Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates.

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full -qubit group, one often resorts to -designs. Unitary -designs mimic the Haar-measure up to -th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an -approximate -design. Strikingly, the number of non-Clifford gates required is independent of the system size - asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the -th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.