Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three

TL;DR

This paper introduces a parallel syndrome extraction method for Calderbank-Shor-Steane codes of distance three, reducing circuit depth and resource use by sharing flag qubits during stabilizer measurements.

Contribution

It generalizes previous small-code methods to larger codes, enabling fault-tolerant parallel stabilizer measurements with shared flag qubits.

Findings

Parallel scheme outperforms existing seven- and nine-qubit schemes

Simulated pseudo-thresholds show improved fault-tolerance

Shared flag qubits reduce circuit depth and resource requirements

Abstract

To perform achieve fault-tolerant quantum computation, one can use flagged syndrome extraction with fewer ancilla qubits. However, it suffers from long circuit depth if one stabilizer is measured at a time. Previously, Reichardt showed that it is possible to measure multiple stabilizers with at most one shared flag qubit for certain small quantum codes. In this paper, we propose a procedure for general Calderbank-Shor-Steane codes of distance three so that multiple $Z$-stabilizers ($X$-stabilizers) can be fault-tolerantly measured in parallel with one shared flag qubit. We simulate the memory and computation pseudo-thresholds for various code schemes. In particular, our parallel scheme based on Shor's nine-qubit code performs better than known seven- and nine-qubit schemes in the literature.