Time-Dependent Wave-Packet Approach to Reactions for Quantum Computation

The gist: This method describes a time-dependent wave-packet approach to reactions for quantum computation, which allows for obtaining the scattering matrix for nuclear or chemical reactions on a finite lattice by utilizing unitary time evolution.

Core Methodology

The central quantity in this approach is defined as a time-dependent overlap between incoming and outgoing wave packets whose Fourier transform corresponds to the scattering matrix at fixed energy (Page 1). This technique is ideal for quantum hardware because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). The method involves preparing initial and final states as wave packets and then performing the unitary time evolution between them (Page 7).

Formalism and State Preparation

The formalism is based on the "time-dependent scattering formalism of Tannor & Weeks [22]" (Page 1). The total system Hamiltonian is given by a sum involving kinetic energy operators, internal Hamiltonians, and intercluster interaction terms (Page 1). The system state is prepared as a product of internal cluster states and a relative Gaussian wave packet: Φk0,α⟩ ≡ ψ(1)α1⟩ ψ(2)α2⟩ ϕrel(k0, r) (Page 2). To reduce complexity, the approach can factorize the preparation between clusters by using separate wave packets for each cluster and then projecting onto a zero CM eigenstate to isolate the desired relative momentum eigenstate (Page 8).

Extracting Cross Sections

The scattering matrix S is related to the dimensionless overlap function Cβ,α(k′0, k0;t) (Page 11). If this overlap function is sampled at sufficiently many times, a subsequent Fourier transform is trivial to perform classically (Page 12). The real and imaginary parts of the overlap function can then be extracted from modified Hadamard tests [23] (Page 5). The scattering amplitude fβ,α(E, θ) can be obtained by relating the S matrix to this overlap function through Fourier transform and the two-dimensional analog of Eq. (12) (Page 10).

Numerical Implementation and Results

The method is demonstrated with a series of numerical examples, for both elastic and inelastic processes, comparing against exact calculations (Page 1). For elastic scattering in two dimensions via a Gaussian potential, the total cross section λ is obtained from the forward-scattering amplitude via the optical theorem: λ = √8π/k Imf(E, 0) (Page 11). The approach shows that the resulting cross section is large enough to be distinguished from the noise because extended plane waves automatically include contributions from all impact parameters (Page 3). Inelastic scattering is handled by extending the formalism to a two-channel problem using a Hamiltonian involving coupling terms, allowing for the calculation of elastic and inelastic differential cross sections (Page 5).

General Applicability

The approach is general because it is agnostic to the nature of the interaction, as long as it is short ranged (Page 11). It can be extended to simulate nuclear and chemical reactions where both target and projectile can be composite systems (Page 1). The use of a finite lattice provides scattering information across the entire range of energies where wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The formalism also allows for the simulation of charged-particle reactions and photo-induced reactions by extending it to include Coulomb interactions (Page 11).

Summary

This paper describes a time-dependent wave-packet approach to scattering that is ideal for quantum computation in the first quantization mapping. The approach, however, is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support (Page 6). The method is ideal for simulations on quantum hardware because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1). This formalism is general—both the target and the projectile can be composite systems, opening the way for simulation of nuclear and chemical reactions (Page 1). The use of wave packets localized to a finite lattice provides scattering information across the entire range of energies where the wave packets have appreciable support, with the spatial extent of the lattice being increased to simulate lower energies (Page 6). The method is ideal for quantum computation because it requires only unitary time evolution, making this approach ideal for simulations on quantum hardware (Page 1).

1. A time-dependent wave-packet approach to reactions for quantum computation allows AI systems to calculate scattering cross sections by preparing initial and final states as wave packets and performing unitary time evolution, making it ideal for simulations on quantum hardware. This enables the calculation of elastic and inelastic cross sections for reactions involving two interacting clusters.

2. The method can be adapted to describe reactions where constituents transition between internal eigenstates, enabling the calculation of inelastic scattering amplitudes by preparing states such that the initial and final states are labeled by quantum numbers α = (α1, α2) and β = (β1, β2), respectively.

3. AI systems can probe scattering through different angles by imparting transverse momentum to the final-state wave packets, allowing for the calculation of differential cross sections dλ/dθ at various energies.

4. The method provides a way to obtain total and differential scattering cross sections for 2 → 2 scattering, including inelastic processes, which can be used to model nuclear or chemical reactions involving composite systems.

5. By utilizing the overlap function Cβ,α(k'0, k0;t), AI can compute the S matrix by sampling it at sufficiently many times (set by the momentum/energy resolution of the lattice), and then performing a Fourier transform that determines the scattering matrix.