Efficient Quantum Algorithm for Filtering Product States

Quantum 8, 1389 (2024).

https://doi.org/10.22331/q-2024-06-27-1389

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 $delta$. 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 $deltasqrt{N}$. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in $mathrm{poly}(N,1/delta)$ 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 $delta$ is independent of the system size $N$. Furthermore, the adiabatic evolution can be implemented with circuit depth $mathcal{O}(N^2delta^{-4})$. 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.