We describe a method for obtaining the scattering matrix for nuclear or chemical reactions on a finite lattice. Aside from the preparation of the initial and final states as wave packets, the only other operation required is unitary time evolution, making this approach ideal for simulations on quantum hardware. The central quantity is a time-dependent overlap between incoming and outgoing wave packets whose Fourier transform corresponds to the scattering matrix at fixed energy, from which one can calculate elastic and inelastic cross sections for reactions involving two interacting clusters. Working in Cartesian coordinates enables an efficient encoding of the problem on quantum hardware via the first quantization mapping, with favorable qubit scaling for describing asymptotic scattering states. Within this framework, we describe a quantum algorithm for probing the scattering amplitude through different angles, including the forward direction, which provides access to the total cross section via the optical theorem. We demonstrate our methods through a series of numerical examples, for both elastic and inelastic processes, comparing against exact calculations. The techniques we describe can more readily be extended to a large number of constituent particles than other existing approaches, once fault-tolerant quantum hardware becomes available.
We describe a method for obtaining the scattering matrix for nuclear or chemical reactions on a finite lattice. Aside from the preparation of the initial and final states as wave packets, the only other operation required is unitary time evolution, making this approach ideal for simulations on quantum hardware. The central quantity is a time-dependent overlap between incoming and outgoing wave packets whose Fourier transform corresponds to the scattering matrix at fixed energy, from which one can calculate elastic and inelastic cross sections for reactions involving two interacting clusters. Working in Cartesian coordinates enables an efficient encoding of the problem on quantum hardware via the first quantization mapping, with favorable qubit scaling for describing asymptotic scattering states. Within this framework, we describe a quantum algorithm for probing the scattering amplitude through different angles, including the forward direction, which provides access to the total cross section via the optical theorem. We demonstrate our methods through a series of numerical examples, for both elastic and inelastic processes, comparing against exact calculations. The techniques we describe can more readily be extended to a large number of constituent particles than other existing approaches, once fault-tolerant quantum hardware becomes available.
The description of quantum many-body dynamics is extremely challenging on classical computers, as it can involve many degrees of freedom. On the other hand, the time evolution of quantum states is a natural application for quantum computers that are designed to efficiently perform unitary transformations. In this paper, we study quantum algorithms for response functions, relevant for describing different reactions governed by linear response. We focus on nuclear-physics applications and consider a qubit-efficient mapping on the lattice, which can efficiently represent the large volumes required for realistic scattering simulations. For the case of a contact interaction, we develop an algorithm for time evolution based on the Trotter approximation that scales logarithmically with the lattice size, and is combined with quantum phase estimation. We eventually focus on the nuclear two-body system and a typical response function relevant for electron scattering as an example. We also investigate ground-state preparation and examine the total circuit depth required for a realistic calculation and the hardware noise level required to interpret the signal.
Simulating the real-time dynamics of particle collisions is a promising application of quantum simulators, because classical methods such as tensor networks struggle to capture the highly entangled states generated in high-energy scattering. Realizing such simulations requires both the preparation of incoming wave packets and the detection of the outgoing scattering products. In this work, we propose protocols that address both challenges on programmable analog and digital quantum simulation platforms. Our state-preparation scheme uses a weakly coupled auxiliary qubit - or, more generally, a customized local quench - to inject a single quasiparticle with well-defined momentum. Because it relies only on conservation of energy, this scheme requires no fine-tuning or prior knowledge about particle eigenstates, making it robust against errors in calibration and implementation. The momenta of scattering products are then extracted, using only local measurements, from the interference pattern that arises when particles are reflected at the system's boundary. We validate our protocols through numerical simulations, first in a simple single-particle model and subsequently in two interacting many-body systems: a Rydberg atom chain and an Ising chain in a mixed field. We demonstrate how high-energy regimes, necessary to access inelastic scattering processes, can be reached through an adiabatic ramp, and how the wave packet shape can be optimized by spatially modulating the Hamiltonian. Finally, we show how the protocol can be generalized to systems with more than one spatial dimension. Our proposal provides a versatile approach to the quantum simulation of scattering phenomena, and is compatible with several quantum simulation platforms that are already experimentally available.
Non-relativistic quantum scattering theory informs our broad understanding of nuclear collision processes. However, detailed theoretical insights into the dynamics of fusion, whereby the colliding nuclei form a compound nucleus, remain elusive with conventional approaches. Fundamentally, understanding the complex dissipative processes inherent to compound nucleus formation and when they occur during the collision remains challenging. This, for example, prevents a consistent description of heavy-ion fusion over a wide range of collision energies and hinders searches for new super-heavy elements. To understand these dynamics, we need time-dependent approaches to nuclear collisions. In this thesis, a non-relativistic, time-dependent wave packet theory for potential scattering of charged quantum particles (such as nuclei) has been developed. It formally extends the application of Tannor and Weeks's (1993) formulation of the scattering matrix theory to systems containing the long-range Coulomb potential, by utilising Dollard's (1964) Møller operators. Using the asymptotic localisation of wave packets, analytical theorems have been formulated that allow convenient numerical application of this method. Numerical tests illustrating the theory mark the first rigorous application of this method in nuclear physics. Further insights are gained through a novel analysis of the wave packet time-correlation function (which underlies the theory) to understand how different dynamical processes, such as the formation of long-lived, quasi-bound states, during scattering contribute to the scattering matrix. This provides guidance on using the time-correlation function to obtain fusion observables, along with the associated challenges. Furthermore, this analysis showed that time-correlation functions can be obtained from static calculations, opening numerous new possibilities for studying the dynamics of nuclear reactions, in general, using this approach. This work opens new avenues for studying the dynamics of compound nucleus formation and other dissipative processes in nuclear reactions.
Because the de Broglie wavelength of ultracold molecules is very large, the cross sections for collisions of molecules at ultracold temperatures are always computed by the time-independent quantum scattering approach. Here, we report the first accurate time-dependent wave packet dynamics calculation for reactive scattering of ultracold molecules. Wave packet dynamics calculations can be applied to molecular systems with more dimensions and provide real-time information on the process of bond rearrangement and/or energy exchange in molecular collisions. Our work thus makes possible the extension of rigorous quantum calculations of ultracold reaction properties to polyatomic molecules and adds a new powerful tool for the study of ultracold chemistry.
Collider experiments play a central role in understanding the subatomic structure of matter, as well as developing and verifying the fundamental theory of elementary particle interactions. However, comprehending scattering processes at a fundamental level in theory remains a significant challenge. The necessary involved time evolution and the with time rapidly increasing bond dimension in Tensor Networks make simulating the scattering process with this classical method challenging. On the other hand, quantum computers hold great promise to efficiently simulate real-time dynamics of lattice field theories. In this work, we take the first step in this direction toward simulating fermionic scattering using a digital-quantum computing approach. Specifically, we propose a method based on Givens rotation to prepare the initial state of the fermionic scattering process, which consists of two fermionic wave packets with opposite momenta. With a time evolution operator based on the underlying Hamiltonian acting on the initial state, the two fermionic wave packets propagate and interact with each other. Using the lattice Thirring model as the test bed and the Qiskit Statevector simulator, we observe an elastic scattering between fermions and antifermions in the strong interaction region. In addition, we clearly observe an entanglement production in the scattering process. We consider our work also as an indispensable step towards a quantum simulation of a scattering process on a real quantum device.
We present a comprehensive end-to-end framework for simulating the real-time dynamics of chemical systems on a fault-tolerant quantum computer, incorporating both electronic and nuclear quantum degrees of freedom. An all-particle simulation is nominally efficient on a quantum computer, but practically infeasible. Hence, central to our approach is the construction of a first-quantized plane-wave algorithm making use of pseudoions. The latter consolidate chemically inactive electrons and the nucleus into a single effective dynamical ionic entity, extending the well-established concept of pseudopotentials in quantum chemistry to a two-body interaction. We explicitly describe efficient quantum circuits for initial state preparation across all degrees of freedom, as well as for block-encoding the Hamiltonian describing interacting pseudoions and chemically active electrons, by leveraging recent advances in quantum rejection sampling to optimize the implementations. To extract useful chemical information, we first design molecular fingerprints by combining density-functional calculations with machine learning techniques, and subsequently validate them through surrogate classical molecular dynamics simulations. These fingerprints are then coherently encoded on a quantum computer for efficient molecular identification via amplitude estimation. We provide an extensive analysis of the cost of running the algorithm on a fault-tolerant quantum computer for several chemically interesting systems. As an illustration, simulating the interaction between NH3 and BF3 (a 40-particle system) requires 808 logical qubits to encode the problem, and approximately 1011 Toffoli gates per femtosecond of time evolution. Our results establish a foundation for further quantum algorithm development targeting chemical and material dynamics.