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arXiv:2608.01769

Quantum Simulation of Nuclear Shell Model Using GCM-Based Methods on NISQ Devices

Durgesh Pandey, Ashutosh Singh, Ankit Kumar Das, P. Arumugam

nucl-thquant-phSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

Based on the Generator Coordinate Method (GCM), we use a Quantum GCM (QuGCM) within a hybrid quantum-classical framework to simulate low-lying eigenstates of nuclear systems on quantum devices. The generator basis states are constructed from Hartree-Fock (HF) reference states, excited via symmetry-adapted unitary coupled-cluster (UCC) operators. These states are prepared as non-orthogonal quantum circuits and measured pairwise to compute the required overlap and Hamiltonian kernels. The resulting data is processed using a classical generalized-eigenvalue solver, following the GCM formalism, to extract the system's energy spectrum. To enhance efficiency and reduce circuit depth, we apply the Adaptive Generator Coordinate Inspired method (ADAPT-GCIM), which iteratively selects generator excitations based on energy gradients, thereby avoiding the need to explore the full Hilbert space. Our implementation is applied to nuclear systems, specifically the deuteron with the Reid68 potential and shell-model Hamiltonians of $^6$Li and $^{38}$Ar. For each system, both the QuGCM and ADAPT-GCIM methods produce energy spectra in agreement with classical diagonalization results, demonstrating robustness even under noise and limited-depth constraints. Additionally, we compare fermionic encoding strategies, specifically Jordan-Wigner (JW) transformations of one-hot (OH) encoding and Gray code (GC) mappings, and show that GC encoding reduces circuit complexity and improves fidelity during multi-reference state preparation. Our findings indicate that QuGCM and ADAPT-GCIM provide a practical and scalable path toward simulating correlated quantum systems, with lesser vulnerability to noise and better compatibility with the limitations of current quantum hardware.

Comments
23 pages, 10 figures

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Physical Review C•Sep 3, 2021•27 citations
Based on the Generator Coordinate Method (GCM), we use a Quantum GCM (QuGCM) within a hybrid quantum-classical framework to simulate low-lying eigenstates of nuclear systems on quantum devices. The generator basis states are constructed from Hartree-Fock (HF) reference states, excited via symmetry-adapted unitary coupled-cluster (UCC) operators. These states are prepared as non-orthogonal quantum circuits and measured pairwise to compute the required overlap and Hamiltonian kernels. The resulting data is processed using a classical generalized-eigenvalue solver, following the GCM formalism, to extract the system's energy spectrum. To enhance efficiency and reduce circuit depth, we apply the Adaptive Generator Coordinate Inspired method (ADAPT-GCIM), which iteratively selects generator excitations based on energy gradients, thereby avoiding the need to explore the full Hilbert space. Our implementation is applied to nuclear systems, specifically the deuteron with the Reid68 potential and shell-model Hamiltonians of 6^66Li and 38^{38}38Ar. For each system, both the QuGCM and ADAPT-GCIM methods produce energy spectra in agreement with classical diagonalization results, demonstrating robustness even under noise and limited-depth constraints. Additionally, we compare fermionic encoding strategies, specifically Jordan-Wigner (JW) transformations of one-hot (OH) encoding and Gray code (GC) mappings, and show that GC encoding reduces circuit complexity and improves fidelity during multi-reference state preparation. Our findings indicate that QuGCM and ADAPT-GCIM provide a practical and scalable path toward simulating correlated quantum systems, with lesser vulnerability to noise and better compatibility with the limitations of current quantum hardware.
Some of the computational limitations in solving the nuclear many-body problem could be overcome by utilizing quantum computers. The nuclear shell-model calculations providing deeper insights into the properties of atomic nuclei, is one such case with high demand for resources as the size of the Hilbert space grows exponentially with the number of particles involved. Quantum algorithms are being developed to overcome these challenges and advance such calculations. To develop quantum circuits for the nuclear shell-model, leveraging the capabilities of noisy intermediate-scale quantum (NISQ) devices. We aim to minimize resource requirements (specifically in terms of qubits and gates) and strive to reduce the impact of noise by employing relevant mitigation techniques. We achieve noise resilience by designing an optimized ansatz for the variational quantum eigensolver (VQE) based on Givens rotations and incorporating qubit-ADAPT-VQE in combination with variational quantum deflation (VQD) to compute ground and excited states incorporating the zero-noise extrapolation mitigation technique. Furthermore, the qubit requirements are significantly reduced by mapping the basis states to qubits using Gray code encoding and generalizing transformations of fermionic operators to efficiently represent manybody states. By employing the noise-resilient protocols, we achieve the ground and excited state energy levels of 38Ar and 6Li with better accuracy. These energy levels are presented for noiseless simulations, noisy conditions, and after applying noise mitigation techniques. Results are compared for Jordan Wigner and Gray code encoding using VQE, qubit-ADAPT-VQE, and VQD. Our work highlights the potential of noise-resilient protocols to leverage the full potential of NISQ devices in scaling the nuclear shell model calculations.
The nuclear shell model is a general way of expressing the many-body nuclear Hamiltonian and deciphering the underlying nuclear structure. In today's era of modern and high-power computation, the primary limitation of the nuclear shell model is the enormous dimensionality of its Hilbert space, which far exceeds available storage capacity and prevents the diagonalization of the full Hamiltonian matrix in that space. Quantum computing offers a scalable solution to bypass this curse of dimensionality. In this work, we introduce a single-run quantum simulation capable of obtaining multiple shell-model eigenstates simultaneously. The nuclear Hamiltonian is transformed from a bit to a qubit basis using the Jordan-Wigner transformation, explicitly preserving fermionic anti-commutation. We employ a Subspace Search Variational Quantum Eigensolver (SSVQE) along with an Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansatz to construct the quantum circuit required to solve the shell-model problem. The ADAPT-SSVQE algorithm uses a symmetry-preserving single and double-excitation operator pool and optimizes a weighted energy sum to obtain the simultaneous convergence of all eigenstates within a targeted MJ subspace, eliminating the need for post-processing efforts to extract excited spectra. We benchmark this approach by solving the problem for two and three identical nucleons in a j = 9/2 orbital, successfully extracting five and ten mutually orthogonal states, respectively, within a 10-qubit active space. The algorithm achieves spectroscopic accuracy, in simulation, relative to exact diagonalization and intrinsically restores total angular momentum (\hat{J}^2) symmetry.
Background: The many-body nature of nuclear physics problems poses significant computational challenges. These challenges become even more pronounced when studying the resonance states of nuclear systems, which are governed by the non-Hermitian Hamiltonian. Quantum computing, particularly for quantum many-body systems, offers a promising alternative, especially within the constraints of current noisy intermediate-scale quantum (NISQ) devices.Purpose: This work aims to simulate nuclear resonances using quantum algorithms by developing a variational framework compatible with non-Hermitian Hamiltonians and implementing it fully on a quantum simulator.Methods: We employ the complex scaling technique to extract resonance positions classically and adapt it for quantum simulations using a two-step algorithm. First, we transform the non-Hermitian Hamiltonian into a Hermitian form by using the energy variance as a cost function within a variational framework. Second, we perform $\ensuremath{\theta}$-trajectory calculations to determine optimal resonance positions in the complex energy plane. To address resource constraints on NISQ devices, we utilize Gray code (GC) encoding to reduce qubit requirements.Results: We first validate our approach using a schematic potential model that mimics a nuclear potential, successfully reproducing known resonance energies with high fidelity. We then extend the method to a more realistic $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\alpha}$ nuclear potential and compute the DDD- and GGG-wave resonance energies with a basis size of N=16N=16N=16, using only four qubits. The quantum simulation results closely match the classical values, demonstrating the feasibility of our approach.Conclusions: This study demonstrates, for the first time, that the complete $\ensuremath{\theta}$-trajectory method can be implemented on a quantum computer without relying on any classical input beyond the Hamiltonian. The results establish a scalable and efficient quantum framework for simulating resonance phenomena in nuclear systems. This work represents a significant step toward quantum simulations of open quantum systems and lays the foundation for future investigations into resonance structures in nuclear, atomic, and molecular physics.
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 sdsdsd shell, reaching an energy relative error below 0.2%0.2\%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%2\%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.
The possibility of using the generator coordinate method (GCM) using hybrid quantum-classical algorithms with reduced quantum resources is discussed. The task of preparing the basis states and calculating the various kernels involved in the GCM is assigned to the quantum computer, while the remaining tasks, such as finding the eigenvalues of a many-body problem, are delegated to classical computers for post-processing the generated kernels. This strategy reduces the quantum resources required to treat a quantum many-body problem. We apply the method to the Lipkin model. Using the permutation symmetry of the Hamiltonian, we show that, ultimately, only two qubits is enough to solve the problem regardless of the particle number. The classical computing post-processing leading to the full energy spectrum can be made using standard generalized eigenvalues techniques by diagonalizing the so-called Hill-Wheeler equation. As an alternative to this technique, we also explored how the quantum state deflation method can be adapted to the GCM problem. In this method, variational principles are iteratively designed to access the different excited states with increasing energies. The methodology proposed here is successfully applied to the Lipkin model with a minimal size of two qubits for the quantum register. The performances of the two classical post-processing approaches with respect to the statistical noise induced by the finite number of measurements and quantum devices noise are analyzed. Very satisfactory results for the full energy spectra are obtained once noise corrections techniques are employed.
We present the quantum simulation of the deuteron to calculate its binding energy using the variational quantum eigensolver, which is based on a hybrid quantum-classical approach. Apart from a commonly used Hamiltonian derived from pionless effective field theory, we consider the interaction which can be easily studied with conventional classical methods but the corresponding operator leading to nonzero off-tridiagonal matrix elements. To map the many-body basis states on the qubit states, three encodings are explored, namely, one-hot, Bravyi-Kitaev, and the Gray code. We perform a generalized transformation for many-body operators in Gray code encoding, and simulate the Hamiltonians with nonzero off-tridiagonal matrix elements. Furthermore, the analyses of the relative efficiency of all encodings and corresponding transformations are carried out using the noise model of a real IBM quantum device. We demonstrate that the Gray code is more efficient for a large basis, irrespective of the form of potential and the presence of hardware noise.