Efficient Quantum Algorithm for Filtering Product States

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 δsqrt{N}. 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 mathcal{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.