Random Quantum Circuits are Approximate 2-designs

Details Random Quantum Circuits are Approximate 2-designs Random Quantum Circuits are Approximate 2-designs

2008-02-13

arxiv.org/abs/0802.1919v3

Comm. Math. Phys. Vol. 291, No. 1, pp. 257--302 (2009)

Physics

Quantum Physics

48 pages, 1 figure. Typo in bibliography fixed

Scientific paper

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.

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