Fernando G. S. L. Brandao, Aram W. Harrow, Michal Horodecki
We prove that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbor two-qubit gates form an approximate unitary poly(n)-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are infinity-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove they constitute a quantum poly(n)-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov et al. We give three applications of the results. First we show that almost every circuit U of size n^k on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size n^[(k+3)/6] that are given oracle access to U; this provides a data-hiding scheme against computationally bounded adversaries. Second we reconsider a recent argument of Masanes et al concerning local equilibration of time-evolving quantum systems, and strengthen the connection between fast equilibration of small subsystems and the circuit complexity of the Hamiltonian. Third we show that in one dimension almost every parallel local circuit of linear depth generates topological order, matching an upper bound to the problem due to Bravyi et al.
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http://arxiv.org/abs/1208.0692
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