Quantum advantage from energy measurements of many-body quantum systems

TL;DR

This paper explores the potential for demonstrating quantum advantage through energy measurements of many-body quantum systems, showing that classical simulation is unlikely in certain regimes and proposing new complexity-theoretic insights.

Contribution

It introduces the concept of energy sampling as a physically motivated problem for quantum advantage, analyzing its complexity across different measurement regimes and linking it to Hamiltonian complexity.

Findings

High-resolution energy sampling is classically hard for certain Hamiltonians.

Quantum circuits with commuting gates can efficiently perform high-resolution energy measurements.

Classical algorithms for low-resolution energy sampling are unlikely if quantum computers are more powerful.

Abstract

The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state -- a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be exponentially fast-forwarded, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.