# 2-Designs and Redundant Syndrome Extraction for Quantum Error Correction

## TL;DR

This paper introduces a method using 2-designs to optimize redundant syndrome extraction in quantum error correction, improving fault tolerance especially when measurement errors are significant.

## Contribution

It applies the mathematical theory of 2-designs to design measurement protocols that enhance syndrome separation, improving quantum error correction performance.

## Key findings

- Design-based redundancy improves error correction cost and performance.
- Simulation results for various quantum codes demonstrate effectiveness.
- Method is particularly beneficial when measurement errors are prevalent.

## Abstract

Imperfect measurement can degrade a quantum error correction scheme. A solution that restores fault tolerance is to add redundancy to the process of syndrome extraction. In this work, we show how to optimize this process for an arbitrary ratio of data qubit error probability to measurement error probability. The key is to design the measurements so that syndromes that correspond to different errors are separated by the maximum distance in the signal space, in close analogy to classical error correction codes. We find that the mathematical theory of 2-designs, appropriately modified, is the right tool for this. Analytical and simulation results for the bit-flip code, the 5-qubit code, and the Steane code are presented. The results show that design-based redundancy protocols show improvement in both cost and performance relative to conventional fault-tolerant error-correction schemes in situations, quite important in practice, where measure errors are common. In the near term, the construction of a fault-tolerant logical qubit with a small number of noisy physical qubits will benefit from targeted redundancy in syndrome extraction.