Average-Case Quantum Advantage with Shallow Circuits
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Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation in the worst-case setting, and left the existence of a similar separation in the average-case setting, which would give a much stronger evidence of the superiority of small-depth quantum computation, as an open problem. In this paper we solve this problem: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a `shallow' quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem with high probability on a fraction 1-δ of the inputs, for some constant δ>0, must have logarithmic depth. The same construction also gives a distribution that can be sampled exactly by shallow quantum circuits, but such that any classical circuit approximating it, even with constant additive error, must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph. A similar result has been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233).
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