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arXiv:2301.05976 · Physical Review C

Quantum simulations in effective model spaces: Hamiltonian-learning variational quantum eigensolver using digital quantum computers and application to the Lipkin-Meshkov-Glick model

Caroline Robin, Martin J. Savage A

Published Aug 18, 2023• 39 citations
arXivHTMLPDF

Abstract

Background: Quantum simulations offer the potential to predict the structure and dynamics of nuclear many-body systems that are beyond the capabilities of classical computing. Generally, preparing the ground state of strongly interacting many-body systems relevant to nuclear physics is, however, inefficient, even using ideal quantum computers. In addition, currently available noisy intermediate-scale quantum (NISQ) era quantum devices possess modest numbers of qubits, limiting the size of quantum many-body systems that can be simulated. In this context, a reformulation of the quantum many-body problems using truncated model spaces and Hamiltonians is desirable to make them more amenable to near-term quantum computers. The importance of symmetries in low-energy theories, including effective field theories (EFTs), lattice quantum chromodynamics (QCD), and effective model spaces for nuclear systems, in particular their interplay with the reduction of active Hilbert spaces, is well known. Lesser known is the fact that the noncommutivity of some symmetries and truncations of the model space can be profitably combined with variational calculations to rearrange the entanglement into localized structures and enable more efficient simulations.Purpose: The goal of the present study is to explore and utilize the noncommutivity of symmetries and model-space truncations of quantum many-body systems important to nuclear physics, particularly in combination with variational algorithms for quantum simulations and effective Hamiltonian learning.Method: We introduce an iterative hybrid classical-quantum algorithm, the Hamiltonian learning variational quantum eigensolver (HL-VQE), that simultaneously optimizes an effective Hamiltonian, thereby rearranging entanglement into the effective model space, and the associated ground-state wave function. Quantum simulations, using classical computers and IBM's superconducting-qubit quantum computers, are performed to demonstrate the HL-VQE algorithm, in the context of the Lipkin-Meshkov-Glick (LMG) model of interacting fermions, where the Hamiltonian transformation corresponds to an orbital rotation. We use a mapping where the number of qubits scales with the $\mathrm{logarithm}$ of the size of the effective model space, rather than the particle number.Results: HL-VQE is found to provide an exponential improvement in LMG-model calculations of the ground-state energy and wave function, compared to naive truncations without Hamiltonian learning, throughout a significant fraction of the Hilbert space. In the context of EFT, this corresponds to counterterms scaling exponentially with the cutoff as opposed to power law. Implementations on IBM's QExperience quantum computers and simulators for one- and two-qubit effective model spaces are shown to provide accurate and precise results, reproducing classical predictions.Conclusions: For a range of parameters defining the LMG model, the HL-VQE algorithm is found to have better scaling of quantum resources requirements than previously explored algorithms. In particular, the HL-VQE scales efficiently over a large fraction of the model space, in contrast to VQE alone. This work constitutes a step in the development of entanglement-driven quantum algorithms for descriptions of nuclear many-body systems. This, in part, leverages the potential of noisy intermediate-scale quantum (NISQ) devices. The exponential scaling of counterterms observed in this study suggests the possibility of more general applicability to other nonperturbative EFTs.

DOI
10.1103/physrevc.108.024313

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Quantum computing opens up new possibilities for the simulation of many-body nuclear systems. As the number of particles in a many-body system increases, the size of the space if the associated Hamiltonian increases exponentially. This presents a challenge when performing calculations on large systems when using classical computing methods. By using a quantum computer, one may be able to overcome this difficulty thanks to the exponential way the Hilbert space of a quantum computer grows with the number of quantum bits (qubits). Our aim is to develop quantum computing algorithms which can reproduce and predict nuclear structure such as level schemes and level densities. As a sample Hamiltonian, we use the Lipkin-Meshkov-Glick model. We use an efficient encoding of the Hamiltonian onto many-qubit systems, and have developed an algorithm allowing the full excitation spectrum of a nucleus to be determined with a variational algorithm capable of implementation on today's quantum computers with a limited number of qubits. Our algorithm uses the variance of the Hamiltonian, ⟨H2⟩−⟨H⟩2\langle H^2\rangle - \langle H\rangle ^2⟨H2⟩−⟨H⟩2, as a cost function for the widely-used variational quantum eigensolver (VQE). In this work we present a variance based method of finding the excited state spectrum of a small nuclear system using a quantum computer, using a reduced-qubit encoding method.
Quantum computing opens up new possibilities for the simulation of many-body nuclear systems. As the number of particles in a many-body system increases, the size of the space if the associated Hamiltonian increases exponentially. This presents a challenge when performing calculations on large systems when using classical computing methods. By using a quantum computer, one may be able to overcome this difficulty thanks to the exponential way the Hilbert space of a quantum computer grows with the number of quantum bits (qubits). Our aim is to develop quantum computing algorithms which can reproduce and predict nuclear structure such as level schemes and level densities. As a sample Hamiltonian, we use the Lipkin-Meshkov-Glick model. We use an efficient encoding of the Hamiltonian onto many-qubit systems, and have developed an algorithm allowing the full excitation spectrum of a nucleus to be determined with a variational algorithm capable of implementation on today’s quantum computers with a limited number of qubits. Our algorithm uses the variance of the Hamiltonian,⟨H⟩2 − ⟨H⟩2, as a cost function for the widely-used variational quantum eigensolver (VQE). In this work we present a variance based method of finding the excited state spectrum of a small nuclear system using a quantum computer, using a reduced-qubit encoding method.
Atomic nuclei are important laboratories for exploring and testing new insights into the universe, such as experiments to directly detect dark matter or explore properties of neutrinos. The targets of interest are often heavy, complex nuclei that challenge our ability to reliably model them (as well as quantify the uncertainty of those models) with classical computers. Hence there is great interest in applying quantum computation to nuclear structure for these applications. As an early step in this direction, especially with regards to the uncertainties in the relevant quantum calculations, we develop circuits to implement variational quantum eigensolver (VQE) algorithms for the Lipkin-Meshkov-Glick model, which is often used in the nuclear physics community as a testbed for many-body methods. We present quantum circuits for VQE for two and three particles and discuss the construction of circuits for more particles. Implementing the VQE for a two-particle system on the IBM Quantum Experience, we identify initialization and two-qubit gates as the largest sources of error. We find that error mitigation procedures reduce the errors in the results significantly, but additional quantum hardware improvements are needed for quantum calculations to be sufficiently accurate to be competitive with the best current classical methods.
In this project we studied a simplified potential from [1] describing nuclear physics from quark and gluon fields, where spatial degrees of freedom have been integrated over. This potential deals with the phase structure of the strong interaction. We created a matrix representation of the Hamiltonian for this theory using discrete quantum mechanics [2]. We then feed this Hamiltonian to IBM's Qiskit package [3]. We run on IBM's 32-qubit quantum simulator, which approximates the ground state energy using the variational quantum eigensolver (VQE) algorithm [4]. This result is then compared with the exact ground state energy, which we can calculate classically since the matrices we use are still small enough. The bosonic Hamiltonian for a 6-qubit representation took over 1000 Pauli factors to represent fully. To improve our performance, we can look for a more efficient representation of this model's Hamiltonian. We will develop a better variational trial wave function and initial state, as well as include the effect of fermions and heavy quarks on the potential.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.
Nuclear lattice effective field theory has become an important framework for quantum many-body calculations in nuclear physics, yet its classical implementation remains increasingly challenging for more general interactions and larger systems. In this work, we develop a quantum-computing framework for a three-dimensional nuclear lattice model. We construct a variational quantum eigensolver framework and systematically compare the Jordan-Wigner and Gray code encodings. Our analysis shows that for the few-body systems considered here, Gray code combined with symmetry reduction yields a substantially more compact qubit representation. Based on this framework, we perform numerical studies for 2H^{2}\mathrm{H}2H, 3H^{3}\mathrm{H}3H, and 4He^{4}\mathrm{He}4He on finite lattices. The calculated ground-state energies exhibit a clear approach toward the corresponding experimental binding energies as the lattice size increases. These results provide a proof-of-principle foundation for future quantum simulations of nuclear many-body problems.