Quantum Simulation of Chemistry via Quantum Fast Multipole Transform
Abstract
Here we describe an approach for simulating quantum chemistry on quantum computers with significantly lower asymptotic complexity than prior work.
The approach uses a real-space first-quantised representation of the molecular Hamiltonian which we propagate using high-order product formulae.
Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with $\widetilde{\cal O}(\eta)$ complexity for a simulation with $\eta$ particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with $t(\eta^{4/3}N^{1/3} + \eta^{1/3} N^{2/3} ) (\eta Nt/\epsilon)^{o(1)}$ gate complexity, where $N$ is the number of grid points, $\epsilon$ is target precision, and $t$ is the duration of time evolution.
This is roughly a speedup by ${\cal O}(\eta)$ over most prior algorithms.
We provide lower complexity than all prior work for $N<\eta^6$ (the only regime of practical interest), with only first-quantised interaction-picture simulations providing better performance for $N>\eta^6$. However, we expect the algorithm to have large constant factors that are likely to limit its practical applicability.
The approach uses a real-space first-quantised representation of the molecular Hamiltonian which we propagate using high-order product formulae.
Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with $\widetilde{\cal O}(\eta)$ complexity for a simulation with $\eta$ particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with $t(\eta^{4/3}N^{1/3} + \eta^{1/3} N^{2/3} ) (\eta Nt/\epsilon)^{o(1)}$ gate complexity, where $N$ is the number of grid points, $\epsilon$ is target precision, and $t$ is the duration of time evolution.
This is roughly a speedup by ${\cal O}(\eta)$ over most prior algorithms.
We provide lower complexity than all prior work for $N<\eta^6$ (the only regime of practical interest), with only first-quantised interaction-picture simulations providing better performance for $N>\eta^6$. However, we expect the algorithm to have large constant factors that are likely to limit its practical applicability.
