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 sharing nature of the GNA also leads to transferability across different system sizes and geometries.
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.
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…
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…
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 …
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…
We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor net…
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…
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.
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) …
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 …
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…
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…
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…
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…
Variational methods have proven to be excellent tools to approximate ground states of complex many body Hamiltonians. Generic tools like neural networks are extremely powerful, but their parameters are not necessarily physically motivated. Thus, an efficient parametrization of the wave-function can become challenging. In this letter we introduce a neural-network based variational ansatz that retains the flexibility of these generic methods while allowing for a tunability with respect to the relevant correlations governing the physics of the system. We illustrate the success of this approach on…
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 …
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.
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 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 new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a Variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any "sign" instability. We show results for various two dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatzs is offered. In particular, our numerical estimates are in …
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…
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 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)…
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…
