance compared to the bare code used in the work of Muyuan Li \emph{et al.}