Efficient unitary designs with a system-size independent number of non-Clifford gates

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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