Unitary Designs from Random Symmetric Quantum Circuits

In this work, we study distributions of unitaries generated by random quantum circuits that contain only gates respecting a symmetry. We develop a unified approach applicable to all symmetry groups and obtain an equation that determines the exact design properties of such distributions. It has been recently shown that the locality of gates imposes various constraints on the set of realizable unitaries, which in general, significantly depend on the symmetry under consideration. Generically, such constraints include restrictions on the relative phases between sectors with inequivalent irreducible representations of the symmetry. We call a set of symmetric gates semi-universal if they realize all unitaries that respect the symmetry, up to such restrictions. For instance, while in the case of Z₂, U (1), and SU (2) symmetries, 2-qubit gates are semi-universal on qubit systems, in the case of SU (d) symmetry with d 3, semi-universality can be achieved only with 3-qudit gates. The failure of semi-universality implies that the distribution generated by the circuit is not even a 2-design for the Haar (uniform) distribution over the group of all unitaries respecting the symmetry. On the other hand, when semi-universality holds, under mild conditions on the gate set, which hold, e. g. , in the case of U (1) and SU (2), the distribution of unitaries generated by the random circuit becomes a t-design for t growing polynomially with the number of qudits with a degree that is determined by the locality of gates. More generally, we present a simple linear equation that determines the maximum value of the integer t, for which the uniform distribution of unitaries generated by the circuits is a t-design, and solve this equation for various examples.