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arXiv:2407.04155 · Physical Review C

Evaluation of phase shifts for nonrelativistic elastic scattering using quantum computers

Francesco Turro, Kyle Wendt, Sofia Quaglioni, Francesco Pederiva, Alessandro Roggero

Published Nov 14, 2024• 10 citations
arXivHTMLPDF

Abstract

Simulations of scattering processes are essential in understanding the physics of our universe. Computing relevant scattering quantities from ab initio methods is extremely difficult on classical devices because of the substantial computational resources needed. This work reports the development of an algorithm that makes it possible to obtain phase shifts for generic nonrelativistic elastic scattering processes on a quantum computer. This algorithm is based on extracting phase shifts from the direct implementation of the real-time evolution. The algorithm is improved by a variational procedure making it more accurate and resistant to the quantum noise. The reliability of the algorithm is first demonstrated by means of classical numerical simulations for different potentials and later tested on existing quantum hardware, specifically on IBM quantum processors.

DOI
10.1103/physrevc.110.054604

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Simulations of scattering processes are essential in understanding the physics of our universe. Computing relevant scattering quantities from ab initio methods is extremely difficult on classical devices because of the substantial computational resources needed. This work reports the development of an algorithm that makes it possible to obtain phase shifts for generic nonrelativistic elastic scattering processes on a quantum computer. This algorithm is based on extracting phase shifts from the direct implementation of the real-time evolution. The algorithm is improved by a variational procedure making it more accurate and resistant to the quantum noise. The reliability of the algorithm is first demonstrated by means of classical numerical simulations for different potentials and later tested on existing quantum hardware, specifically on IBM quantum processors.
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We present a method to extract the phase shift of a scattering process using the real-time evolution in the early and intermediate stages of the collision in order to estimate the time delay of a wave packet. This procedure is convenient when using noisy quantum computers for which the asymptotic out-state behavior is unreachable. We demonstrate that the challenging Fourier transforms involved in the state preparation and measurements can be implemented in 1+11+11+1 dimensions with current trapped ion devices and IBM quantum computers. We compare quantum computations of the time delays obtained in the one-particle quantum mechanics limit and the scalable quantum field theory formulation with accurate numerical results. We discuss the finite volume effects in the Wigner formula connecting time delays to phase shifts. The results reported involve two- and four-qubit calculations, and we discuss the possibility of larger scale computations in the near future.
We investigate the feasibility of extracting infinite volume scattering phase shift on quantum computers in a simple one-dimensional quantum mechanical model, using the formalism established in Ref.~\cite{Guo:2023ecc} that relates the integrated correlation functions (ICF) for a trapped system to the infinite volume scattering phase shifts through a weighted integral. The system is first discretized in a finite box with periodic boundary conditions, and the formalism in real time is verified by employing a contact interaction potential with exact solutions. Quantum circuits are then designed and constructed to implement the formalism on current quantum computing architectures. To overcome the fast oscillatory behavior of the integrated correlation functions in real-time simulation, different methods of post-data analysis are proposed and discussed. Test results on IBM hardware show that good agreement can be achieved with two qubits, but complete failure ensues with three qubits due to two-qubit gate operation errors and thermal relaxation errors.
We propose a hybrid quantum-classical framework to solve the elastic-scattering phase shift of two well-bound nuclei in an uncoupled channel. Within this framework, we develop a many-body formalism in which the continuum scattering states of the two colliding nuclei are regulated by a weak external harmonic-oscillator potential with varying strength. Based on our formalism, we propose an approach to compute the eigenenergies of the low-lying scattering states of the relative motion of the colliding nuclei as a function of the oscillator strength of the confining potential. Utilizing the modified effective range expansion, we extrapolate the elastic-scattering phase shift of the colliding nuclei from these eigenenergies to the limit when the external potential vanishes. In our hybrid approach, we leverage the advantage of quantum computing to solve for these eigenenergies from a set of many-nucleon Hamiltonian eigenvalue problems. These eigenenergies are inputs to classical computers to obtain the phase shift. We demonstrate our framework with two simple problems, where we implement the rodeo algorithm to solve the relevant eigenenergies with the IBM Qiskit quantum simulator. The results of both the spectra and the elastic-scattering phase shifts agree well with other theoretical results.
We introduce a new method to calculate phase shifts on noisy intermediate scale quantum (NISQ) hardware platforms using a wave packet edge time delay. The method uses the early and intermediate stages of the collision because the standard method based on the asymptotic out-state behavior is unreachable using today’s NISQ platforms. The calculation was implemented on a 4-site transverse Ising model in one spatial dimension with and without a potential interaction. A time evolution operator describing the progression of the system was constructed and transmission and reflection coefficients were calculated based on the identified quantum Fourier transformed momentum states. The detailed analysis of the phase shift calculations on both IBM superconducting transmon and University of Maryland ion trap quantum computers shows the platform independence of the methodology. This successful implementation of this wave packet preparation and projection on momentum eigenstates can now be performed with actual quantum computing hardware platforms. This method provides a procedure for calculating phase shifts and opens the possibility of using noisy intermediate scale quantum devices to perform real-time quantum mechanics and quantum field theory scattering calculations.
Real-time scattering calculations on a noisy intermediate scale quantum (NISQ) quantum computer are disrupted by errors that accumulate throughout the circuits. To improve the accuracy of such physics simulations, one can supplement the application circuits with a recent error mitigation strategy known as noiseless output extrapolation (NOX). We tested these error mitigation protocols on a transverse field Ising model and improved upon previous phase shift calculations. Our proof-of-concept 4-qubit application circuits were run on several IBM quantum computing hardware architectures. We introduce metrics that show between 22% and 73% error reduction for circuit depths ranging from 13 to 37 hard cycles, confirming that the NOX technique applies to circuits with a broad range of failure rates. We also observed an approximate 28% improvement in the accuracy of the time delay calculation of the scattering phase shift. These observations on different cloud-accessible devices confirm that NOX improves performance even when circuits are executed in substantially time-separated batches. Finally, we provide a heuristic method to obtain systematic error bars on the mitigated results, compare them with empirical errors and discuss their effects on phase shift estimates.