Syndrome Measurement Order for the [[7,1,3]] Quantum Error Correction Code

TL;DR

This paper investigates how the order of syndrome measurements affects the accuracy of the [[7,1,3]] quantum error correction code, providing guidance for optimizing fault-tolerant quantum computing based on error environments.

Contribution

It analyzes different syndrome measurement orders and compares Shor and Steane states, revealing environment-dependent optimal strategies for improved fidelity.

Findings

Optimal syndrome order varies with error environment

Steane states often yield higher fidelity than Shor states

Order of syndrome measurements impacts error correction accuracy

Abstract

In this work we explore the accuracy of quantum error correction depending of the order of the implemented syndrome measurements. CSS codes require bit-flip and phase flip-syndromes be measured separately. To comply with fault tolerant demands and to maximize accuracy this set of syndrome measurements should be repeated allowing for flexibility in the order of their implementation. We examine different possible orders of Shor state and Steane state syndrome measurements for the [[7,1,3]] quantum error correction code. We find that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on the error environment. We also compare the fidelity when syndrome measurements are done with Shor states versus Steane states and find that Steane states generally, but not always, lead to final states with higher fidelity. Together, these results allow a quantum computer programmer to choose the optimal syndrome measurement scheme based on the system's error environment.