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arXiv:2110.06098 · Physical Review C

Variational approaches to constructing the many-body nuclear ground state for quantum computing

Ionel Stetcu, Alessandro Baroni, Joseph Carlson

Published Jun 21, 2022• 63 citations
arXivHTMLPDF

Abstract

We explore the preparation of specific nuclear states on gate-based quantum hardware using variational algorithms. Large-scale classical diagonalizations of the nuclear shell model have reached sizes of ${10}^{9}--{10}^{10}$ basis states but are still severely limited by computational resources. Quantum computing can, in principle, solve such systems exactly with exponentially fewer resources than classical computing. Exact solutions for large systems require many qubits and large gate depth, but variational approaches can effectively limit the required gate depth. We use the unitary coupled cluster approach to construct approximations of the ground-state vectors, later to be used in dynamics calculations. The testing ground is the phenomenological shell model space, which allows us to mimic the complexity of the internucleon interactions. We find that often one needs to minimize over a large number of parameters, using a large number of entanglements that makes the application on existing hardware challenging. Prospects for rapid improvements with more capable hardware are, however, very encouraging.

DOI
10.1103/physrevc.105.064308

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We explore the preparation of specific nuclear states on gate-based quantum hardware using variational algorithms. Large-scale classical diagonalizations of the nuclear shell model have reached sizes of 109−−1010{10}^{9}--{10}^{10}109−−1010 basis states but are still severely limited by computational resources. Quantum computing can, in principle, solve such systems exactly with exponentially fewer resources than classical computing. Exact solutions for large systems require many qubits and large gate depth, but variational approaches can effectively limit the required gate depth. We use the unitary coupled cluster approach to construct approximations of the ground-state vectors, later to be used in dynamics calculations. The testing ground is the phenomenological shell model space, which allows us to mimic the complexity of the internucleon interactions. We find that often one needs to minimize over a large number of parameters, using a large number of entanglements that makes the application on existing hardware challenging. Prospects for rapid improvements with more capable hardware are, however, very encouraging.
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.
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.
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.
Quantum computing is one of the most promising technologies of the near future, and the simulation of quantum many-body systems is a natural application. In this work, we present classical simulations of the ground states of light atomic nuclei within the p shell, from He 6 to B 10 , calculated within the nuclear shell model. We compare the performance of two leading variational quantum eigensolver algorithms: the unitary coupled cluster (UCC) and the adaptive derivative-assembled pseudo-Trotter (ADAPT) methods, introducing a new metric to quantify the use of quantum resources in each simulation. We find that Slater determinants are the most useful reference states for both approaches. Our analysis suggests that ADAPT is more efficient for nuclei close to magic numbers, while UCC tends to require fewer resources toward the mid shell. This work lays the groundwork for robust benchmarking of quantum algorithms in nuclear structure studies.
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.