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• W2891685619 abstract "We prove that $poly(t) cdot n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n qudits are approximate $t$-designs in various measures. These include the monomial measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was $poly(t)cdot n$ due to Brandao-Harrow-Horodecki (BHH) for $D=1$. We also improve the scrambling and decoupling bounds for spatially local random circuits due to Brown and Fawzi. One consequence of our result is that assuming the polynomial hierarchy (PH) is infinite and that certain counting problems are $#P$-hard on average, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under the assumption that PH is infinite. However, to show the hardness of approximate sampling using this strategy requires that the quantum circuits have a property called anti-concentration, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Thus our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent proposal by the Google Quantum AI group to perform such a sampling task with 49 qubits on a two-dimensional lattice and confirms their conjecture that $O(sqrt n)$ depth suffices for anti-concentration. We also prove that anti-concentration is possible in depth O(log(n) loglog(n)) using a different model." @default.
• W2891685619 created "2018-09-27" @default.
• W2891685619 creator A5008766821 @default.
• W2891685619 creator A5032563888 @default.
• W2891685619 date "2023-05-04" @default.
• W2891685619 modified "2023-10-18" @default.
• W2891685619 title "Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates" @default.
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• W2891685619 doi "https://doi.org/10.1007/s00220-023-04675-z" @default.
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