Anticoncentration theorems for schemes showing a quantum speedup

Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

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Abstract

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.

Most near-term proposals for demonstrating a quantum speedup are based on sampling from the output probability distribution of certain random circuits. The technique most often used to prove the speedup for these tasks requires complexity-theoretic conjectures about the sampled distribution to be assumed, one of them being that the distribution `anticoncentrates'. Here, we prove this conjecture for two-designs as well as for a scheme based on the constant-time evolution of a translation-invariant Ising model. Our results give rise to a range of schemes exhibiting a quantum speedup that satisfy this condition by construction, covering a number of experimentally relevant settings, including random universal circuits, diagonal unitaries, Clifford+T circuits and instances of quantum simulation schemes.

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