A new wavefunction which improves the Gutzwiller-type local ansatz method has been proposed to describe the correlated electron system. The ground-state energy, double occupation number, momentum distribution function, and quasiparticle weight have been calculated for the half-filled band Hubbard model in infinite dimensions. It is shown that the new wavefunction improves the local-ansatz approach (LA) proposed by Stollhoff and Fulde. Especially, calculated momentum distribution functions show a reasonable momentum dependence. The result qualitatively differs from those obtained by the LA and the Gutzwiller wavefunction. Furthermore, the present approach combined with the projection operator method CPA is shown to describe quantitatively the excitation spectra in the insulator regime as well as the critical Coulomb interactions for a gap formation in infinite dimensions.
We propose a one-parameter variational ansatz to describe the tunneling-driven Abelian to non-Abelian transition in bosonic $ν=1/2+1/2$ fractional quantum Hall bilayers. This ansatz, based on exact matrix product states, captures the low-energy physics all along the transition and allows to probe its characteristic features. The transition is continuous, characterized by the decoupling of antisymmetric degrees of freedom. We futhermore determine the tunneling strength above which non-Abelian statistics should be observed experimentally. Finally, we propose to engineer the inter-layer tunneling…
The momentum-dependent local-ansatz (MLA) wavefunction describes well correlated electrons in solids in both the weak and strong interaction regimes. In order to apply the theory to the realistic system, we have extended the MLA to the first-principles version using the tight-binding LDA+U Hamiltonian. We demonstrate for the paramagnetic Fe that the first-principles MLA can describe a reasonable correlation energy gain and suppression of charge fluctuations due to electron correlations. Furthermore, we show that the MLA yields a distinct momentum dependence of the momentum distribution, and th…
A fast time propagation method for nonequilibrium Green's functions based on the generalized Kadanoff--Baym Ansatz (GKBA) is applied to a lattice system with a symmetry-broken equilibrium phase, namely an excitonic insulator. The adiabatic preparation of a correlated symmetry-broken initial state from a Hartree--Fock wave function within GKBA is assessed by comparing with a solution of the imaginary-time Dyson equation. We find that it is possible to reach a symmetry-broken correlated initial state with nonzero excitonic order parameter by the adiabatic switching procedure. We discuss under wh…
The Bethe ansatz equations are presented for Bariev's correlated electron chain with boundaries. This is achieved by using the coordinate space Bethe ansatz method.
The wavefunction method provides us with a useful tool to describe electron correlations in solids at the ground state. In this paper we review the recent development of the momentum-dependent local ansatz wavefunction (MLA). It is constructed by taking into account two-particle excited states projected onto the local orbitals, and the momentum-dependent amplitudes of these states are chosen as variational parameters. The MLA describes accurately correlated electron states from the weak to the intermediate Coulomb interaction regime in infinite dimensions, and works well even in the strongly c…
Momentum dependent local-ansatz wavefunction approach (MLA) to the correlated electron systems in solids has been further developed to solve best a self-consistent equation for variational parameters at non half-filling. With use of the improved variational scheme we performed the numerical calculations for the non-half-filled band Hubbard model on the hypercubic lattice in infinite dimensions. We verified that the self-consistent scheme significantly improves the correlation energy and the momentum distribution as compared with the original scheme in the MLA. We also demonstrate that the theo…
Deep neural networks have been shown as a potentially powerful ansatz in variational Monte Carlo for solving quantum many-body problems. We propose two improvements in this direction. The first is graph neural ansatz (GNA), which is a variational wavefunction universal to arbitrary geometry. GNA results in accurate ground-state energies on 2D Kagome lattices, triangular lattices, and randomly connected graphs. Secondly, we design a distributed workflow on multiple accelerators to scale up the computation. We compute Kagome lattices with sizes up to 432 sites on 128 TPU cores. The parameter sha…
A variational ansatz for momentum eigenstates of translation invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically non-trivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-1/2 XXZ antiferromagnet and the spin-1 Heisenberg antiferromagnet and we obtain surprisingly accurate results.
The question about the existence of Bogoliubov's quasiparticles in the BCS wave functions underneath Gutzwiller's projection is of importance to strongly correlated systems. We develop a method to examine the two-particle excitations of Gutzwiller-projected BCS wave functions by using the variational Monte Carlo approach. We find that the exact Gutzwiller-projected quasiparticle (GQP) dispersions are quantitatively reproduced by the Gutzwiller-projected Bogoliubov quasiparticles (GBQP) except the regions where d-wave Cooper pairing is strong. Since GQP still shows higher energy than GBQP near …
We present a theoretical study on the response properties to an external electric field of strongly correlated one-dimensional metals. Our investigation is based on the recently developed Bethe-Ansatz local density approximation (BALDA) to the density functional theory formulation of the Hubbard model. This is capable of describing both Luttinger liquid and Mott-insulator correlations. The BALDA calculated values for the static linear polarizability are compared with those obtained by numerically accurate methods, such as exact (Lanczos) diagonalization and the density matrix renormalization g…
We propose a variational approximation to the ground state energy of a one-dimensional gas of interacting bosons on the continuum based on the Bethe Ansatz ground state wavefunction of the Lieb-Liniger model. We apply our variational approximation to a gas of dipolar bosons in the single mode approximation and obtain its ground state energy per unit length. This allows for the calculation of the Tomonaga-Luttinger exponent as a function of density and the determination of the structure factor at small momenta. Moreover, in the case of attractive dipolar interaction, an instability is predicted…
We have extended the momentum-dependent local-ansatz (MLA) wavefunction method to the first-principles version using the tight-binding LDA+U Hamiltonian for the description of correlated electrons in the real system. The MLA reduces to the Rayleigh-Schrödinger perturbation theory in the weak correlation limit, and describes quantitatively the ground state and related low-energy excitations in solids. The theory has been applied to the paramagnetic Fe. The role of electron correlations on the energy, charge fluctuations, amplitude of local moment, momentum distribution functions, as well as the…
Following the Dirac-Frenkel time-dependent variational principle, dynamics of a one-dimensional Holstein polaron is probed by employing the Davydov D2 Ansatz with two sets of variational parameters, one for each constituting particle in the exciton-phonon system, and a simplified variant of the Davydov D1 Ansatz, also known as the $\tilde{D}$ Ansatz, with an additional set of phonon displacement parameters. A close examination of variational outputs from the two trial states reveals fine details of the polaron structure and intricacies of dynamic exciton-phonon interactions. Superradiance cohe…
We describe how to implement the time-dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems. This is achieved by confining the nonuniformity to a (dynamically growable) finite region with fixed boundary conditions. The suppression of unphysical quasiparticle reflections from the boundary of the nonuniform region is also discussed. Using this algorithm we study the dynamics of localized excitations in infinite systems, which we illustrate in the case of the spin-1 anti-ferromagnetic Heisenberg model and the $ϕ^4$ model.
We introduce a variational wavefunction for many-body ground states that involves imaginary time evolution with two different Hamiltonians in an alternating fashion with variable time intervals. We successfully apply the ansatz on the one- and two-dimensional transverse-field Ising model and systematically study its scaling for the one-dimensional model at criticality. We find the total imaginary time required scales logarithmically with system size, in contrast to the linear scaling in conventional Quantum Monte Carlo. We suggest this is due to unique dynamics permitted by alternating imagina…
The electronic band structure of many compounds, e.g., carbon-based structures, can exhibit essentially no dispersion. Models of electrons in flat-band lattices define non-perturbative strongly correlated problems by default. We construct a set of Jastrow-correlated ansatz wavefunctions to capture the low energy physics of interacting particles in flat bands. We test the ansatz in a simple Coulomb model of spinless electrons in a honeycomb ribbon. We find that the wavefunction accurately captures the ground state in a transition from a crystal to a uniform quantum liquid.
Many simulation tasks require that one first prepare a system's Gibbs state. We present a family of quantum circuits for variational preparation of thermal Gibbs states on a quantum computer; we call them the thermal multi-scale entanglement renormalization ansatz (TMERA). TMERA circuits transform input qubits to wavepacket modes localized to varying length scales and approximate a systems Gibbs state as a mixed state of these modes. The TMERA is a based on the deep multi-scale entanglement renormalization ansatz (DMERA); a TMERA modifies a ground-state DMERA circuit by preparing each input qu…
Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here we put forward a variational Monte Carlo approach for the multi-scale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with non-unitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just log(L) …
Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the Variational Quantum Eigensolver (VQE) has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this manuscript, we try to answer the question: how much of the underlying physics of a 1D Hubbard chain is described …
We present a comparison between a number of recently introduced low-memory wave function optimization methods for variational Monte Carlo in which we find that first and second derivative methods possess strongly complementary relative advantages. While we find that low-memory variants of the linear method are vastly more efficient at bringing wave functions with disparate types of nonlinear parameters to the vicinity of the energy minimum, accelerated descent approaches are then able to locate the precise minimum with less bias and lower statistical uncertainty. By constructing a simple hybri…
We present a variational treatment of the ground state of the 2-leg t-J ladder, which combines the dimer and the hard-core boson models into one effective model. This model allows us to study the local structure of the hole pairs as a function of doping. A second order recursion relation is used to generate the variational wave function, which substantially simplifies the computations. We obtain good agreement with numerical density matrix renormalization group results for the ground state energy in the strong coupling regime. We find that the local structure of the pairs depends upon whether …
The variational quantum eigensolver (VQE) combines classical and quantum resources in order simulate classically intractable quantum states. Amongst other variables, successful VQE depends on the choice of variational ansatz for a problem Hamiltonian. We give a detailed description of a quantum circuit version of the 2D isometric tensor network (isoTNS) ansatz which we call qisoTNS. We benchmark the performance of qisoTNS on two different 2D spin 1/2 Hamiltonians. We find that the ansatz has several advantages. It is qubit efficient with the number of qubits allowing for access to some exponen…
Variational quantum algorithms (VQAs) are promising methods to demonstrate quantum advantage on near-term devices as the required resources are divided between a quantum simulator and a classical optimizer. As such, designing a VQA which is resource-efficient and robust against noise is a key factor to achieve potential advantage with the existing noisy quantum simulators. It turns out that a fixed VQA circuit design, such as the widely-used hardware efficient ansatz, is not necessarily robust against imperfections. In this work, we propose a genome-length-adjustable evolutionary algorithm to …
An adaptive variational quantum imaginary time evolution (AVQITE) approach is introduced that yields efficient representations of ground states for interacting Hamiltonians on near-term quantum computers. It is based on McLachlan's variational principle applied to imaginary time evolution of variational wave functions. The variational parameters evolve deterministically according to equations of motions that minimize the difference to the exact imaginary time evolution, which is quantified by the McLachlan distance. Rather than working with a fixed variational ansatz, where the McLachlan dista…
Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail. In this paper, we focus on a special family of quantum circuits called the Hamiltonian Variational Ansatz (HVA), which takes inspiration from the quantum approximation optimization algorithm and adiabatic quantum computation. Through the study of its entanglement spectrum and energy gradient statistics, we f…
We introduce a variational ansatz based on Gaussian states for (1+1)-dimensional lattice gauge models. To this end we identify a set of unitary transformations which decouple the gauge degrees of freedom from the matter fields. Using our ansatz, we study static aspects as well as real-time dynamics of string breaking in two (1+1)-dimensional theories, namely QED and two-color QCD. We show that our ansatz captures the relevant features and is in excellent agreement with data from numerical calculations with tensor networks.
We investigate the spectral properties of an open interacting system by solving the Generalized Kadanoff-Baym Ansatz (GKBA) master equation for the single-particle density matrix, namely the time-diagonal lesser Green's function. To benchmark its validity, we compare the solution obtained within the GKBA with the solution of the Dyson equation (equivalently the full Kadanoff-Baym equations). In both approaches, we treat the interaction within the self-consistent second-order Born approximation, whereas the GKBA still retains the retarded propagator calculated at the Hartree-Fock level. We cons…
A variational Monte Carlo algorithm for lattice gauge theories with continuous gauge groups: a study of (2+1)-dimensional compact QED with dynamical fermions at finite density
We propose here a single Pfaffian correlated variational ansatz, that dramatically improves the accuracy with respect to the single determinant one, while remaining at a similar computational cost. A much larger correlation energy is indeed determined by the most general two electron pairing function, including both singlet and triplet channels, combined with a many-body Jastrow factor, including all possible spin-spin spin-density and density-density terms. The main technical ingredient to exploit this accuracy is the use of the Pfaffian for antisymmetrizing an highly correlated pairing funct…
Representing a target quantum state by a compact, efficient variational wave-function is an important approach to the quantum many-body problem. In this approach, the main challenges include the design of a suitable variational ansatz and optimization of its parameters. In this work, we address both of these challenges. First, we define the variational class of Computational Graph States (CGS) which gives a uniform framework for describing all computable variational ansatz. Secondly, we develop a novel optimization scheme, supervised wave-function optimization (SWO), which systematically impro…
We provide a detailed formulation of the recently proposed variational approach [Y. Ashida et al., Phys. Rev. Lett. 121, 026805 (2018)] to study ground-state properties and out-of-equilibrium dynamics for generic quantum spin-impurity systems. Motivated by the original ideas by Tomonaga, Lee, Low, and Pines, we construct a canonical transformation that completely decouples the impurity from the bath degrees of freedom. By combining this transformation with a Gaussian ansatz for the fermionic bath, we obtain a family of variational many-body states that can efficiently encode the strong entangl…
We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a Noisy, Intermediate Scale Quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the Variational Quantum Eigensolver algorithm. This iterative scheme, dubbed Natural Orbitalization (NOization)…
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that instead of fixing an ansatz upfront, this algorithm grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an a…
We introduce a method to investigate the static and dynamic properties of both Abelian and non-Abelian lattice gauge models in 1+1 dimensions. Specifically, we identify a set of transformations that disentangle different degrees of freedom, and apply a simple Gaussian variational ansatz to the resulting Hamiltonian. To demonstrate the suitability of the method, we analyze both static and dynamic aspects of string breaking for the U(1) and SU(2) gauge models. We benchmark our results against tensor network simulations and observe excellent agreement, although the number of variational parameter…
Quantum Monte Carlo methods are accurate and promising many body techniques for electronic structure calculations which, in the last years, are encountering a growing interest thanks to their favorable scaling with the system size and their efficient parallelization, particularly suited for the modern high performance computing facilities. The ansatz of the wave function and its variational flexibility are crucial points for both the accurate description of molecular properties and the capabilities of the method to tackle large systems. In this paper, we extensively analyze, using different va…
