Unitary 2-designs, variational quantum eigensolvers, and barren plateaus

Are randomised quantum circuits inducing unitary 2-designs a reasonable thing to hope for in the near future? It seems so, following a paper presented by Saeed Mehraban at QIP this year. With Harrow, Mehraban showed that short-depth random circuits involving nearest-neighbour unitary gates (where proximity is defined by a cubic lattice structure) produce approximate unitary $t$-designs. This is precisely the experimental model of Google’s Quantum AI group, who are working with a 49-qubit 2-dimensional lattice of qubits.

The main application of these randomised circuits is to prove ‘quantum supremacy’. But can they do anything useful? In the attached note, I discuss an application to building a variational quantum eigensolver, which runs into the problem of ‘barren plateaus’ recently highlighted by McClean et al. (See within for references.)