Implementing Clifford Gates on Stabilizer Codes via Measurement

TL;DR

This paper presents a measurement-based method to implement Clifford gates fault-tolerantly in stabilizer codes, enabling efficient logical gate operations crucial for universal quantum computing.

Contribution

It generalizes code rewiring techniques to any stabilizer code, providing a fault-tolerant way to implement Clifford gates with minimal measurements, including in the 15-qubit Reed-Muller code.

Findings

Fault-tolerant Clifford gates via measurements in stabilizer codes

Implementation of logical Hadamard in 15-qubit Reed-Muller code with minimal observables

Applicable to toric and LDPC codes for universal quantum computation

Abstract

We describe a method to use measurements and correction operations in order to implement the Clifford group in a stabilizer code, generalising a result from [Bombin,2011] for topological subsystem colour codes. In subsystem stabilizer codes of distance at least $3$ the process can be implemented fault-tolerantly. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only three observables. This is an alternative to the method proposed by [Paetznick and Reichardt, 2013] to generate a set of gates which is universal for quantum computing for this code. The construction is inspired by the description of code rewiring from [Colladay and Mueller, 2018]. Inspired by the code rewiring strategy of [Colladay and Mueller, 2018], we describe a method to use measurements and correction operations in order to implement the Clifford group in the code space of any stabilizer code, and we specify a sufficient set of conditions under which the distance of the code is preserved throughout. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only two observables, providing the only non-transversal gate required for universality. Furthermore this approach is applicable to the toric code and quantum LDPC code