A framework for symmetric quantum circuits: Semi-universality of 3-qudit SU(d)-invariant gates

Quantum circuits with symmetry-respecting gates have attracted broad interest in quantum information science. While recent work has developed a theory for circuits with Abelian symmetries, revealing important distinctions between Abelian and non-Abelian cases, a comprehensive framework for non-Abelian symmetries has been lacking. In this work, we develop novel techniques and a powerful framework that is particularly useful for understanding circuits with non-Abelian symmetries. Using this framework we settle an open question on quantum circuits with SU(d) symmetry. We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries, up to certain constraints on the relative phases between sectors with inequivalent representation of symmetry. Furthermore, we prove that these gates achieve full universality when supplemented with 3 ancilla qudits. Interestingly, we find that studying circuits with 3-qudit gates is also useful for a better understanding of circuits with 2-qudit gates. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits.