# Fault-tolerant conversion between adjacent Reed-Muller quantum codes based on gauge fixing

## TL;DR

This paper presents a fault-tolerant method for converting between adjacent Reed-Muller quantum codes, optimizing stabilizer measurements and enabling efficient, scalable conversions crucial for universal quantum computation.

## Contribution

It introduces a novel, resource-efficient fault-tolerant conversion scheme between Reed-Muller codes that scales linearly with code parameters, improving upon previous exponential methods.

## Key findings

- Only 7-8 weight-4 stabilizers needed per conversion
- Conversion method handles single-qubit errors effectively
- Scales linearly with code parameter m, unlike previous exponential approaches

## Abstract

We design forward and backward fault-tolerant conversion circuits, which convert between the Steane code and the 15-qubit Reed-Muller quantum code so as to provide a universal transversal gate set. In our method, only 7 out of total 14 code stabilizers need to be measured, and we further enhance the circuit by simplifying some stabilizers; thus, we need only to measure eight weight-4 stabilizers for one round of forward conversion and seven weight-4 stabilizers for one round of backward conversion. For conversion, we treat random single-qubit errors and their influence on syndromes of gauge operators, and our novel single-step process enables more efficient fault-tolerant conversion between these two codes. We make our method quite general by showing how to convert between any two adjacent Reed-Muller quantum codes $\overline{\textsf{RM}}(1,m)$ and $\overline{\textsf{RM}}\left(1,m+1\right)$, for which we need only measure stabilizers whose number scales linearly with m rather than exponentially with m obtained in previous work. We provide the explicit mathematical expression for the necessary stabilizers and the concomitant resources required.