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Nuclear structure problem

Overview

The structure of nuclei can be approximately described using the shell model (see [1] for an overview), a phenomenological model with parameters fitted to experimental observations. However, high accuracy descriptions of nuclear structure, exotic nuclei, accurate scattering cross sections, or non-equilibrium phenomena require a first-principles treatment. Describing the properties of nuclei from first principles (e.g., lattice quantum chromodynamics simulations) is beyond the reach of analytic and current computational capabilities for all but the simplest nuclei [2]. Nevertheless, we can integrate out the short-range physics to obtain effective field theories (EFTs) that describe the interactions of nucleons. The prototypical example is chiral effective field theory, which describes the interactions of nucleons and virtual pions. The parameters of the EFT can be inferred from experiments (in the future it may also be possible to determine the parameters directly from lattice QCD calculations), resulting in a many-body Hamiltonian that describes the formation and potential decay of nuclei.

Actual end-to-end problem(s) solved

The EFT provides a many-body Hamiltonian describing how nucleons interact. Classical techniques to find the eigenstates and eigenenergies of this Hamiltonian include coordinate-space methods (e.g., quantum Monte Carlo methods) as well as projecting onto a basis set and using techniques such as perturbation theory or coupled cluster [3]. In this sense, the problem is similar to the electronic structure problem in quantum chemistry. A common problem is to prepare the ground state of a collection of nucleons, in order to compute nuclear binding energies and determine if a given nucleus is stable (for example, determining the long lifetime of \(^{14}\)C [4, 5]). Simulations can also be used to calculate scattering cross sections, which are used to analyze experiments on nucleus-neutrino scattering [6], beta decay, and nuclear reactions. Reactions such as nuclear fission and nuclear fusion can also be studied using explicitly time-dependent approaches [7], although these have higher computational costs than static calculations. Simulating both fusion and fission reactions has a number of use cases, such as an improved understanding of nuclear astrophysics, where reactions commonly occur at energies too high or too low to be replicated in experiments [8].

Dominant resource cost/complexity

The quantum computing approaches to date have ported much of the machinery from quantum algorithms for the electronic structure problem [9]. The nuclear structure problem can be tackled by projecting the Hamiltonian onto a single-particle basis (often harmonic oscillator eigenstates) [3]. In second quantization, a qubit is required for each single-particle basis function included. The EFT can be expanded to higher orders in the coupling parameter; it is typical to retain at least 3-nucleon couplings caused by the pion, and higher-order terms could also be included. Including the 3-nucleon coupling results in a Hamiltonian with \(\mathcal{O}\left( N^6 \right)\) terms, which can be contrasted with the \(\mathcal{O}\left( N^4 \right)\) scaling of the electronic structure Hamiltonian. As such, algorithms that scale with the number of terms (e.g., product formulae) may have a higher cost for nuclear structure calculations than electronic structure problems of a similar size. Nevertheless, an exact comparison depends on a number of other factors (dependent upon the algorithm used), such as the commutativity of the Hamiltonian terms, structure of the coefficients, and the energy scales in the problem.

Quantum algorithms that prepare energy eigenstates scale either as \(1/\gamma\) (where \(\gamma\) is the overlap of the initial state with the desired eigenstate) [10], or with the minimum gap size along an adiabatic path (see adiabatic state preparation) [11]. If we are only interested in measuring the energy of the state, this can be obtained using the quantum phase estimation algorithm, which also projects the system into the corresponding energy eigenstate. The cost of this approach scales as \(\mathcal{O}\left( 1/\gamma^2 \right)\). Once the desired state has been prepared, observables can be measured to precision \(\epsilon\) with complexity \(\mathcal{O}\left( 1/\epsilon^2 \right)\) (direct sampling) or \(\mathcal{O}\left( 1/\epsilon \right)\) (amplitude estimation).

The above algorithms for preparing states (and related algorithms for performing time evolution in dynamics simulations) require access to the Hamiltonian, which introduces a dependence on the norm of the Hamiltonian or the number of terms (or both). These costs have not yet been elucidated for nuclear structure calculations.

Existing error corrected resource estimates

We are not aware of any error corrected resource estimates for problems in nuclear physics. For an initial investigation into the cost of nucleus-neutrino scattering, see [6].

Caveats

For quantum algorithms to be efficient, we must be able to prepare an initial state that has only polynomially vanishing overlap with the desired state. This is the same problem that afflicts quantum algorithms for the electronic structure problem. For simulations of nuclear dynamics, it may be necessary to work with a basis set that is sufficiently flexible to account for the varying positions of the nuclei.

The parameter values of the EFT are obtained from fits to experimental data, and so may introduce systematic inaccuracies into the nuclear structure calculation.

Comparable classical complexity and challenging instance sizes

Classical approaches use similar techniques to those developed for the electronic structure problem, such as perturbation theory, Monte Carlo methods, or coupled cluster. Refs. [5, 3] provide an excellent overview of state-of-the-art approaches. Classical methods can provide excellent agreement with experiments for the binding energies of small nuclei with 20-50 nucleons [3]. As a further example, recent high-accuracy simulations of the \(^{100}\)Sn nucleus have enabled improved agreement between theory and experiment for observed \(\beta\)-decay rates [12]. Time-dependent simulations of dynamics or non-equilibrium phenomena are more challenging and are an active area of research [7, 8].

Speedup

The majority of classical approaches for the nuclear structure problem scale polynomially with system size, but introduce controllable errors due to the use of approximations (e.g., truncating the expansion in coupled cluster methods) [3]. For quantum computers to achieve exponential speedups, we require the identification of systems where (1) Classical methods must exponentially increase their resources to obtain accurate results and (2) It is efficient to prepare an initial state for the quantum calculation that only has polynomially decaying overlap with the desired state. There have recently been initial investigations into whether these requirements coexist in chemical systems [13]. We are not aware of similar work in nuclear physics.

NISQ implementation

Almost all of the work to date on applying quantum computing to the nuclear structure problem has focused on variational algorithms, such as [14, 15, 16]. There is currently no evidence that near-term quantum devices will be able to implement sufficiently deep circuits to achieve advantage over their classical counterparts with these methods.

Outlook

Further research is required to determine the fault-tolerant resources for solving nuclear structure problems on quantum computers. While the problem is inherently similar to the electronic structure problem in quantum chemistry, it is necessary to adapt known algorithms to the nuclear setting, and to understand and optimize their scaling for classically challenging problems. The simulation of nuclear reaction dynamics appears a particularly interesting target, which has not yet received a thorough reformulation suitable for quantum simulation.

Bibliography

  1. David J. Dean. Beyond the nuclear shell model. Physics Today, 60(11):48–53, 2007. URL: https://doi.org/10.1063/1.2812123, arXiv:https://doi.org/10.1063/1.2812123, doi:10.1063/1.2812123.

  2. Christian W. Bauer, Zohreh Davoudi, A. Baha Balantekin, Tanmoy Bhattacharya, Marcela Carena, Wibe A. de Jong, Patrick Draper, Aida El-Khadra, Nate Gemelke, Masanori Hanada, Dmitri Kharzeev, Henry Lamm, Ying-Ying Li, Junyu Liu, Mikhail Lukin, Yannick Meurice, Christopher Monroe, Benjamin Nachman, Guido Pagano, John Preskill, Enrico Rinaldi, Alessandro Roggero, David I. Santiago, Martin J. Savage, Irfan Siddiqi, George Siopsis, David Van Zanten, Nathan Wiebe, Yukari Yamauchi, Kübra Yeter-Aydeniz, and Silvia Zorzetti. Quantum simulation for high-energy physics. PRX Quantum, 4:027001, 5 2023. arXiv: https://arxiv.org/abs/2204.03381. URL: https://link.aps.org/doi/10.1103/PRXQuantum.4.027001, doi:10.1103/PRXQuantum.4.027001.

  3. Heiko Hergert. A guided tour of ab initio nuclear many-body theory. Frontiers in Physics, 2020. arXiv: https://arxiv.org/abs/2008.05061. URL: https://www.frontiersin.org/articles/10.3389/fphy.2020.00379, doi:10.3389/fphy.2020.00379.

  4. P. Maris, J. P. Vary, P. Navrátil, W. E. Ormand, H. Nam, and D. J. Dean. Origin of the anomalous long lifetime of \(^14\mathrm C\). Physical Review Letters, 106:202502, 5 2011. arXiv: https://arxiv.org/abs/1101.5124. URL: https://link.aps.org/doi/10.1103/PhysRevLett.106.202502, doi:10.1103/PhysRevLett.106.202502.

  5. G Hagen, T Papenbrock, M Hjorth-Jensen, and D J Dean. Coupled-cluster computations of atomic nuclei. Reports on Progress in Physics, 77(9):096302, 9 2014. arXiv: https://arxiv.org/abs/1312.7872. URL: https://dx.doi.org/10.1088/0034-4885/77/9/096302, doi:10.1088/0034-4885/77/9/096302.

  6. Alessandro Roggero, Andy C. Y. Li, Joseph Carlson, Rajan Gupta, and Gabriel N. Perdue. Quantum computing for neutrino-nucleus scattering. Physical Review D, 101:074038, 4 2020. arXiv: https://arxiv.org/abs/1911.06368. URL: https://link.aps.org/doi/10.1103/PhysRevD.101.074038, doi:10.1103/PhysRevD.101.074038.

  7. Michael Bender, Rémi Bernard, George Bertsch, Satoshi Chiba, Jacek Dobaczewski, Noël Dubray, Samuel A Giuliani, Kouichi Hagino, Denis Lacroix, Zhipan Li, Piotr Magierski, Joachim Maruhn, Witold Nazarewicz, Junchen Pei, Sophie Péru, Nathalie Pillet, Jørgen Randrup, David Regnier, Paul-Gerhard Reinhard, Luis M Robledo, Wouter Ryssens, Jhilam Sadhukhan, Guillaume Scamps, Nicolas Schunck, Cédric Simenel, Janusz Skalski, Ionel Stetcu, Paul Stevenson, Sait Umar, Marc Verriere, Dario Vretenar, Michał Warda, and Sven Åberg. Future of nuclear fission theory. Journal of Physics G: Nuclear and Particle Physics, 47(11):113002, 10 2020. arXiv: https://arxiv.org/abs/2005.10216. URL: https://dx.doi.org/10.1088/1361-6471/abab4f, doi:10.1088/1361-6471/abab4f.

  8. Petr Navrátil and Sofia Quaglioni. Ab initio nuclear reaction theory with applications to astrophysics. In Handbook of Nuclear Physics, pages 1–46. Springer, 2022. doi:10.1007/978-981-15-8818-1\_7-1.

  9. Paul D Stevenson. Comments on quantum computing in nuclear physics. International Journal of Unconventional Computing, 2023. URL: https://openresearch.surrey.ac.uk/esploro/outputs/other/Comments-on-Quantum-Computing-in-Nuclear/99641066502346.

  10. Lin Lin and Yu Tong. Near-optimal ground state preparation. Quantum, 4:372, 2020. arXiv: https://arxiv.org/abs/2002.12508. doi:10.22331/q-2020-12-14-372.

  11. Kianna Wan and Isaac Kim. Fast digital methods for adiabatic state preparation. arXiv: https://arxiv.org/abs/2004.04164, 2020.

  12. P. Gysbers, G. Hagen, J. D. Holt, G. R. Jansen, T. D. Morris, P. Navrátil, T. Papenbrock, S. Quaglioni, A. Schwenk, S. R. Stroberg, and K. A. Wendt. Discrepancy between experimental and theoretical β-decay rates resolved from first principles. Nature Physics, 15(5):428–431, 5 2019. arXiv: https://arxiv.org/abs/1903.00047. URL: https://doi.org/10.1038/s41567-019-0450-7, doi:10.1038/s41567-019-0450-7.

  13. Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M. Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kastoryano, Ryan Babbush, John Preskill, David R. Reichman, Earl T. Campbell, Edward F. Valeev, Lin Lin, and Garnet Kin-Lic Chan. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nature Communications, 14(1):1952, 2023. arXiv: https://arxiv.org/abs/2208.02199. URL: https://doi.org/10.1038/s41467-023-37587-6, doi:10.1038/s41467-023-37587-6.

  14. E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski. Cloud quantum computing of an atomic nucleus. Physical Review Letters, 120:210501, 5 2018. arXiv: https://arxiv.org/abs/1801.03897. URL: https://link.aps.org/doi/10.1103/PhysRevLett.120.210501, doi:10.1103/PhysRevLett.120.210501.

  15. Hsuan-Hao Lu, Natalie Klco, Joseph M. Lukens, Titus D. Morris, Aaina Bansal, Andreas Ekström, Gaute Hagen, Thomas Papenbrock, Andrew M. Weiner, Martin J. Savage, and Pavel Lougovski. Simulations of subatomic many-body physics on a quantum frequency processor. Physical Review A, 100:012320, 7 2019. arXiv: https://arxiv.org/abs/1810.03959. URL: https://link.aps.org/doi/10.1103/PhysRevA.100.012320, doi:10.1103/PhysRevA.100.012320.

  16. I. Stetcu, A. Baroni, and J. Carlson. Variational approaches to constructing the many-body nuclear ground state for quantum computing. Physical Review C, 105:064308, 6 2022. arXiv: https://arxiv.org/abs/2110.06098. URL: https://link.aps.org/doi/10.1103/PhysRevC.105.064308, doi:10.1103/PhysRevC.105.064308.