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Abstract

We introduce a quantum algorithm to efficiently prepare states with an
arbitrarily 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 $\delta\sqrt{N}$. 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
$\mathcal{O}(N^2\delta^{-4})$. 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.