Fault-tolerant quantum error correction for Steane's seven-qubit color code with few or no extra qubits

TL;DR

This paper introduces a faster, more qubit-efficient fault-tolerant error correction method for Steane's seven-qubit code by extracting multiple syndromes simultaneously and using two code blocks to avoid extra qubits.

Contribution

It extends flagged syndrome extraction to measure multiple syndromes at once and demonstrates error correction with no additional qubits using two code blocks.

Findings

Multiple syndrome extraction is achievable without extra qubits.

Two code blocks can be combined into a larger code for error correction.

The method conforms to planar geometry constraints.

Abstract

Steane's seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as "flagged" syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane's error-correction method, and also conforms to planar geometry constraints. We further show that Steane's code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.