Kavli Affiliate: David Gross

| First 5 Authors: Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross

| Summary:

Many quantum information protocols require the implementation of random

unitaries. Because it takes exponential resources to produce Haar-random

unitaries drawn from the full $n$-qubit group, one often resorts to

$t$-designs. Unitary $t$-designs mimic the Haar-measure up to $t$-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 $O(t^{4}log^{2}(t)log(1/varepsilon))$ many

non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an

$varepsilon$-approximate $t$-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 $t$-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.

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