Clifford Group and Unitary Designs in the Presence of Symmetry

In this talk, we present the symmetric generalization of the well-known statement that the Clifford group is a unitary 3-design, with the symmetric extension of unitary designs. Concretely, we show that a symmetric Clifford group is a symmetric unitary 3-design, if and only if the symmetry constraint is essentially given by some Pauli subgroup. Moreover, we also present a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For a comprehensive understanding, we also take physically relevant U(1) and SU(2) symmetries as examples of non-Pauli symmetries, and show that the symmetric Clifford groups are only symmetric unitary 1-designs and not 2-designs under those symmetries. Finally, for the verification of our results, we present the numerical results about the frame potentials, which measure the difference with respect to randomness between the uniform ensemble over a symmetric unitary group and its subgroup. This work is expected to open up a new perspective into quantum information processing techniques such as randomized benchmarking, and to provide a deep understanding to many-body systems such as monitored random circuits.