Nuclear scattering phase shifts with factorized geometric-time RODEO on a quantum processor

Technical Summary: Nuclear Scattering Phase Shifts with Factorized Geometric-Time RODEO on a Quantum Processor

Problem Statement
Reconstructing nuclear scattering phase shifts from quantum-computed spectra presents a significant challenge. While quantum eigensolvers can estimate discrete energy levels in a confined system (a "trap"), connecting these discrete states to continuum observables (free-space scattering) requires classical post-processing, specifically the Modified Effective Range Expansion (MERE). The performance of a quantum circuit in this context cannot be judged solely by the root-mean-square (RMS) error of individual energy levels. Spectral errors can be amplified, suppressed, or reorganized during the subsequent energy interpolation and zero-confinement extrapolation. Therefore, a resource-reduced quantum circuit must be benchmarked by its ability to preserve the information necessary for the final scattering reconstruction, rather than by spectral fidelity alone.

Methodology
The authors reconstruct elastic s-wave neutron–proton (np) phase shifts using data from the IBM Aachen quantum processor. The study compares two implementations of the RODEO (Repetitive Operation for Discrete Energy Optimization) algorithm:

1. Dynamic Direct R10: A baseline implementation using ten cycles with fixed Gaussian-quantile time steps. For each query energy, the controlled evolution is re-synthesized and compiled separately.
2. Factorized Geometric-Time RODEO (FG–R6): A compressed implementation using six cycles with geometrically sampled time steps. This method employs exact query-phase separation, where the Hamiltonian evolution is fixed to a reference energy ($E_{ref}$) and compiled once. The query-dependent phase ($\Delta E = E_q - E_{ref}$) is bound to the ancilla qubit after compilation.

Both methods utilize a three-qubit register (two system qubits and one recycled ancilla) on the IBM Aachen backend. The physical model is a schematic s-wave np square-well interaction. Classically constructed four-dimensional Okubo–Lee–Suzuki (OLS) effective Hamiltonians are used to represent the system. The experiment measures four positive-energy levels at five different trap strengths ($\omega = 3.6$ to $4.0$ MeV), yielding twenty spectral inputs. These inputs are fed into a classical MERE analysis to extrapolate to free space and calculate phase shifts over a 0.1–30.0 MeV energy grid.

Key Contributions

• Circuit Compression via Factorization: The paper introduces the FG–R6 circuit, which combines ancilla reuse with a factorized time-evolution strategy. By separating the query phase from the Hamiltonian evolution, the circuit reduces the number of controlled-evolution cycles from ten to six and eliminates the need to recompile the evolution operator for every query energy.
• Hardware Benchmarking: The study provides a complete hardware-derived benchmark for nuclear scattering, moving beyond single-level tests to a full set of twenty spectral inputs required for free-space reconstruction.
• Resource vs. Accuracy Trade-off Analysis: The work demonstrates that reducing circuit depth and two-qubit gate counts does not necessarily degrade the accuracy of the final scattering observable, even if the RMS error of the intermediate energy levels increases.

Results

• Resource Reduction: Relative to the Dynamic Direct R10 implementation, the FG–R6 implementation reduces the median compiled circuit depth by 33.1% (from 459 to 307) and the two-qubit gate count by 37.2% (from 113 to 71).
• Energy Uncertainty: FG–R6 achieves a smaller median finite-shot energy uncertainty (0.534 keV) compared to R10 (0.610 keV), despite having a larger median fitted amplitude and a larger RMS center offset ($\Delta E_c$) relative to the reference energy (1.648 keV vs. 0.793 keV).
• Scattering Phase Shift Accuracy: On the full 0.1–30.0 MeV grid, the maximum deviation of the reconstructed phase shift from the exact-energy MERE reference decreases significantly from 0.842° (R10) to 0.163° (FG–R6). The maximum deviation from the analytical square-well solution also drops from 0.981° to 0.156°.
• Energy-Dependent Performance: The improvement is driven primarily by the low-energy extrapolation region (below 10 MeV). In the 10.0–30.0 MeV common interpolation subset, the Direct R10 implementation actually yields slightly smaller residuals than FG–R6.
• Fit-Form Sensitivity: The study notes that fit-form sensitivity remains appreciable. The spread in results across different polynomial orders for the energy and trap fits is substantial (e.g., a half-range of 20.12° on the full grid for R10 vs. 6.75° for FG–R6), indicating that the choice of extrapolation model significantly impacts the final result.

Significance and Claims
The paper claims that this reduced-space benchmark successfully connects RODEO circuit compression to nuclear continuum observables. The primary significance lies in demonstrating that circuit performance must be assessed through the reconstruction of the scattering observable and its specific energy range, rather than by spectral RMS errors alone. The FG–R6 implementation is presented as a validated accuracy–resource compromise for the specific spectrum and window tested, rather than a universal minimum-cycle solution.

The authors explicitly state that these results do not demonstrate quantum advantage or many-body scalability. The hardware data do not separately determine the effects of phase compilation versus time sampling, nor do they establish independent-run reproducibility. The work serves as a starting point for extending these methods to larger model spaces, realistic interactions, and few-body reaction dynamics, such as resonant higher-partial-wave scattering (e.g., neutron–alpha scattering) and systems involving long-range Coulomb interactions.