Anticoncentration theorems for schemes showing a quantum speedup

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.

Sample Complexity of Device-Independently Certified “Quantum Supremacy”

“…For the case of the random universal circuits of Ref. [15] there is evidence that the output distribution of fixed instances is essentially given by an exponential (Porter-Thomas) distribution P PT whose second moment is given by [20,Eq. (8)]…”

Quantum advantage of unitary Clifford circuits with magic state inputs

“…The gates used in these circuits are from {CZ, X 1/2 , Y 1/2 , T }. In [17] it is shown that circuits from this set anticoncentrate if they are chosen as follows: let G = {CZ, X 1/2 , X −1/2 , Y 1/2 , Y −1/2 , T, T † } (i.e. the previous set closed under inverses).…”

“…This result has been shown in the recent works [6] and [24] (and our results were developed independently concurrently) but only for a single particular hardness conjecture. Furthermore both papers prove the anticoncentration conjecture by using the fact that random Clifford circuits form a k-design for suitable k. The idea of using k-designs to prove anticoncentration conjectures is explored in [17]. In this paper, we use a different approach.…”

Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States

“…By magic state injection, circuits with this gate set can be efficiently simulated by Clifford circuits with magic state |T inputs, where |T = 1 √ 2 (|0 + e iπ/4 |1 ). It has been shown that postCM = postBQP [13], and thus output probabilities are #P-hard approximate up to some constant relative error [24][25][26]. However, if there is some independent depolarizing error acting on each input magic state, e.g., the input state on each register is (1 − ε)|T T | + ε I 2 , then Theorem 1 implies directly that there exists a classical algorithm to approximate the output probability up to l 1 norm δ in time n O(log(1/δ )/ε) for a large fraction of the CM circuits with noisy inputs.…”