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

• URL: http://arxiv.org/abs/2002.09524v3
• Date: Sun, 11 Jun 2023 20:42:07 GMT
• Title: Efficient unitary designs with a system-size independent number of non-Clifford gates
• Authors: Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross, Ingo Roth
• Abstract summary: It takes exponential resources to produce Haar-random unitaries drawn from the full $n$-qubit group. Unitary $t-designs mimic the Haar-$-th moments. We derive novel bounds on the convergence time of random Clifford circuits to the $t$-th moment of the uniform distribution on the Clifford group.
• Score: 2.387952453171487
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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