Quantum-classical algorithm for Ewald summation based computation of long-range electrostatics

TL;DR

This work tackles the computational bottleneck of long-range electrostatics in large biomolecular systems by introducing a quantum-classical hybrid algorithm that uses Ewald-type splitting and computes the Fourier (reciprocal-space) term $E^L$ on a quantum processor with the Quantum Fourier Transform. The classical terms $E^S$, $E^{self}$, and $E^{dip}$ are handled on conventional hardware, while $E^L$ benefits from the potential exponential speedup of the QFT. The authors demonstrate that the method can achieve relative accuracies better than $0.1\%$ and exhibit a quantum advantage when the number of charges $N$ exceeds the grid size $M$, with $E^L$ contributing a small but critical fraction of the total energy. They also discuss practical considerations for near-term devices, including gate costs, repetitions $K$, and suggestions for further improvements such as employing classical shadows to boost accuracy. The work extends the application of QFT-based techniques to computational physics, chemistry, and biology, offering a path toward integrating quantum acceleration into all-atom molecular dynamics simulations.

Abstract

Numerical exploration of large-size real biological systems requires computational power far exceeding that of modern classical computers. In computational molecular science, calculation of long-range electrostatic interactions between charged atoms - the strongest interactions in condensed phases, is a major bottleneck. Here, we propose a quantum algorithm for fast yet accurate computation of Coulomb electrostatic energy for a system of point charges. The algorithm employs the Ewald method based decomposition of electrostatic energy E into several energy terms, of which "the Fourier component" of E is computed in the algorithm proposed on a quantum device, utilizing the power of Quantum Fourier Transform. We demonstrate the algorithm's quantum advantage for a range of systems of point charges in the three-dimensional space when the number of charges (system size) N exceeds the number of grid points M, and show that the numerical error is rather small <0.1%. The algorithm can be implemented in running the all-atom Molecular Dynamics simulations on a quantum computer, thereby expanding the scope of applications of QFT methods in computational physics, chemistry, and biology.

Figures (5)

• Figure 1: The workflow of computation of the $E^L$-component of electrostatic energy $E$ (Eq. \ref{['eq:eq5']}), which consists of the following three steps: state initiation, $d$-dimensional QFT, and measurement.
• Figure 2: Relative importance and computational time for electrostatic energy contributions. Panel A: Contribution of different energy terms $E^{S}$, $E^{L}$, $E^{\mathrm{self}}$, and $E^{\mathrm{dip}}$ to the total electrostatic energy $E$ (Eq. (\ref{['eq:eq5']})) profiled as functions of the number of point charges (system size) $N$. The dashed and solid lines correspond to change configurations in which charges are mixed and separated, respectively (see the inset). Panel B: Computational time associated with the calculation of $E^{S}$, $E^{L}$, $E^{\mathrm{self}}$, and $E^{\mathrm{dip}}$ as a function of $N$. Calculations were carried out in a 32$\times$32$\times$32 grid in 3$d$-space of point charges. Color denotation is explained in the graphs.
• Figure 3: Quantum circuit for Quantum Fourier Transform. The states $|x_1\rangle, |x_2\rangle, \ldots, |x_n\rangle$ comprise the computational basis set. The gate $H$ denotes Hadamard gate, and $R_n$ denotes the phase gate, which rotates the state by $2\pi / 2^n$ radians around the $z$-axis (see Methods).
• Figure 4: Profiles of computational time (averages and standard deviations) associated with calculation of the $E^L$ energy term (Eq. (\ref{['eq:eq5']})). Compared as a function the number of point charges (system size) $N$ is the computational time for the calculation of $E^L$-energy term using the classical algorithm and the quantum algorithm proposed. Calculations were carried out on a 32$\times$32$\times$32 grid in 3$d$-space. Color denotation is explained in the graph.
• Figure 5: Numerical accuracy of electrostatic energy calculation for classical and hybrid quantum-classical Ewald method based algorithms. Profiled as a function of the system size $N$ is relative error ${(|E_{ex}-E_{app}|}) /{E_{ex}}$ (averages and standard deviations), where $E_{ex}$ is the exact electrostatic energy calculated using direct summation (see Eqs. (\ref{['eq:eq3']})), and $E_{app}$ is the approximate energy obtained with the classical and quantum-classical algorithms. Calculations were performed on a 32$\times$32 grid in 2$d$-space (dotted lines) and on a 16$\times$16$\times$16 grid in 3$d$-space (solid lines).

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof