Hamiltonian operator approximation for energy measurement and ground state preparation

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 propagators using a differential representation. The proposed approach, named Hamiltonian operator approximation (HOA), is highly suitable for 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, and perform energy measurements. Benchmarking the measurement scheme, we discuss the relevance of the discretization step size, stencil order, number of measurements, 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. We find that for Heisenberg model of increasing system size of twelve and more spins our approach outperforms variational methods, both in terms of the gate depth and the number of measurements.