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Random Quantum Circuits are Approximate 2-designs
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Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.
Forward citations
Cited by 3 Pith papers
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Strong unitary designs in optimal depth and space
For every fixed k and error tolerance, strong approximate unitary k-designs are constructed in optimal Theta(log n) depth on the n system qubits using random perfect-matching layers.
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Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms
Clebsch-Gordan transforms give exact compressed oracles for Haar-random unitary group actions, with efficient circuits for U(d).
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Randomized Benchmarking in the Analogue Setting
Analogue randomized benchmarking (ARB) measures the average error rate per time evolution for a family of Hamiltonians on an analogue quantum simulator, demonstrated in classical simulations of XY spin chains.
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