Approximate t-designs by random quantum circuits with nearly optimal depth

Presenting Author: Saeed Mehraban, Massachusetts Institute of Technology
Contributing Author(s): Aram Harrow

We prove that poly(t) 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) 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.

(Session : from 5:00pm - 7:00 pm)


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