Calculating scattering phase shifts requires computing up to, a scaling challenge that has long limited accurate modeling of particle interactions. Researchers have now demonstrated a new method for simulating quantum scattering using wave packets localized within a discrete lattice volume, building upon the time-dependent scattering formalism of Tannor and Weeks. This approach allows for the computation of total and differential cross sections across a range of energies, and overcomes limitations of current quantum algorithms that struggle with reactions beyond simple elastic scattering. The work directly links a first-quantized lattice description to quantum hardware, opening a path to simulating complex reactions including those with internal particle structure.
Lattice Discretization and Challenges in Quantum Scattering
The spatial extent of a lattice directly impacts the simulation of low-energy quantum scattering, as a larger lattice is required to accurately model the asymptotic form of scattering wave functions. This work demonstrates a method for simulating scattering using wave packets localized on a finite lattice, enabling the extraction of scattering information across a broad energy range where the wave packets maintain significant support. Researchers found that a 64×64 spatial lattice, with a constant lattice spacing, was sufficient for modeling two-nucleon interactions governed by a Gaussian potential.
The formalism employed is compatible with a first-quantized, lattice-based description suitable for implementation on quantum hardware; each particle’s position and quantum numbers are encoded using qubits. This approach bypasses limitations of current quantum algorithms, which struggle to generalize beyond elastic scattering to more complex reactions like capture or breakup, or scattering involving composite systems with internal structure.
By representing particles directly on a lattice, the method offers a pathway to simulate a wider range of nuclear reactions. The study successfully computed total and differential cross sections by using a Fourier transform to isolate specific energy components of the S matrix. Choosing appropriate asymptotic times for the wave packets and lattice parameters is important for accurate results, as these factors determine the simulation’s evolution time and the probing of the overlap function.
The overlap function, a central quantity in the simulation, exhibits exponential damping at longer times for fast reactions, but can display long-lived behavior in scenarios involving resonances or compound states, necessitating a larger lattice for resolution. The accuracy of the computed cross sections was demonstrated with relative errors typically less than 1% within a specific energy range of 5 to 40 MeV.
The differential cross section, dλ/dθ, was calculated at scattering energies of 10, 20, and 30 MeV, showing the method’s ability to resolve angular distributions. Distinguishing between different scattering channels is also possible; preparing wave packets in specific initial and final states allows for the computation of both elastic (S_0,0) and inelastic (S_1,0) scattering matrices and their corresponding differential cross sections. This method extends beyond simple two-body scattering, as the researchers also considered a system where initial and final wave packets were prepared in different quantum number channels.
The ability to model both elastic and inelastic scattering, alongside the method’s compatibility with quantum hardware, represents a step toward simulating complex nuclear reactions. The limits of integration within the calculations can be adjusted to optimize the simulation based on the scattering problem, wave packets and lattice size, offering further control over the process. The work demonstrates that the chosen wave packets have appreciable support for scattering energies between 5 and 40 MeV, validating the method’s range of applicability.
Time-Dependent Wave Packets for Cross Section Calculation
The computed correlation function incorporates contributions from all incident energies present within the initial and final wave packets, resulting in a signal strong enough for reliable detection above background noise. Extending plane waves in the x and y dimensions automatically includes contributions to the cross section from all impact parameters, further amplifying the signal strength and improving measurement precision. Propagation of these initial and final wave packets is limited by computational feasibility, requiring careful consideration of asymptotic time choices and lattice parameters for accurate results, as detailed in supplementary material.
Forward scattering, defined by a specific angular relationship, provides a key benchmark for validating the method’s accuracy. After evolving the Møller wave packets, and, researchers obtain the overlap function, a central component in determining scattering behavior.
Performing a Fourier transform on this overlap function allows for the calculation of the forward-scattering matrix S(E,0) at any energy E where the wave packets exhibit significant support, effectively mapping the scattering process across a range of energies. This matrix is then related to the forward-scattering amplitude f(E,0) through a two-dimensional analog of a known equation, enabling the determination of the total cross section using the optical theorem.
The method’s accuracy was demonstrated using an attractive Gaussian potential in two spatial dimensions, where the real and imaginary parts of the overlap function were calculated for forward scattering. This range is significant because the chosen wave packets maintain appreciable support within it, ensuring reliable calculations.
By preparing both in the zero-threshold channel, the elastic scattering matrix and the corresponding differential cross section can be obtained. Conversely, initiating the wave packet in the zero-threshold channel and finalizing it in a finite-threshold channel yields the inelastic scattering matrix and the inelastic differential cross section .
Analysis of the forward limit reveals expected behavior; the inelastic cross section vanishes at the threshold energy. The total integrated cross section, derived from the forward-scattering amplitude via the optical theorem, also aligns with previous results, demonstrating the wave-packet formalism’s consistent performance when wave packets have appreciable support.
First-Quantization Mapping to Qubit Encoding
The method detailed in this work builds upon the time-dependent scattering formalism initially proposed by Tannor and Weeks, adapting it for implementation on quantum computing hardware. This approach diverges from conventional methods by directly linking a first-quantized lattice description of scattering constituents to qubit encoding, a strategy that circumvents the computational scaling challenges associated with increasing particle numbers and quantum states.
The researchers developed a framework where each particle’s position within the lattice, alongside its quantum numbers like spin and isospin, is represented by a specific number of qubits, enabling a direct mapping of the scattering problem onto a quantum computer. This encoding scheme allows for the calculation of both total and differential cross sections, a capability not easily generalized in existing quantum algorithms.
Current quantum algorithms often struggle with reactions extending beyond purely elastic scattering, failing to efficiently model scenarios like capture or breakup reactions, or scattering between composite systems possessing internal structure. The team’s formalism addresses this limitation by explicitly incorporating the possibility of transitions between internal eigenstates of the scattering constituents, expanding the scope of simulatable reactions.
The method relies on propagating wave packets localized within the discrete lattice volume, allowing for information about the reaction process to be gathered across a range of scattering energies through a single simulation. Extracting accurate scattering data requires computing phase shifts in partial waves up to a limit defined by, where k represents the scattering momentum and is the range of the potential.
A key aspect of their approach is the use of absolute Cartesian coordinates, simplifying the mapping to first-quantized encoding on quantum hardware, in contrast to the Jacobi coordinates employed in the original Tannor and Weeks proposal. By introducing momentum , along the transverse directions, we are able to probe scattering through different angles, a refinement of earlier methods. These tests require O(ε^-2) measurements, where ε denotes the targeted error, and the number of time samples needed depends on the system and the simulation box size.
While propagating wave packets to infinite time is impractical, the team found that evolving them to a finite, asymptotic time ∞, sufficiently large to suppress interactions but small enough to avoid boundary effects, provides an accurate approximation. This allows for the replacement of infinite time limits in the Møller operators with this finite value, streamlining the computational process. The team acknowledges that the current work focuses on short-ranged interactions characteristic of the strong nuclear force, but notes that extending the S-matrix scattering formalism to include Coulomb interactions is feasible and key for simulating charged-particle reactions.
Momentum Eigenstates Enable Angular Scattering Analysis
By manipulating the momentum of wave packets, researchers have enabled detailed analysis of angular scattering, a capability previously limited by computational demands. This method bypasses the need to calculate up to, a scaling challenge inherent in traditional scattering simulations, by using the inherent momentum distribution within localized wave packets.
Each wave packet, containing a mixture of different momentum eigenstates, provides a comprehensive view of the reaction process across a spectrum of scattering energies, streamlining the computational process. This ability to examine transitions between states of definite relative momentum, normalized with respect to energy, is important for accurately modeling complex interactions.
The asymptotic scattering state is obtained via energy projection of the Møller-evolved wave packet, using quantities k, , and E that are intrinsically related. Numerical examples, restricted to two spatial dimensions due to computational complexity, demonstrate the method’s accuracy. The total cross section is computed at evenly spaced scattering energies, even those that do not align with lattice eigenenergies, showing the method’s versatility. While the initial demonstrations use a central potential, the framework extends to interactions allowing for the exchange of angular momentum or other quantum numbers between projectile and target.
This adaptability is further demonstrated through the extension of the model to a two-channel scattering problem, incorporating a Hamiltonian with interactions designed to avoid Feshbach resonances. The researchers emphasize that this approach is not limited to simple two-particle interactions. Both the target and projectile can be composite systems, expanding the scope of simulations. A detailed error analysis of the formalism is available in the Supplemental Material, providing a comprehensive assessment of its accuracy and limitations.
The computational challenge of accurately modeling quantum scattering stems from the scaling of calculations with momentum and potential range; specifically, computing scattering phase shifts requires calculations up to, a limit that demands increasing computational resources as the system’s complexity grows. The core of this approach lies in the ability to represent quantum states using qubits, encoding particle positions within a spatial lattice alongside intrinsic quantum numbers such as spin and isospin.
This encoding allows for the direct translation of the scattering problem into a series of quantum operations, streamlining the computational process. Unlike prior quantum algorithms limited to elastic scattering, this formalism extends to inelastic processes where the scattering constituents can transition between internal eigenstates. The method’s adaptability is demonstrated through the ability to model systems beyond simple two-particle interactions, accommodating both the target and projectile as composite entities with internal structure.
A key innovation is the introduction of momentum along transverse directions, x and y, to the propagating wave packet. By imparting momentum in these directions, the simulation can probe scattering through different angles, providing a more complete picture of the interaction. This contrasts with earlier implementations that restricted momentum to the direction of propagation, limiting the angular resolution of the results. The researchers use Møller operators to define asymptotic scattering states, projecting the energy of the Møller-evolved wave packet to obtain the S matrix, which encapsulates the transition amplitude between these states.
The S matrix is then used to calculate cross sections, providing a measure of the probability of a particular scattering event occurring. The method relies on calculating the overlap function between initial and final wave packets, a quantity that can be efficiently obtained through Hadamard tests on a quantum computer.
These tests require a number of measurements scaling with the desired error tolerance, but the signal strength is enhanced by the extended nature of the wave packets, which incorporate contributions from all impact parameters. The formalism also allows for the treatment of systems where the initial and final states differ in their internal quantum numbers, expanding the range of phenomena that can be investigated.
The ability to accurately simulate scattering processes across a range of energies and angles represents a step toward using quantum computers for nuclear physics and beyond, offering a new tool for understanding the fundamental forces that govern matter.
👉 More information
🗞 Time-Dependent Wave-Packet Approach to Reactions for Quantum Computation
✍️ Evan Rule and Ionel Stetcu
🧠 DOI: http://link.aps.org/doi/10.1103/kwgf-jdbn
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