Hamiltonian operator approximation for energy measurement and ground state preparation

TL;DR

This paper introduces Hamiltonian operator approximation (HOA), a method for energy measurement and ground state preparation on analog quantum simulators, outperforming some variational methods for larger systems.

Contribution

The paper presents HOA, a novel approach to approximate Hamiltonians as sums of propagators, optimized for analog quantum simulators and large-scale quantum systems.

Findings

HOA effectively measures energy in analog quantum simulators.

HOA can prepare ground states with high precision for complex models.

HOA outperforms variational methods for systems with 12 or more spins.

Abstract

The Hamiltonian operator plays a central role in quantum theory being a generator of unitary quantum dynamics. Its expectation value describes the energy of a quantum system. Typically being a non-unitary operator, the action of the Hamiltonian is either encoded using complex ancilla-based circuits, or implemented effectively as a sum of Pauli string terms. Here, we show how to approximate the Hamiltonian operator as a sum of propagators using a differential representation. The proposed approach, named Hamiltonian operator approximation (HOA), is designed to benefit analog quantum simulators, where one has direct access to simulation of quantum dynamics, but measuring separate circuits is not possible. We describe how to use this strategy in the hybrid quantum-classical workflow for performing energy measurements. Benchmarking the measurement scheme, we discuss the relevance of the discretization step size, stencil order, number of shots, and noise. We also use HOA to prepare ground states of complex material science models with direct iteration and quantum filter diagonalization, finding the lowest energy for the 12-qubit Hamiltonian of hydrogen chain H$_6$ with $10^{-5}$ Hartree precision using $11$ time-evolved reference states. The approach is compared to the variational quantum eigensolver, proving HOA beneficial for systems at increasing size corresponding to noisy large scale quantum devices. We find that for Heisenberg model with twelve or more spins our approach may outperform variational methods, both in terms of the gate depth and the total number of measurements.