Clifford Group and Unitary Designs under Symmetry

TL;DR

This paper extends the concept of unitary designs to symmetric cases, characterizing when symmetric Clifford groups form 3-designs under certain symmetries and exploring their properties under others, with implications for quantum information.

Contribution

It generalizes the Clifford group as a symmetric unitary 3-design under Pauli symmetries and provides a construction method, also analyzing symmetry constraints like U(1) and SU(2).

Findings

Symmetric Clifford group is a symmetric 3-design under Pauli subgroup symmetries.

Under U(1) and SU(2) symmetries, it is a 1-design but not a 2-design.

Numerical verification via frame potentials supports theoretical results.

Abstract

We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design. Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup. We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries. Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group. This work will open a new perspective into quantum information processing such as randomized benchmarking, and give a deep understanding to many-body systems such as monitored random circuits.