Leveraging Automorphisms of Quantum Codes for Fault-Tolerant Quantum Computation

TL;DR

This paper explores how automorphisms of quantum codes can be exploited to implement certain quantum gates fault-tolerantly by permuting physical qubits, potentially reducing operational overhead in quantum computing.

Contribution

It introduces a method to use quantum code automorphisms for fault-tolerant gate implementation, providing conditions and examples for codes with large automorphism groups.

Findings

Automorphisms enable fault-tolerant implementation of some gates.

Conditions identified for codes with large automorphism groups.

Examples include codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], [[31,11,5]].

Abstract

Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and [[31,11,5]].