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The Journal of Physical Chemistry Letters

Real-Time Simulation of Low-Energy Quantum Scattering with Quantum Circuits

Joshua M. Courtney, Phillip C. Stancil

Published Sep 8, 2026• 0 citations
Publisher

Abstract

Simulating chemical dynamics such as atomic and molecular scattering is a natural target for quantum computation, where conventional (classical) hardware faces exponential cost. We present a product-formula quantum algorithm that models single-channel helium-helium scattering by real-time evolution of a Gaussian wavepacket, requiring no precomputation of eigenstates. Interaction with the Lennard-Jones potential and centrifugal barrier is applied unitarily with closed-form circuits, with the evolution processed directly with error-free quantum circuit compilation. Tracking the wave function results in converged phase shifts, from which the deflection function, scattering amplitude, and differential cross section are calculated. Identical-boson integral cross sections agree with time-independent Numerov and log-derivative benchmarks to 6.1%, with a localized residual in s-wave scattering. The construction realizes any potential of the form ∑ir-mi with the same gate count as a single 1/r term, scales efficiently to multichannel dynamics, and provides a ground-truth operator for benchmarking approximate quantum simulation.

DOI
10.1021/acs.jpclett.6c02134

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Simulating chemical dynamics such as atomic and molecular scattering is a natural target for quantum computation, where conventional (classical) hardware faces exponential cost. We present a product-formula quantum algorithm that models single-channel helium-helium scattering by real-time evolution of a Gaussian wavepacket, requiring no precomputation of eigenstates. Interaction with the Lennard-Jones potential and centrifugal barrier is applied unitarily with closed-form circuits, with the evolution processed directly with error-free quantum circuit compilation. Tracking the wave function results in converged phase shifts, from which the deflection function, scattering amplitude, and differential cross section are calculated. Identical-boson integral cross sections agree with time-independent Numerov and log-derivative benchmarks to 6.1%, with a localized residual in s-wave scattering. The construction realizes any potential of the form ∑ir-mi with the same gate count as a single 1/r term, scales efficiently to multichannel dynamics, and provides a ground-truth operator for benchmarking approximate quantum simulation.
We explore how the fundamental problems in quantum molecular dynamics can be modeled using classical simulators (emulators) of quantum computers and the actual quantum hardware available to us today. The list of problems we tackle includes propagation of a free wave packet, vibration of a harmonic oscillator, and tunneling through a barrier. Each of these problems starts with the initial wave packet setup. Although Qiskit provides a general method for initializing wave functions, in most cases it generates deep quantum circuits. While these circuits perform well on noiseless simulators, they suffer from excessive noise on quantum hardware. To overcome this issue, we designed a shallower quantum circuit for preparing a Gaussian-like initial wave packet, which improves the performance of real hardware. Next, quantum circuits are implemented to apply the kinetic and potential energy operators for the evolution of a wave function over time. The results of our modeling on classical emulators of quantum hardware agree perfectly with the results obtained using the traditional (classical) methods. This serves as a benchmark and demonstrates that the quantum algorithms and Qiskit codes we developed are accurate. However, the results obtained on the actual quantum hardware available today, such as IBM's superconducting qubits and IonQ's trapped ions, indicate large discrepancies due to hardware limitations. This work highlights both the potential and challenges of using quantum computers to solve fundamental quantum molecular dynamics problems.
Molecular quantum dynamics simulations that treat both electrons and nuclei quantum mechanically are crucial for predicting chemical reactions. With classical computation, a full wave-function representation of both types of particles requires resources that grow exponentially with their number. Oscillator-qubit processors, including trapped-ion and circuit QED devices, represent electronic states with qubits and nuclear vibrations with oscillators. Their native operations produce linear vibronic interactions, from which programmable anharmonic interactions can be built. Product-formula approaches build the corresponding phase gates by repeating small noncommuting phase steps, and quantum signal processing (QSP) approaches approximate each gate directly with a Fourier series. Simulations apply many such gates in sequence, so accumulated cost and error limit how long the dynamics can be followed. We propose near-optimal quantum primitives for molecular quantum dynamics simulation, covering the representations of both types of particles and their interactions. We synthesize nonlinear and multimode bosonic phase gates using generalized Jacobi-Anger (GJA) expansions within QSP, with a cost that grows near-optimally in both interaction strength and precision. Our interaction model and gate synthesis each have their own accuracy setting. Chosen together, these settings make our multimode benchmark circuits about one-third shorter than those of the previous direct QSP approach. We then simulate energy exchange between the stretching and bending vibrations of a simplified carbon dioxide model. With circuit-QED noise, a short simulation narrowly misses our accuracy criteria at qubit and oscillator lifetimes near 2 ms and 5 ms. These lifetimes mark a concrete checkpoint on the way to simulating anharmonic multimode molecular quantum dynamics on near-term oscillator-qubit processors.
Quantum simulation has begun to penetrate the field of quantum chemistry in hopes of efficiently calculating ground state energies and approximating real-time evolution. With modern research highlighting nonadiabatic dynamics, tunably approximating deep circuits representing potential landscapes becomes crucial for simulating real quantum systems. Variationally approximating unitaries allows for shallower circuits and accuracy tunable to hardware fidelity, so long as the observable quantities are preserved. We show the variational compression of Trotter terms preserve reaction rate coefficients via classical emulation of a hybrid quantum-classical optimization method, as well as fast-forwarded adiabatic dynamics on quantum hardware. Compressed circuits can be incorporated with product-formula-based time evolution to approximate dynamics of a particle in two coupled harmonic potentials, allowing tunability when removing high-cost qubit interactions. Approximate rate coefficients are recovered after substituting terms in a nonadiabatic dynamic process, giving proof-of-principle for observable preservation under variational optimization. Attention is paid to minimizing qubit and gate-count resources.
We present hybrid quantum–classical algorithms to compute time-dependent Møller wavepacket correlation functions via digital quantum simulation. Reactant and product channel wavepackets are encoded as qubit states, evolved under a discretized molecular Hamiltonian, and their correlation is reconstructed using both a modified Hadamard test and a multi-fidelity estimation (MFE) protocol. The method is applied to the collinear H + H2 exchange reaction on a London–Eyring–Polanyi–Sato potential energy surface. Quantum-estimated correlation functions show quantitative agreement with high-resolution classical wavepacket simulations across the full time domain, reproducing both short-time scattering peaks and long-time oscillatory dynamics. The ancilla-free MFE protocol achieves matching results with reduced circuit depth. These results provide a proof of principle that digital quantum circuits can be used to accurately calculate the wavepacket correlation functions for a benchmark chemical reaction system.
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.