Representations of the multi-qubit Clifford group

TL;DR

This paper characterizes the irreducible subrepresentations of the tensor representations of the multi-qubit Clifford group, which is crucial for understanding its structure and applications in quantum information processing.

Contribution

It provides a complete characterization of all irreducible subrepresentations of the two-copy tensor representation of the multi-qubit Clifford group.

Findings

Identified all irreducible subrepresentations of the two-copy representation.

Facilitated reduction in the number of samples needed for randomized benchmarking.

Enhanced understanding of the Clifford group's tensor representations.

Abstract

The Clifford group is a fundamental structure in quantum information with a wide variety of applications. We discuss the tensor representations of the $q$-qubit Clifford group, which is defined as the normalizer of the $q$-qubit Pauli group in $U(2^q)$. In particular, we characterize all irreducible subrepresentations of the two-copy representation $\varphi^{\otimes2}$ of the Clifford group on the matrix space $\mathbb{C}^{d\times d}\otimes \mathbb{C}^{d\times d}$ with $d=2^q$. In an upcoming companion paper we applied this result to cut down the number of samples necessary to perform randomised benchmarking, a method for characterising quantum systems.