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

TL;DR

This work addresses the challenge of simulating nuclear shell-model Hamiltonians on near-term quantum devices by mapping each Slater determinant (SD) to a qubit, thereby enabling shallower variational quantum eigensolver (VQE) circuits at the expense of more qubits. Using the Jordan–Wigner transformation, the SD-based qubit Hamiltonian is formulated as $H_{qubit}=\sum_{m} H_{mm} \frac{(I_m - Z_m)}{2} + \sum_{m<n} H_{mn} \frac{(X_m X_n + Y_m Y_n)}{2}$, and state preparation relies on single-excitation Givens rotations connecting SD configurations, with double excitations required by two-body terms. The approach is demonstrated across Li isotopes, $^{18}$F, and heavy nuclei $^{210}$Po and $^{210}$Pb, showing substantially reduced circuit depth and Pauli-term counts, and, after Zero-Noise Extrapolation (ZNE) with two-qubit gate folding, energies within about 4% of shell-model results for all seven nuclei. The results underscore the viability of SD-based encodings for near-term nuclear quantum simulations and point to scalable directions, including three-nucleon forces and alternative SD encodings, as quantum hardware advances beyond a few tens of qubits.

Abstract

In this work, we introduce a new qubit mapping strategy for the Variational Quantum Eigensolver (VQE) applied to nuclear shell model calculations, where each Slater determinant (SD) is mapped to a qubit, rather than assigning qubits to individual single-particle states. While this approach may increase the total number of qubits required in some cases, it enables the construction of simpler quantum circuits that are more compatible with current noisy intermediate-scale quantum (NISQ) devices. We apply this method to seven nuclei: Four lithium isotopes $^{6-9}$Li from the \textit{p}-shell, $^{18}$F from the \textit{sd}-shell, and two heavier nuclei ($^{210}$Po, and $^{210}$Pb). We run circuits representing their ground states on a noisy simulator (IBM's \textit{FakeFez} backend) and quantum hardware ($ibm\_pittsburgh$). For heavier nuclei, we demonstrate the feasibility of simulating $^{210}$Po and $^{210}$Pb as 22- and 29-qubit systems, respectively. Additionally, we employ Zero-Noise Extrapolation (ZNE) via two-qubit gate folding to mitigate errors in both simulated and hardware-executed results. Post-mitigation, the best results show less than 4 \% deviation from shell model predictions across all nuclei studied. This SD-based qubit mapping proves particularly effective for lighter nuclei and two-nucleon systems, offering a promising route for near-term quantum simulations in nuclear physics.

Figures (11)

• Figure 1: Single particle states of the $p$-shell. $l, j, m$ quantum numbers given in usual nuclear notation. The indexing from 0 to 11 are used for qubit mapping.
• Figure 2: Single excitation Givens rotation in terms of basic quantum gates. Adapted from single_ex.
• Figure 3: Double excitation Givens rotation in terms of basic quantum gates. Adapted from double_ex.
• Figure 4: Double excitation ansatz for $^6$Li (1$^+$).
• Figure 5: Single excitation ansatz for $^6$Li ($1^+$).