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arXiv:2609.34298

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

Myeong-Hwan Mun, Jubin Park, Myung-Ki Cheoun, Eunja Ha

nucl-thquant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

We reconstruct elastic $s$-wave neutron-proton ($np$) phase shifts from trapped spectra measured on IBM Aachen using a compressed RODEO circuit. For a schematic square-well interaction represented by classically constructed four-dimensional effective Hamiltonians, four positive-energy levels at five trap strengths supply twenty inputs to classical modified effective range expansion (MERE) extrapolation to free space. The six-cycle factorized geometric-time RODEO implementation (FG-R6) combines ancilla reuse, exact query-phase separation, post-compilation binding, and six numerically optimized geometric evolution times. Numerical cycle-count tests support this choice within the adopted local spectral tolerances. At the same three-qubit width and equal shot budgets, FG-R6 reduces median compiled depth and two-qubit-gate count by 33.1% and 37.2% relative to ten-cycle dynamic direct RODEO (direct R10), increases the median fitted amplitude, and lowers median finite-shot energy uncertainty from 0.610 to 0.534 keV. For this dataset's primary fit on the 0.1-30.0 MeV grid, the maximum central phase-shift deviation from exact-energy MERE decreases from $0.842^\circ$ to $0.163^\circ$, despite a larger root-mean-square (RMS) trapped-energy deviation. The full-grid gain arises mainly in the low-energy extrapolation region; direct R10 has slightly smaller residuals relative to the same reference on the 10.0-30.0 MeV common energy-interpolation subset. The FG-R6 central curve differs from the analytical square-well solution by at most $0.156^\circ$. Fit-form sensitivity remains appreciable and is assessed separately from finite-shot uncertainty. This reduced-space benchmark connects RODEO circuit compression to nuclear continuum observables and shows why circuit performance must be assessed through the scattering reconstruction and its energy range, not spectral RMS errors alone.

Comments
17 pages, 11 figures, 15 tables

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We propose a hybrid quantum-classical framework to solve the elastic-scattering phase shift of two well-bound nuclei in an uncoupled channel. Within this framework, we develop a many-body formalism in which the continuum scattering states of the two colliding nuclei are regulated by a weak external harmonic-oscillator potential with varying strength. Based on our formalism, we propose an approach to compute the eigenenergies of the low-lying scattering states of the relative motion of the colliding nuclei as a function of the oscillator strength of the confining potential. Utilizing the modified effective range expansion, we extrapolate the elastic-scattering phase shift of the colliding nuclei from these eigenenergies to the limit when the external potential vanishes. In our hybrid approach, we leverage the advantage of quantum computing to solve for these eigenenergies from a set of many-nucleon Hamiltonian eigenvalue problems. These eigenenergies are inputs to classical computers to obtain the phase shift. We demonstrate our framework with two simple problems, where we implement the rodeo algorithm to solve the relevant eigenenergies with the IBM Qiskit quantum simulator. The results of both the spectra and the elastic-scattering phase shifts agree well with other theoretical results.
A smaller statistical error bar need not mean a more accurate molecular energy. We evaluate noiselessly optimized molecular variational quantum eigensolver states on IBM Aachen and Pittsburgh, two 156-qubit Heron processors (provider-reported revision 3). Only energy evaluation runs on hardware, with parameters, layouts and compiled circuits frozen and dynamical decoupling disabled. A five-molecule pilot, three-molecule isolated cohort in three windows and known-state controls comprise 110 jobs and 28.2 billed quantum processing unit (QPU) minutes; the isolated cohort spans 30.2 h within a 31.7 h campaign. Raw 1024 O2 and BeH2 mean residuals exceed returned uncertainty by factors 12-32. Pittsburgh BeH2 cross-job standard deviation is 5.6 times mean returned uncertainty. The zero-noise extrapolation (ZNE) package reduces O2 absolute residuals by 49.7% on Aachen and 83.5% on Pittsburgh at about fourfold charge. On Pittsburgh, twirled readout error extinction (TREX) removes 80 millihartree (mHa) per Raw 1024 job charge versus 50 mHa for ZNE. Controls reveal qubit-124 preparation or readout errors 5.4-5.6 times calibration; 13 mitigated energy point estimates fall outside the mapped spectrum, with no Raw counterpart. These results support workload checks and repeated measurements before additional sampling.
Hardware noise and finite sampling perturb the fidelity estimates forming a quantum-kernel Gram matrix. We measured how far three hardware-reconstructed Gram matrices depart from an exact statevector reference for one frozen four-qubit ZZ feature map on N=24 indoor air-quality windows, executed on ibm_fez at 1024 shots per circuit in three single, non-interleaved jobs: baseline, dynamical decoupling alone, and gate twirling alone. All were complete, finite, and positive-semidefinite. Off-diagonal root-mean-squared error (RMSE) against the reference was 0.0878, 0.0864, and 0.0427; full-matrix centered kernel alignment (CKA) ranged 0.933-0.989 and the post hoc diagonal-excluded (U-centered) CKA 0.816-0.986. The gate-twirled job deviated least on every reported geometry axis; its baseline contrasts are deletion-stable for the Spearman, mean-absolute-error, RMSE, and full-matrix CKA diagnostics, while the Pearson and diagonal-excluded contrasts fall just below that convention. Dynamical decoupling was not separated from the baseline. The observed error exceeded both finite-shot reference scales, so, under those sampling-only models, sampling does not explain it. Centered kernel-target alignment did not track reconstruction fidelity and stayed at or below each label-permutation reference: implementation fidelity and task relevance are distinct diagnostic axes. All configuration-level statements describe three realized jobs on one backend; no mitigation-efficacy, classifier-superiority, forecasting, or quantum-advantage claim is made.
We benchmark two error-mitigation strategies for digital quantum simulation of a long-range interacting fermionic model on IBM Heron r2 superconducting processors. The testbed is a minimal exactly solvable system—two spinless electrons on a 6-site ring with repulsive 1/r{\text{1/}}r1/r chord-distance Coulomb interactions—whose ground state exhibits a smooth Wigner-dimer localization crossover quantified by the antipodal occupation probability P3(U){P_3}(U)P3​(U) . Circuit parameters of a particle-number-conserving variational ansatz are optimized classically; the quantum processors execute only final energy and configuration measurements. We compare (i) zero-noise extrapolation (ZNE) applied to the full Hamiltonian with (ii) a hybrid estimator in which the off-diagonal kinetic energy is measured with ZNE while the diagonal interaction energy is reconstructed from particle-number post-selected bitstrings. Across eight interaction strengths U/t∈[5,75]U{\text{/}}t \in [5,75]U/t∈[5,75] , five measurement sessions on two devices (ibm_fez, ibm_marrakesh), and three circuit depths, the hybrid estimator achieves, in the interacting regime U/t⩾15U{\text{/}}t \geqslant 15U/t⩾15 , relative ground-state energy errors of 4.3–14% that decrease systematically with UUU —an order of magnitude better than full-Hamiltonian ZNE (29–85%)—and reproduces the localization crossover in P3(U){P_3}(U)P3​(U) . At U/t=5U{\text{/}}t = 5U/t=5 , where E0≈0.08t{E_0} \approx 0.08{\kern 1pt} tE0​≈0.08t renders the relative error ill-conditioned, the absolute hybrid error is 0.62t0.62{\kern 1pt} t0.62t , in line with the rest of the scan. A depth scan at fixed U/t=45U{\text{/}}t = 45U/t=45 shows that the shallowest circuit wins despite a 1.8% ansatz expressivity penalty, quantifying the expressivity–noise trade-off. Shot-noise uncertainties, recovered exactly from per-noise-factor data, are below one percentage point in relative energy error for the hybrid estimator (versus 7–14 percentage points for ZNE at equal shot budget) and an order of magnitude smaller than the measured inter-session calibration spread. The exact variational bound Eexp⩾Etheory{E_{{\text{exp}}}} \geqslant {E_{{\text{theory}}}}Eexp​⩾Etheory​ is satisfied in all 27 full-Hamiltonian hardware energy measurements. All data were acquired with under nine minutes of QPU time on the free IBM Quantum Open Plan.
Qudits, the multi-level generalization of qubits, provide a natural extension of the binary paradigm in quantum computation and offer new opportunities to enhance algorithmic performance. Beyond their direct applicability to the simulation of multi-level quantum systems, higher-dimensional ancillae can improve sampling efficiency in quantum algorithms by enabling the simultaneous implementation of multiple control operations, thereby reducing circuit complexity. In this work, we pursue three main objectives. First, we present a formulation of the Rodeo algorithm employing a general ddd-level ancilla qudit. Second, we introduce the concept of the \emph{Rodeo kernel}, defined as a two-frequency interferometer, which acts as a spectral filter in the energy domain. Finally, we propose a microcanonical protocol for the Rodeo algorithm. This protocol enables the estimation of entropic quantities through a single energy sweep and admits a natural interpretation as a Gaussian convolution of the density of states. To support the theoretical analysis, we perform numerical evaluations of the corresponding quantum circuit using ancilla qudits of dimensions three, four, and five. The simulations are performed for the one-dimensional Ising model, considering both spin-$\sfrac{1}{2}$ and spin-111 particles. The ancilla qutrit implementation exhibits an 18%18\%18% reduction in fluctuations compared to the qubit implementation. Our results show that qudits provide a framework for spectral analysis and thermodynamic characterization of multi-level quantum systems.