Closing gaps of a quantum advantage with short-time Hamiltonian dynamics

TL;DR

This paper rigorously proves two key conjectures for Hamiltonian quantum simulators, demonstrating their potential for quantum advantage by establishing anticoncentration and average-case hardness, thus strengthening the case for their classical intractability.

Contribution

It proves anticoncentration and average-case hardness for 2D translation-invariant, constant depth Hamiltonian quantum simulators, closing theoretical gaps in quantum advantage claims.

Findings

Proved anticoncentration for specific quantum architectures.

Established average-case hardness based on new techniques.

Provided strongest evidence for classical intractability of these simulators.

Abstract

Demonstrating a quantum computational speedup is a crucial milestone for near-term quantum technology. Recently, quantum simulation architectures have been proposed that have the potential to show such a quantum advantage, based on commonly made assumptions. The key challenge in the theoretical analysis of this scheme - as of other comparable schemes such as boson sampling - is to lessen the assumptions and close the theoretical loopholes, replacing them by rigorous arguments. In this work, we prove two open conjectures for these architectures for Hamiltonian quantum simulators: Anticoncentration of the generated probability distributions and average-case hardness of exactly evaluating those probabilities. The latter is proven building upon recently developed techniques for random circuit sampling. For the former, we develop new techniques that exploit the insight that approximate 2-designs for the unitary group admit anticoncentration. We prove that the 2D translation-invariant, constant depth architectures of quantum simulation form approximate 2-designs in a specific sense, thus obtaining a significantly stronger result. Our work provides the strongest evidence to date that Hamiltonian quantum simulation architectures are classically intractable.