Exhaustive Optimisation of Automorphism Groups for Stabiliser Codes

TL;DR

This paper presents a comprehensive framework for optimizing the implementation of logical operations in small stabiliser quantum codes by exploiting automorphism groups, code equivalence, and conjugacy classes.

Contribution

It introduces a novel method leveraging automorphism groups and code equivalence to systematically find all optimal logical operation implementations for small stabiliser codes.

Findings

Generated a complete table of optimal logical operations for codes with n ≤ 7, k ≤ 2.

Provided physical circuit implementations for all small stabiliser codes.

Enhanced methods for fault-tolerant logical operation realization.

Abstract

An important measure of utility for a quantum code is the identification of which logical operations can be implemented fault-tolerantly on its codespace. We introduce a framework which leverages the automorphism groups of associated classical codes, the choice of logical basis and exploitation of code equivalence to construct all distinct implementable realisations of each valid logical operation for a given $[[n,k,d]]$ code. We establish conjugacy classes and group transversals (unrelated to transversality) as key explanatory concepts. We subsequently motivate and calculate two figures-of-merit that can be optimised with this framework. Our results yield a table of optimal logical operations and their corresponding physical circuits for all small stabiliser codes with $n \leq 7$ and $k \leq 2$, drawn from quantum databases. This exhaustive table of results provides the optimal physical implementations of logical operations which may be advantageous for both magic state cultivation and experimental purposes.