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arXiv:2306.08885 · Physical Review A

Shadow-based quantum subspace algorithm for the nuclear shell model

Ruyu Yang, Tianren Wang, Bing-Nan Lu, Ying Li, Xiaosi Xu

Published Jan 21, 2025• 2 citations
arXivHTMLPDF

Abstract

In recent years, researchers have been exploring the applications of noisy intermediate-scale quantum (NISQ) computation in various fields. One important area in which quantum computation can outperform classical computers is the ground-state problem of a many-body system, e.g., the nucleus. However, using a quantum computer in the NISQ era to solve a meaningful-scale system remains a challenge. To calculate the ground energy of nuclear systems, we propose a new algorithm that combines classical shadow and subspace diagonalization techniques. Our subspace is composed of matrices, with the basis of the subspace being the classical shadow of the quantum state. We test our algorithm on nuclei described by the Cohen-Kurath shell model and USD shell model. We find that the accuracy of the results improves as the number of shots increases, following the Heisenberg scaling.

DOI
10.1103/physreva.111.012620

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•Nov 4, 2025•1 citation
In recent years, researchers have been exploring the applications of noisy intermediate-scale quantum (NISQ) computation in various fields. One important area in which quantum computation can outperform classical computers is the ground-state problem of a many-body system, e.g., the nucleus. However, using a quantum computer in the NISQ era to solve a meaningful-scale system remains a challenge. To calculate the ground energy of nuclear systems, we propose a new algorithm that combines classical shadow and subspace diagonalization techniques. Our subspace is composed of matrices, with the basis of the subspace being the classical shadow of the quantum state. We test our algorithm on nuclei described by the Cohen-Kurath shell model and USD shell model. We find that the accuracy of the results improves as the number of shots increases, following the Heisenberg scaling.
The exact diagonalization of the nuclear shell model scales exponentially, leading to severe memory bottlenecks in classical high-performance computing (HPC). While hybrid quantum algorithms like the Variational Quantum Eigensolver (VQE) aim to overcome these limits, their deep quantum circuits and iterative feedback loops are susceptible to substantial noise inherent in the current Noisy Intermediate-Scale Quantum (NISQ) hardware. This noise renders several algorithms, such as the VQE, impractical for large-scale calculations despite sophisticated noise-mitigation techniques. As a pragmatic approach tolerant to these issues, we apply the Sample-based Quantum Diagonalization (SQD) framework to nuclear shell models for the first time. Using 38Ar^{38}\text{Ar}38Ar as a benchmark to confirm the numerical accuracy, we extend SQD to 32Mg^{32}\text{Mg}32Mg, solving a nuclear shell-model Hamiltonian whose underlying Hilbert space cannot be directly diagonalized using conventional classical methods in a given HPC system. We present a systematic comparison of SQD with standard variational quantum schemes and exact classical solvers. By leveraging NISQ hardware connected via the cloud to classical HPC clusters, the SQD-based scheme could outperform conventional supercomputers in memory scaling and total execution time, enabling more rigorous large-scale shell model calculations.
The nuclear shell model is one of the prime many-body methods to study the structure of atomic nuclei, but it is hampered by an exponential scaling on the basis size as the number of particles increases. We present a shell-model quantum circuit design strategy to find nuclear ground states by exploiting an adaptive variational quantum eigensolver algorithm. Our circuit implementation is in excellent agreement with classical shell-model simulations for a dozen of light and medium-mass nuclei, including neon and calcium isotopes. We quantify the circuit depth, width and number of gates to encode realistic shell-model wavefunctions. Our strategy also addresses explicitly energy measurements and the required number of circuits to perform them. Our simulated circuits approach the benchmark results exponentially with a polynomial scaling in quantum resources for each nucleus. This work paves the way for quantum computing shell-model studies across the nuclear chart and our quantum resource quantification may be used in configuration-interaction calculations of other fermionic systems.
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.
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{}^{6-9}6−9 Li from the p -shell, 18{}^{18}18 F from the sd -shell, and two heavier nuclei ( 210{}^{210}210 Po, and 210{}^{210}210 Pb). We run circuits representing their ground states on a noisy simulator (IBM’s FakeFez backend) and quantum hardware ( ibm_pittsburghibm\_pittsburghibm_pittsburgh ). For heavier nuclei, we demonstrate the feasibility of simulating 210{}^{210}210 Po and 210{}^{210}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 binding energy 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.
In this work, we compare three qubit-mapping strategies to study the structure of the nuclear ground state within the shell model description employing the Variational Quantum Eigensolver (VQE) approach. Although the initial point for different mappings is a Hamiltonian matrix in many-body particle basis or Slater determinant (SD) basis, the structure of the trial wavefunction and resource counts are different for each mapping. These three mappings are tested for a mid ppp-shell nucleus 10^{10}10B and compared the quantum resources required to find the ground state for each mapping. Further, we extend the qubit-efficient mapping to study the ground state of one more mid ppp-shell nucleus 12^{12}12C. We run circuits up to 26-qubits representing their ground states on a noisy simulator (IBM's FakeFez backend) and quantum hardware (ibm_fezibm\_fezibm_fez). The best post-error mitigated results from the hardware for 10^{10}10B ground state is obtained following SD to qubit mapping with a percent error of 0.21 \%. The percent errors for the same state following cSD and pnSD mapping are 3.37 and 8.88 \%, respectively. On the other hand, following the cSD mapping, the post-error mitigated ground state energy of 12^{12}12C is 6.82 \% away from the exact result. We further evaluate the fidelity of the VQE wavefunctions obtained from hardware with respect to the shell model wavefunctions for the cSD mapping. This cSD mapping can be useful for scaling the VQE algorithm for complex nuclei across different mass regions in terms of qubit efficiency.
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^{6-9}6−9Li from the \textit{p}-shell, 18^{18}18F from the \textit{sd}-shell, and two heavier nuclei (210^{210}210Po, and 210^{210}210Pb). We run circuits representing their ground states on a noisy simulator (IBM's \textit{FakeFez} backend) and quantum hardware (ibm_pittsburghibm\_pittsburghibm_pittsburgh). For heavier nuclei, we demonstrate the feasibility of simulating 210^{210}210Po and 210^{210}210Pb 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. PACS numbers: 21.60.Cs, 21.30.Fe, 21.10.Dr, 27.20.+n