Quantum advantage from energy measurements of many-body quantum systems

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.

Dynamics of single-mode nonclassicalities and quantum correlations in the Jaynes-Cummings model

“…The pioneering works of Feynman [1,2], Bennett and Brassard [3], Deutsch [4], Shor [5], and others have strongly established that using quantum resources in computation, communication, and metrology, one can obtain certain advantages usually referred to as quantum advantage. Subsequent theoretical and experimental works including Google's recent experimental demonstration of quantum advantage [6] have further established this fact ( [7][8][9][10][11][12] are a few recent examples). A common feature of all these works is that quantum advantage requires nonclassical states, or in other words, the quantum states having no classical analog.…”

Computational advantage of quantum random sampling

“…Quantities that first come to mind here are measurements of k-point correlation functions of the type b † i b j . First steps towards this have been taken by Novo et al (2021) where it has been shown that one can run a Stockmeyer argument for the task of reproducing the statistics of an energy measurement of a local Hamiltonian. Deviating from the mindset of quantum random sampling Baez et al (2020) have shown a quantum advantage for the estimation of dynamical structure factors, bringing the insight that performing measurements on quantum states arising from time evolution under local Hamiltonians is BQP complete (Nagaj, 2012;Nagaj and Wocjan, 2008;Vollbrecht and Cirac, 2008) closer to reality.…”