Quantum Mapping Simplifies Nuclear Shell Model Circuits

Published in Discover Quantum Science, the study demonstrates that mapping each Slater Determinant (SD) to a single qubit, rather than assigning qubits to individual single-particle states, enables the construction of simpler quantum circuits. This approach is particularly compatible with current noisy intermediate-scale quantum (NISQ) devices, which are highly sensitive to circuit complexity and error rates. The team applied this method to seven different nuclei, ranging from light lithium isotopes to heavy polonium and lead isotopes, to evaluate the feasibility of the approach on both simulators and real quantum hardware.

The nuclear shell model serves as a fundamental paradigm for describing the structure of atomic nuclei. In its Configuration Interaction formalism, the model describes nuclear states through the mutual interaction of nucleons moving in a basis of single-particle orbitals. While the model is efficient for nuclei close to magic numbers, the combinatorial increase in the Hilbert space with the number of valence particles often renders many nuclear systems inaccessible to classical computers. Quantum computation offers a potential solution by leveraging the exponential scaling of the multi-qubit Hilbert space, allowing for the efficient representation of highly entangled states that are difficult to simulate classically.

Traditional approaches to mapping the nuclear shell model to qubits typically assign each single-particle state to a qubit. However, this standard mapping often results in prohibitively deep circuits due to the need for double-excitation Givens rotations to account for two-body interactions. The new method recasts the Hamiltonian in a basis of Slater Determinant configurations. In this framework, the Hamiltonian is expressed in terms of many-particle matrix elements, allowing the use of single-excitation Givens rotations. This change in encoding trades an increase in the total number of required qubits for a substantial reduction in circuit depth and gate count, which is a critical advantage for current hardware where error rates demand simple circuits.

The study focused on seven nuclei: four lithium isotopes ($^{6-9}$Li) from the p-shell, $^{18}$F from the sd-shell, and two heavier nuclei, $^{210}$Po and $^{210}$Pb. For the lithium isotopes, the number of valence particles in the p-shell varies from two to five. The researchers enumerated the allowed configurations consistent with the ground state spin and parity for each isotope. For example, the ground state of $^{6}$Li, which has spin J=1, is represented by eight specific configurations in the SD basis. This requires an eight-qubit system, compared to the twelve qubits needed in the single-particle basis. The simplicity of the circuit is evident in the resource counts, where the SD basis requires significantly fewer two-qubit gates and fewer Pauli terms in the Hamiltonian representation.

To determine the optimal gate parameters, the team performed VQE runs using three different classical optimizers: COBYLA, SLSQP, and SPSA. The SLSQP optimizer showed the fastest convergence for $^{6}$Li and $^{7}$Li, reaching the exact ground state binding energies in approximately 250 iterations. However, for the neutron-rich isotopes $^{8}$Li and $^{9}$Li, as well as the heavier nuclei, the COBYLA optimizer provided better convergence. The optimized circuits were then transpiled for the IBM FakeFez backend, a noisy simulator based on the 156-qubit ibm_fez quantum computer, and executed on the ibm_pittsburgh quantum hardware. The transpilation process involved level three optimization within the IBM compiler to minimize the number of two-qubit operations.

In the noisy simulations, the results for $^{6}$Li and $^{7}$Li showed an underbinding of approximately 7% compared to the reference shell model energies. In contrast, the neutron-rich isotopes $^{8}$Li and $^{9}$Li exhibited lower errors, with binding energies only 2.22% and 1.46% less bound than the reference values. The researchers conjectured that the large component of the initialized qubit configuration in the exact solution for these isotopes contributed to the lower error rates, as less of the wave function amplitude was generated by the more error-prone two-qubit gates. Hardware executions on ibm_pittsburgh generally mirrored the noisy simulation results for the lighter isotopes, though $^{8}$Li and $^{9}$Li showed overbinding rather than underbinding on the hardware.

A direct comparison between the SD mapping and the single-particle mapping was conducted for $^{6}$Li and $^{18}$F. The SD mapping with its single-excitation ansatz required significantly lower resources and resulted in fewer errors in both noisy simulation and hardware runs. For $^{6}$Li, the single-excitation results were less than 10% off from the exact results, whereas the double-excitation results showed underbinding of around 55% in noisy simulation and 65% on hardware. This performance gap highlights the efficiency of the SD approach in mitigating the impact of hardware noise on the final energy calculations.

The study also explored the feasibility of simulating heavier nuclei, specifically $^{210}$Po and $^{210}$Pb, which are beyond the $^{208}$Pb core. Using the KHPE shell model interaction, the full m-scheme dimensions for these nuclei are 62 and 99 Slater determinants, respectively. However, by considering only those SDs that are combinations of two time-reversed single-particle states, the researchers reduced the system size to 22 qubits for $^{210}$Po and 29 qubits for $^{210}$Pb. The noisy simulation for $^{210}$Po yielded a binding energy of -9.438 MeV, which was about 8% more than the reference value. Hardware executions for both heavy nuclei showed larger deviations, with $^{210}$Pb being 85% more bound than the exact shell model result, indicating a high susceptibility to hardware errors due to the deeper circuits.

To address these errors, the team implemented Zero-Noise Extrapolation (ZNE) using two-qubit gate folding. This technique involves adding pairs of CZ gates that act as the identity operator to scale the noise level. By executing the circuits at different noise scales (1, 3, and 5) and extrapolating to the zero-noise limit, the researchers significantly improved the accuracy of the results. For the hardware results, the error-mitigated binding energies for all nuclei fell within a 4% error range compared to the shell model predictions. Notably, for $^{210}$Pb, which had a raw hardware error of 85%, the linear ZNE reduced the error to just 1.19%.

The findings suggest that the SD-based qubit mapping is a promising route for near-term quantum simulations in nuclear physics. The method is particularly effective for lighter nuclei and two-nucleon systems, where the trade-off between qubit count and circuit depth is favorable. The reduction in circuit depth and the decrease in the number of Pauli terms to be measured make this approach well-suited for current NISQ devices. Furthermore, the strategy offers a potential pathway for including three-nucleon forces within the same circuit design in future studies.

However, the method has limitations. The primary constraint is that each configuration in the full wave function requires its own qubit, meaning the qubit count can become very large for nuclei with many valence particles, scaling with the m-scheme dimension. Additionally, the advantage of low circuit depth becomes less relevant as fault-tolerant quantum computers become available, where deeper circuits can be managed more effectively. For more complex nuclei that exceed current hardware limits, alternative encodings, such as Gray code schemes, may be necessary, though these increase the number of Pauli terms to be measured.

The study concludes that while the SD mapping increases the qubit requirement in some cases, it provides a viable alternative encoding that is appropriate for current hardware where error rates are a dominant factor. The ability to achieve sub-4% accuracy after error mitigation across a range of nuclear masses demonstrates the practical utility of this approach. As quantum hardware continues to advance toward utility-scale devices with over 100 qubits, strategies that trade increased qubit count for reduced gate complexity and shallower circuit depth will likely play a significant role in scalable quantum simulations in nuclear physics.

Subject of Research: A low-circuit-depth quantum computing approach to the nuclear shell model

Article Title: A low-circuit-depth quantum computing approach to the nuclear shell model

Article References: Sarma, C., & Stevenson, P. D. (2026). A low-circuit-depth quantum computing approach to the nuclear shell model. Discover Quantum Science, 2(1), Article 6. https://doi.org/10.1007/s44464-026-00009-9

Image Credits: AI Generated

DOI: 10.1007/s44464-026-00009-9

Keywords: low-circuit-depth, quantum, computing, approach, nuclear, shell, model, scientific research