Unitary Designs from Random Symmetric Quantum Circuits

This talk explores distributions of unitaries generated by random quantum circuits with symmetry-respecting gates. We present a unified approach for all symmetry groups, introducing an equation that determines the exact design properties of these distributions. Recent findings show that gate locality imposes constraints on realizable unitaries, which significantly depend on the symmetry. We introduce the concept of "semi-universality" for symmetric gate sets, which realize all symmetry-respecting unitaries up to certain phase restrictions. While 2-qubit gates achieve semi-universality for Z₂, U(1), and SU(2) symmetries in qubit systems, SU(d) symmetry (d≥3) requires 3-qudit gates. Crucially, failure of semi-universality precludes the circuit-generated distribution from being even a 2-design for the Haar distribution over symmetry-respecting unitaries. However, when semi-universality holds, under mild conditions satisfied by U(1) and SU(2), the distribution becomes a t-design with t growing polynomially with qudit number. We present a linear equation determining the maximum integer t_max for which the uniform distribution of circuit-generated unitaries is a t-design for all t ≤ t_max. For U(1), SU(2), and cyclic groups, we determine exact t_max values as a function of qubit number and gate locality. For SU(d), we provide exact t_max values for up to 4-qudit gates.