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arXiv:2404.08581 · Physical Review C

Emulating the generator coordinate method with extended eigenvector continuation for the Lipkin-Meshkov-Glick model

Q. Y. Luo, Xilin Zhang, Li‐Hui Chen, J. M. Yao

Published Jul 2, 2024• 5 citations
arXivHTMLPDF

Abstract

We present a benchmark study of the generator coordinate method (GCM) combined with eigenvector continuation (EC) in two different schemes for the low-lying states of the Lipkin-Meshkov-Glick (LMG) model, where the interaction strength is treated as a controlling parameter, simulating quantum many-body systems with the phase transition from noncollective to collective states. We demonstrate that the ${\mathrm{EC}}_{\mathrm{kmax}}$ scheme accurately reproduces the low-lying states of the LMG model. In this scheme, the EC basis consists of the wave functions of low-lying states up to the ${k}_{\mathrm{max}}$-th state of sampling Hamiltonians. Compared to ${\mathrm{EC}}_{1}$, which only includes the wave functions of the $k\mathrm{th}$ state of sampling Hamiltonians for the $k\mathrm{th}$ state of a target Hamiltonian, the ${\mathrm{EC}}_{\mathrm{kmax}}$ scheme exhibits significantly improved efficiency and accuracy. This study suggests the potential utilization of the extended EC scheme as an efficient emulator for GCM calculations.

DOI
10.1103/physrevc.110.014309

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We present a benchmark study of the generator coordinate method (GCM) combined with eigenvector continuation (EC) in two different schemes for the low-lying states of the Lipkin-Meshkov-Glick (LMG) model, where the interaction strength is treated as a controlling parameter, simulating quantum many-body systems with the phase transition from noncollective to collective states. We demonstrate that the ECkmax{\mathrm{EC}}_{\mathrm{kmax}}ECkmax​ scheme accurately reproduces the low-lying states of the LMG model. In this scheme, the EC basis consists of the wave functions of low-lying states up to the kmax{k}_{\mathrm{max}}kmax​-th state of sampling Hamiltonians. Compared to EC1{\mathrm{EC}}_{1}EC1​, which only includes the wave functions of the kthk\mathrm{th}kth state of sampling Hamiltonians for the kthk\mathrm{th}kth state of a target Hamiltonian, the ECkmax{\mathrm{EC}}_{\mathrm{kmax}}ECkmax​ scheme exhibits significantly improved efficiency and accuracy. This study suggests the potential utilization of the extended EC scheme as an efficient emulator for GCM calculations.
Abstract The Lipkin–Meshkov–Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the main features of real physical systems. In the present contribution, we explicitly review the fact that different many-body approximations commonly used in different fields in physics clearly fail to describe the exact LMG solution. With similar assumptions as those adopted for the LMG model, we propose a new Hamiltonian based on a general two-body interaction. The new model (extended LMG) is not only more general than the original LMG model and, therefore, has a potentially larger spectrum of applicability, but also the physics behind its exact solution can be much better captured by common many-body approximations. At the basis of this improvement lies a new term in the Hamiltonian that depends on the number of constituents and polarizes the system; the associated symmetry breaking is discussed, together with some implications for the study of more realistic systems.
For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin–Meshkov–Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.
The simulation of large-scale quantum systems is one of the most sought-after applications of quantum computers. Of particular interest for near-term demonstrations of quantum computational advantage are analog quantum simulations, which employ analog controls instead of digitized gates. Most analog quantum simulations to date, however, have been performed using qubit-based processors, despite the fact that many physical systems are more naturally represented in terms of qudits (i.e., ddd-level systems). Motivated by this, we present an experimental realization of the Lipkin-Meshkov-Glick (LMG) model using an analog simulator based on a single superconducting transmon qudit with up to d=9d = 9d=9 levels. This is accomplished by moving to a rotated frame in which evolution under any time-dependent local field and one-axis twisting can be realized by the application of multiple simultaneous drives. Combining this analog drive scheme with universal control and single-shot readout of the qudit state, we provide a detailed study of five finite-size precursors of quantum criticality in the LMG model: dynamical phase transitions, closing of the energy gap, Kibble-Zurek-like dynamics, statistics of the order parameter, and excited-state phase transitions. For each experiment we devise a protocol for extracting the relevant properties which does not require any prior knowledge of the system eigenstates, and can therefore be readily extended to higher dimensions or more complicated models. Our results cement high-dimensional transmon qudits as an exciting path towards simulating many-body physics.
An atomic nucleus is a quantum system of interacting nucleons and constitutes a problem difficult to solve exactly. For this reason, a diversity of approximate resolution methods has been designed, and Projected Generator Coordinate Method (PGCM) is one of them. The strong point of PGCM is to construct a physically inspired small dimensional space, in which an approximate solution of the nuclear many-body problem is easily found. However the numerical cost of PGCM space computation make this method inadapted for sensibility analysis of nuclear observables with restect to parametrisation of the interaction model, this analysis requiring an huge number of PGCM computations. In order to make this type of study possible, this thesis explore the concept of PGCM emulator. In this work, a combination of PGCM with Eigenvector Continuation (EC) is constructed and studied. This combination (the PGCM-EC emulator) takes advantage of mathematical similarities between PGCM and EC, and above all of the decomposition of the hamiltonian as a linear combination of parameter-independent terms. The latter property is used to concentrate the heavier numerical cost in the computation of parameter-independent quantities (the elementary kernels), and open the feasability of massive PGCM emulations, the price being having first-handedly computed the costly elementary kernels. Limits of the emulator are also explored, by introducing the concept of over-training, which is exactly a consequence of the aproximativeness of a PGCM computation. Eventually this thesis demonstrates the possibility to emulate millions of PGCM computations with an error on collective spectroscopy less than 3%, and with a low numerical cost fraction of 1% of the million PGCM calculations cost.
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 logarithm\mathrm{logarithm}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.