Quantum-inspired Bayesian probability algorithm for nuclear mass predictions

TL;DR

The paper addresses the challenge of accurate nuclear mass predictions by augmenting theoretical models with a quantum-inspired Bayesian probability (QIBP) framework. It maps mass residuals $δ$ into Hilbert-space wave functions, derives Schrödinger potentials, and constructs Boltzmann-based priors and likelihoods to obtain a Bayesian posterior $p(δ|Z_t,N_t)$ for refined mass residuals. Across WS4 and HFB models, QIBP yields substantial reductions in mass residuals and demonstrates robust extrapolation to unknown nuclei, as well as improved $Q_α$ predictions and clearer/accountable shell effects. This work demonstrates the feasibility and value of quantum machine learning approaches in nuclear physics and points to potential extensions to nuclear reactions and astrophysical processes.

Abstract

In this study, a novel quantum-inspired Bayesian probability (QIBP) algorithm, informed by quantum dynamics, is proposed to improve the predictions of nuclear mass from theoretical models. Within the QIBP framework, residuals between the theoretical and experimental mass values are mapped into wave functions in Hilbert space. The corresponding potentials are obtained by solving the Schrödinger equation. Assuming that the residuals follow a Boltzmann distribution, the prior and likelihood probability density functions (PDFs) can be obtained from potentials. Finally, the Bayesian theorem is applied to derive the posterior PDF for estimating the target nuclear mass residuals. Global optimization and extrapolation analyses indicate that the QIBP algorithm effectively captures quantum effects and subtle patterns, which are not fully incorporated into theoretical models, thereby providing reliable predictions. In addition, the extrapolation based on the synthetic experimental set further evaluates the performance and applicability of the QIBP algorithm across the entire nuclear chart. Furthermore, the QIBP algorithm is applied to predict $α$-decay energies of Ra and Es isotopes, and the shell effects manifested in these isotopes are analyzed. This study validates the feasibility of quantum machine learning in nuclear mass research, and demonstrates that the proposed algorithm can accurately describe nuclear masses, with potential applications in other areas of nuclear physics.

Figures (5)

• Figure 1: (a) The raw nuclear mass residuals $\delta$ (MeV) from the WS4 model. (b) The nuclear mass residuals from the WS4 model after refinement by the QIBP algorithm. (c) The same as (a), but for the HFB model. (d) The same as (b), but for the HFB model after refinement by the QIBP algorithm.
• Figure 2: (a) Prior PDFs for Pb isotopes with $98 \le N \le 102$, obtained by the CBP estimator. (b) Prior PDFs for the same nuclei, calculated using the QIBP algorithm.
• Figure 3: One-neutron separation energies $S_{\mathrm{n}}$ (MeV) for Pb isotopes. Brown circles denote experimental data, blue diamonds with error bars represent the $S_{\mathrm{n}}$ values and corresponding uncertainties obtained using the QIBP algorithm, and red circles and brown diamonds correspond to the results from the CBP estimator and the WS4 model, respectively.
• Figure 4: (a) Improvement ratio $\Delta\delta/\delta^{\mathrm{pre}}$ for each nucleus in the validation set obtained by the QIBP algorithm, with the values mapped to colors according to the color bar on the right. The black crosses denote nuclei with $\Delta\delta/\delta^{\mathrm{pre}} \ge 80\%$. The “synthetic experimental set” corresponds to nuclear masses predicted by the WS4 model, while the theoretical model to be optimized is the HFB model. The learning set includes 2253 nuclei from AME2020 with $Z \ge 20$ and $N \ge 20$, which are marked by khaki circles. The validation set comprises 2297 nuclei within the drip line determined by the HFB model, covering $20 \le Z \le 110$ and $20 \le N \le 160$. (b) Same as (a), but for the CBP estimator.
• Figure 5: (a) $\alpha$-decay energies $Q_\alpha$ (MeV) for the Ra isotope chain. The blue sphere represent experimental values, the red curve shows the original predictions from the WS4 model, the blue curve shows the results after optimization by the QIBP algorithm. The blue dotted line denotes the $Q_\alpha$ values obtained by optimizing WS4 with the CBP estimator. (b) The same as (a), but for the Es isotope chain.