Qudit circuits with SU(d) symmetry: Locality imposes additional conservation laws

On the universality of Sn -equivariant k-body gates

“…Still, despite the great promise of GQML there are still many open questions regarding its full capabilities. As an example, the seminal works of [17][18][19][20], have started to analyze restrictions to universality and their connections to the type of gates used. Moreover, it is has been documented that quantum noise can be quite detrimental for equivariant models [50][51][52] and that reducing a quantum learning model's expressive power too much can 3.…”

“…As an example, the Lie algebra g 2 studied in this paper has a relatively small dimension deficit of ⌊n/2⌋ compared to the full S n -invariant algebra u Sn (d). The question of whether few-bodyness (or locality) of symmetry-equivariant qubit gates could enforce extra linear or quadratic symmetries has only begun to be studied, for instance in [19,20], and such possibilities would cause the DLA dimension deficiency to be much larger. We thus believe that a more general investigation into the failure modes for gate sets with additional constraints (beyond symmetry-equivariance) to generate the full symmetry-invariant algebra will be a fruitful area of study.…”

“…This question was first studied in [17], where the authors found that, as opposed to the well-known result which states that any unitary can be decomposed in one-and two-qubit gates, certain equivariant unitaries cannot be decomposed in equivariant k-local gates. These result were then extended in [18][19][20] to rotationally-invariant circuits, tensor product representations of groups acting on qudits, and tensor product representations of Abelian symmetries.…”

Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits

“…Constraints imposed by locality-In the case of qubit systems with SU(2) symmetry the restrictions imposed by locality are limited to constraints on the relative phases between the subspaces with different irreps of symmetry (Interestingly, in the case of qudits with SU(d) symmetry for d ≥ 3, there are stronger constraints [41]). In terms of realizable Hamiltonians, as stated in theorem 1 below, this amounts to constraints on the inner products of the Hamiltonian with the projectors to subspaces {H j }, denoted by {Π j : j = j min , • • • , j max }.…”