Efficient Quantum Algorithm for Filtering Product States

• URL: http://arxiv.org/abs/2312.13892v3
• Date: Tue, 25 Jun 2024 09:31:14 GMT
• Title: Efficient Quantum Algorithm for Filtering Product States
• Authors: Reinis Irmejs, Mari Carmen Bañuls, J. Ignacio Cirac,
• Abstract summary: We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators.
• Score: 0.3277163122167433
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\delta$. 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.

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