Lipkin Model Research Articles

Extended Lipkin model: Proposal for implementation in a quantum platform and machine learning analysis of its phase diagram

Extended Lipkin model: Proposal for implementation in a quantum platform and machine learning analysis of its phase diagram

Quantum Mixtures and Information Loss in Many-Body Systems

In our study, we investigate the phenomenon of information loss, as measured by the Kullback–Leibler divergence, in a many-fermion system, such as the Lipkin model. Information loss is introduced as the number N of particles increases, particularly when the system is in a mixed state. We find that there is a significant loss of information under these conditions. However, we observe that this loss nearly disappears when the system is in a pure state. Our analysis employs tools from information theory to quantify and understand these effects.

Solving the Lipkin model using quantum computers with two qubits only with a hybrid quantum-classical technique based on the generator coordinate method

Solving the Lipkin model using quantum computers with two qubits only with a hybrid quantum-classical technique based on the generator coordinate method

Simulating excited states of the Lipkin model on a quantum computer

We simulate the excited states of the Lipkin model using the recently proposed Quantum Equation of Motion (qEOM) method. The qEOM generalizes the EOM on classical computers and gives access to collective excitations based on quasi-boson operators $\hat{O}^\dagger_n(\alpha)$ of increasing configuration complexity $\alpha$. We show, in particular, that the accuracy strongly depends on the fermion to qubit encoding. Standard encoding leads to large errors, but the use of symmetries and the Gray code reduces the quantum resources and improves significantly the results on current noisy quantum devices. With this encoding scheme, we use IBM quantum machines to compute the energy spectrum for a system of $N=2, 3$ and $4$ particles and compare the accuracy against the exact solution. We found that the results of the approach with $\alpha = 2$, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with $\alpha = 1$, which corresponds to the random phase approximation (RPA), but the SRPA is more amenable to noise for large coupling strengths. Thus, the proposed scheme shows potential for achieving higher spectroscopic accuracy by implementations with higher configuration complexity, if a proper error mitigation method is applied.

Large cumulant eigenvalue as a signature of exciton condensation

The Bose-Einstein condensation of excitons into a single quantum state is known as exciton condensation. Exciton condensation, which potentially supports the frictionless flow of energy, has recently been realized in graphene bilayers and van der Waals heterostructures. Here we show that exciton condensates can be predicted from a combination of reduced density matrix theory and cumulant theory. We show that exciton condensation occurs if and only if there exists a large eigenvalue in the cumulant part of the particle-hole reduced density matrix. In the thermodynamic limit we show that the large eigenvalue is bounded from above by the number of excitons. In contrast to the eigenvalues of the particle-hole matrix, the large eigenvalue of the cumulant matrix has the advantage of providing a size-extensive measure of the extent of condensation. Here we apply this signature to predict exciton condensation in both the Lipkin model and molecular stacks of benzene. The computational signature has applications to the prediction of exciton condensation in both molecules and materials.

Analysis of quantum correlations within the ground state of a three-level Lipkin model

The performance of beyond mean field methods in solving the quantum many body problem for fermions is usually characterized by the correlation energy measured with respect to the underlying mean field value. In this paper we address the issue of characterizing the amount of correlations associated to different approximations from a quantum information perspective. With this goal in mind, we analyze the traditional Hartree-Fock (HF) method with spontaneous symmetry breaking, the HF with symmetry restoration and the generator coordinate method (GCM) in a exactly solvable fermion model known as the three-level Lipkin model. To characterize correlations including entanglement and beyond we use the quantum discord between different partition orbitals. We find that for physically motivated partitions, the quantum discord of the exact ground state is reasonably well reproduced by the different approximations. However, other partitions create "fake quantum correlations" in order to capture quantum correlations corresponding to partitions for which the Hartree-Fock solution fails. Those are removed and redistributed through a symmetry restoration process.

A possible trial beyond the conventional random phase approximation for the su(2)-Lipkin model including the case of nonclosed shell system

Concerning the [Formula: see text]-Lipkin model, the calculation of the excitation energy to the first excited state gives rise to the following fact: The two results based on the exact treatment and the conventional random phase approximation (RPA) are in unexpected disagreement. In order to remove this discrepancy, the conventional RPA is renewed. With the aim of this renewal, the terms, which cannot be dealt with in the conventional one, are estimated. The basic framework of this new form is not reorganized from that of the conventional one. In addition, the [Formula: see text]-Lipkin model itself is also modified so as to be applicable to the case with nonclosed shell system. To this modification, one more [Formula: see text]-algebra, which has been already proposed by the present authors, is applied. Through the use of this algebra, the [Formula: see text]-Lipkin model is applicable to the case with any total fermion number permitted in the model.

Quantum computing for the Lipkin model with unitary coupled cluster and structure learning ansatz * *A.C. appreciates the financial support of Advanced Leading Graduate Course for Photon Science, the University of Tokyo. H.L. acknowledges the JSPS Grant-in-Aid for Early-Career Scientists (18K13549) and the JSPS Grant-in-Aid for Scientific Research (S) (20H05648)

We report a benchmark calculation for the Lipkin model in nuclear physics with a variational quantum eigensolver in quantum computing. Special attention is paid to the unitary coupled cluster (UCC) ansatz and structure learning (SL) ansatz for the trial wave function. Calculations with both the UCC and SL ansatz can reproduce the ground-state energy well; however, it is found that the calculation with the SL ansatz performs better than that with the UCC ansatz, and the SL ansatz has even fewer quantum gates than the UCC ansatz.

Symmetry projection to coupled-cluster singles plus doubles wave function through the Monte Carlo method

A new method for calculating the symmetry-projected energy of coupled-cluster singles and doubles (CCSD) wave function through the Monte Carlo method is proposed. We present benchmark calculations in considering the three-level Lipkin model which is a simple and minimal model with two phases: spherical and deformed ones. It is demonstrated that this new method gives good ground state energy and low-lying spectra.

Lipkin model on a quantum computer

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.

The Hellmann–Feynman theorem at finite temperature

We present a simple derivation of the Hellmann–Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples, which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model, and the Lipkin model. We show that the Hellmann–Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free energy is available, but there is no direct access to the thermal average of the operators themselves.

Applications of Time-Dependent Density-Matrix Approach

The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of $^{40}$Ca and $^{48}$Ca and the other the fusion reactions of $^{16}$O + $^{16}$O.

Probing an excited-state quantum phase transition in a quantum many-body system via an out-of-time-order correlator

As a measure of information scrambling and quantum chaos, out-of-time-ordered correlator (OTOC) plays more and more important role in many different fields of physics. In this work, we verify that the OTOC can also be used as a prober of the excited-state quantum phase transition (ESQPT) in a quantum many body system. By using the exact diagonalization method, we examine the dynamical properties of OTOC in the Lipkin model, which undergoes an ESQPT. We demonstrate that the OTOC exhibits a remarkable distinct evolution behaviors in different phases of ESQPT. Therefore, the presence of an ESQPT in the quantum many body system can be clearly signaled by the different dynamical behaviors of the OTOC. In particular, we show that the steady state value of the OTOC serves as the order parameter of the ESQPT. Our results highlight the connections between the OTOC and ESQPT, which enable one to use OTOC for experimental tests ESQPTs in quantum many body systems.

Fermionic entanglement in the Lipkin model

We examine the fermionic entanglement in the ground state of the fermionic Lipkin model and its relation with bipartite entanglement. It is first shown that the one-body entanglement entropy, which quantifies the minimum distance to a fermionic Gaussian state, behaves similarly to the mean-field order parameter and is essentially proportional to the total bipartite entanglement between the upper and lower modes, a quantity meaningful only in the fermionic realization of the model. We also analyze the entanglement of the reduced state of four single-particle modes (two up-down pairs), showing that its fermionic concurrence is strongly peaked at the phase transition and behaves differently from the corresponding up-down entanglement. We finally show that the first measures and the up-down reduced entanglement can be correctly described through a basic mean-field approach supplemented with symmetry restoration, whereas the concurrence requires at least the inclusion of RPA-type correlations for a proper prediction. Fermionic separability is also discussed.

Coupled self-consistent random-phase approximation equations for even and odd particle numbers: Tests with solvable models

Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent RPA (SCRPA) equations already known from the literature. From the equations of the odd particle number case the single particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both, even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and the 1D Hubbard model give very good results.

Truncation scheme of time-dependent density-matrix approach III

The time-dependent density-matrix theory (TDDM) where the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices is truncated by approximating a three-body density matrix with one-body and two-body density matrices is applied to the Lipkin model. It is shown that in the large $N$ limit the ground state in TDDM approaches the exact solution. Various truncation schemes for the three-body density matrix are also tested for an extended three-level Lipkin model.

Adiabatic approach to large-amplitude collective motion with the higher-order collective-coordinate operator

We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations we propose in this paper.

On the difference between variational and unitary coupled cluster theories.

There have been assertions in the literature that the variational and unitary forms of coupled cluster theory lead to the same energy functional. Numerical evidence from previous authors was inconsistent with this claim, yet the small energy differences found between the two methods and the relatively large number of variational parameters precluded an unequivocal conclusion. Using the Lipkin Hamiltonian, we here present conclusive numerical evidence that the two theories yield different energies. The ambiguities arising from the size of the cluster parameter space are absent in the Lipkin model, particularly when truncating to double excitations. We show that in the symmetry adapted basis under strong correlation, the differences between the variational and unitary models are large, whereas they yield quite similar energies in the weakly correlated regime previously explored. We also provide a qualitative argument rationalizing why these two models cannot be the same. Additionally, we study a generalized non-unitary and non-hermitian variant that contains excitation, de-excitation, and mixed operators with different amplitudes and show that it works best when compared to the traditional, variational, unitary, and extended forms of coupled cluster doubles theories.

Exceptional points near first- and second-order quantum phase transitions.

We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.

A practical scheme for constructing the minimum-weight states of the su(n)-Lipkin model in arbitrary fermion number

With the aim of performing an argument supplement to the previous paper by the present authors, in this paper, a practical scheme for constructing the minimum weight states of the su(n)-Lipkin model in arbitrary fermion number is discussed. The idea comes from the following two points : (i) consideration on the property of one-fermion transfer induced by the su(n)-generators in the Lipkin model and (ii) use of the auxiliary su(2)-algebra presented by the present authors. The form obtained under the points (i) and (ii) is simple.