Representations of the multi-qubit Clifford group

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 ${\phi }^{\otimes 2}$ 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.

21 pages, 2 figures, see also related work by Zhu, Kueng, Grassl and Gross. Third version has substantially improved notation and organisation. Fixed mistake in the proof of lemma 5 (lemma 4 in v1,v2), result unchanged. Also substantially improved the characterisation of several vector spaces (lemmas 7,8,9). Added a tree diagram showing inclusions of all vector spaces in the paper

References in corpus (8)

• Randomized Benchmarking of Quantum Gates
• Robust randomized benchmarking of quantum processes
• Evenly distributed unitaries: on the structure of unitary designs
• Multiqubit Clifford groups are unitary 3-designs
• The Clifford group fails gracefully to be a unitary 4-design
• Multi-qubit Randomized Benchmarking Using Few Samples
• Low rank matrix recovery from Clifford orbits
• Distinguishing quantum states using Clifford orbits

Cited by in corpus (14)

• Theory of quantum system certification: a tutorial
• Recovering quantum gates from few average gate fidelities
• Real Randomized Benchmarking
• Efficient unitary designs with a system-size independent number of non-Clifford gates
• Multi-qubit Randomized Benchmarking Using Few Samples
• Quantum circuits for exact unitary $t$-designs and applications to higher-order randomized benchmarking
• Guaranteed recovery of quantum processes from few measurements
• Thrifty shadow estimation: re-using quantum circuits and bounding tails
• Efficient Unitarity Randomized Benchmarking of Few-qubit Clifford Gates
• Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes
• Efficient classical simulation and benchmarking of quantum processes in the Weyl basis
• Robust Estimation of Nonlinear Properties of Quantum Processes
• On character table of Clifford groups
• Optimizing Circuit Reusing and its Application in Randomized Benchmarking