A time-dependent wave-packet approach to reactions for quantum computation

Abstract

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.

Figures (2)

• Figure 1: Elastic two-body scattering in two spatial dimensions by an attractive Gaussian potential [see Eq. \ref{['eq:test_potential']}]. (a) Overlap function $\mathcal{C}(t)\equiv \mathcal{C}(\mathbf{k}_0',\mathbf{k}_0;t)$ for forward scattering, $\mathbf{k}_0'=\mathbf{k}_0=(0,k_{0,y})$. Solid black and dashed red lines denote, respectively, the real and imaginary parts of the overlap function. (b) Total cross section $\lambda$ as a function of energy, obtained from the forward-scattering amplitude via the optical theorem. The red shaded region shows the relative energy profile of the wave packets (arbitrary normalization). (c) Relative error $|\Delta\lambda|/\lambda$ between the wave-packet and variable-phase results. (d) Differential cross section $d\lambda/d\theta$ at three different scattering energies, $E=5,10,$ and $20$ MeV. In panels (b) and (d), solid lines denote the exact numerical result obtained via the variable phase method alma991072475789706532MARTINAZZO2003187 (adapted to two spatial dimensions); data points were obtained via the time-dependent wave-packet formalism.
• Figure 2: Two-channel scattering cross sections: (a) Elastic (blue) and inelastic (green) differential cross sections evaluated in the forward limit $\theta=0$. (b) Total (elastic + inelastic) integrated cross section $\lambda_\mathrm{tot}$ obtained via the optical theorem. Solid lines were calculated via the variable-phase method; points correspond to the time-dependent wave-packet formalism. Vertical dashed line denotes the excited state threshold energy, $\Delta=13$ MeV.