Reduced-basis methods provide a powerful framework for building efficient and accurate emulators. Although widely applied in many fields to simplify complex models, reduced-basis methods have only been recently introduced into nuclear physics. In this Letter we build an emulator to study the single-particle structure of atomic nuclei. By scaling a suitable mean-field Hamiltonian, a ``universal'' reduced basis is constructed capable of accurately and efficiently reproduce the entire single-particle spectrum of a variety of nuclei. Indeed, the reduced-basis model reproduces both ground- and excited-state energies as well as the associated wave functions with remarkable accuracy. Our results bode well for more demanding applications that use Bayesian optimization to calibrate nuclear energy density functionals.
Reduced-basis methods provide a powerful framework for building efficient and accurate emulators. Although widely applied in many fields to simplify complex models, reduced-basis methods have only been recently introduced into nuclear physics. In this Letter we build an emulator to study the single-particle structure of atomic nuclei. By scaling a suitable mean-field Hamiltonian, a ``universal'' reduced basis is constructed capable of accurately and efficiently reproduce the entire single-particle spectrum of a variety of nuclei. Indeed, the reduced-basis model reproduces both ground- and excited-state energies as well as the associated wave functions with remarkable accuracy. Our results bode well for more demanding applications that use Bayesian optimization to calibrate nuclear energy density functionals.
Recent advances in both theoretical and computational methods have enabled large-scale, precision calculations of the properties of atomic nuclei. With the growing complexity of modern nuclear theory, however, also comes the need for novel methods to perform systematic studies and quantify the uncertainties of models when confronted with experimental data. This study presents an application of such an approach, the reduced basis method, to substantially lower computational costs by constructing a significantly smaller Hamiltonian subspace informed by previous solutions. Our method shows comparable efficiency and accuracy to other dimensionality reduction techniques on an artificial three-body bound system while providing a richer representation of physical information in its projection and training subspace. This methodological advancement can be applied in other contexts and has the potential to greatly improve our ability to systematically explore theoretical models and thus enhance our understanding of the fundamental properties of nuclear systems.
In recent years, reduced-basis methods (RBMs) have been adapted to the many-body eigenvalue problem and they have been used, largely in nuclear physics, as fast emulators able to bypass expensive direct computations while still providing highly accurate results. This work is meant to show that the RBM is an efficient and accurate emulator for the strong correlations induced by the pairing interaction in a variety of finite systems like ultrasmall superconducting grains, interacting topological superfluids, and mesoscopic hybrid superconductor-semiconductor devices, all of which require an expensive, beyond-mean-field, particle-number conserving description. These systems are modeled by the number-conserving Richardson pairing Hamiltonian and its appropriate generalizations. Their ground state is solved for exactly using the density matrix renormalization group. The reduced basis is assembled iteratively from a small number of exact ground-state vectors, well-chosen from across the relevant parameter space using a fast estimate of the emulation error and a greedy local optimization algorithm. The RBM emulation is found to accurately describe the weak-to-strong pairing crossover in small grains, the third-order topological phase transition of the interacting Richardson-Kitaev chain, and the complex charge stability diagram of a hybrid quantum dot--superconductor device. For all considered systems the emulation error decreases exponentially with the reduced-basis dimensionality. The number of basis vectors necessary to reach a fixed emulation accuracy only shows a modest linear growth with increasing system size. RBMs are confirmed to be cheap and accurate emulators for the widely encountered superconducting phenomena. Capable of providing orders of magnitude computational speedup with respect to approaches based only on traditional many-body solvers, they open new possibilities in building and solving models of interacting many-body systems and in better interfacing them with experimental design and data analysis.
Emulating the Green's function using reduced-order equations is challenging because of the amount of information it contains. This information includes the one-particle spectroscopy over the entire energy spectrum, the one-body density matrix, and the ground-state energy. Nevertheless, the computational speedup offered by emulators is essential for applications requiring repeated model evaluations, which are ubiquitous in many-body physics. These include parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. In this Letter, we introduce a systematically improvable emulator based on eigenvector continuation and moment-matching. The resulting approximation, which we refer to as an ``eigenvector-continued Green's function,'' can be obtained by solving linear equations, enabling an efficient separation of the computational workload into offline and online stages.
The nuclear shell model accurately describes the structure and dynamics of atomic nuclei. However, the exponential scaling of the basis size with the number of degrees of freedom hampers a direct numerical solution for heavy nuclei. In this work, we present a quantum annealing protocol to obtain nuclear ground states. We propose a tailored driver Hamiltonian that preserves a large gap and validate our approach in a dozen nuclei with basis sizes up to 10^5 10 5 using classical simulations of the annealing evolution. We explore the relation between the spectral gap and the total time of the annealing protocol, assessing its accuracy by comparing the fidelity and energy relative error to classical benchmarks. While the nuclear Hamiltonian is non-local and thus challenging to implement in current setups, the estimated computational cost of our annealing protocol on quantum circuits is polynomial in the single-particle basis size, paving the way to study heavier nuclei.
Quantum computing is increasingly offering concrete solutions toward the simulation of nuclear structure, with the potential to overcome the exponential scaling that limits classical diagonalization methods in large spaces. A particularly efficient encoding scheme, based on collective like-nucleon pairing modes, reduces the qubit requirements by half and avoids the non-local operator strings of standard fermion-to-qubit mappings. While this quasiparticle framework provides accurate results for semimagic nuclei, it does not adequately describe open-shell systems where proton-neutron correlations become important. In this work, we apply Brillouin-Wigner perturbation theory to systematically improve the quasiparticle description of open-shell nuclei in the sd shell, reaching an energy relative error below 0.2% compared to the nuclear shell model. Furthermore, to make the effective Hamiltonian suitable for quantum simulation, we introduce a mean-field Hartree-Fock approximation of the non-quasiparticle resolvent, achieving ground-state energies typically within 2% of the exact shell-model result. This represents a systematic improvement over the bare quasiparticle Hamiltonian while remaining within the reach of near-term quantum devices.
We present the reduced basis method as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The method uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connect some of the results in the eigenvector continuation literature to the formalism of reduced basis methods and show how these methods can be applied to a broad set of problems. As we illustrate, the possible success of the formalism on such problems can be diagnosed beforehand by a principal component analysis. We apply the reduced basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for 48Ca, achieving speed-ups of more than $\ifmmode\times\else\texttimes\fi{}150$ and $\ifmmode\times\else\texttimes\fi{}250$, respectively, when compared to traditional solvers. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.