Efficient Quantum Algorithm for Filtering Product States

Abstract

We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width δ. Given a local Hamiltonian on N qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to δN. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in poly(N,1/δ) time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width δ is independent of the system size N. Furthermore, the adiabatic evolution can be implemented with circuit depth O(N2δ−4). Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.