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  • Open Access

Quantum Simulation of Electronic Structure via Quantum Fast Multipole Method

Dominic W. Berry1,*, Kianna Wan2,3, Andrew D. Baczewski4, Elliot C. Eklund5, Arkin Tikku5, and Ryan Babbush2,†

  • *Contact author: dominic.berry@mq.edu.au
  • †Contact author: ryanbabbush@gmail.com
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PRX Quantum 7, 033025 – Published 10 August, 2026

DOI: https://doi.org/10.1103/b2vn-zf6c

Abstract

Here we describe an approach for simulating electronic structure on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantized representation of the molecular Hamiltonian, which we propagate using high-order product formulas. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with O˜(η) complexity for a simulation with η 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(η4/3N1/3+η1/3N2/3)(ηNt/ϵ)o(1) gate complexity, where N is the number of grid points, ϵ is target precision, and t is the duration of time evolution. This is roughly a speedup by O(η) over most prior algorithms. We provide lower complexity than all prior work for N<η7 (the regime of practical interest), with only first-quantized interaction-picture simulations providing better performance for N>η7. As with the classical fast multipole method, large particle numbers η≳103 would be needed to realize this advantage.

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Popular Summary

The simulation of quantum systems is one of the leading applications of quantum computers, with methods based on Trotter formulas providing some of the best performance. For first-quantized simulation, a bottleneck is the calculation of the Coulomb potential, because it involves a sum over each pair of charges. This sum can be quite costly for simulations with large numbers of charges, in comparison to the calculation of kinetic energy, which is linear in the number of charges. In classical computing, there are ways of speeding up this calculation by fast summation methods, like the fast multipole method. These give complexity close to linear in the number of charges, which would speed up the quantum algorithm by a corresponding factor. The problem is that a direct translation of the classical method into quantum gates would not yield a speedup. This is because it would require data access according to the values in quantum registers. That has a cost corresponding to the number of data registers, which here would be the number of charges, negating the speedup. Here, we show how to solve that problem and provide an algorithm for fast summation with complexity near linear in the number of particles, similar to the classical algorithm. Our method leverages coherent sorting networks applied to shifted Morton orderings, enabling the data accesses to be in a prechosen set of locations. We expect that these methods could be applied to solve the coherent data access problem in other quantum algorithms.

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PRX Quantum 7, 033025– Published 10 August, 2026
  • Received 8 December 2025
  • Revised 8 April 2026
  • Accepted 1 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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