Abstract Projected variational wavefunctions such as the Gutzwiller, many-body correlator and Jastrow ansatzes have provided crucial insight into the nature of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two or more spatial dimensions. However, these ansatzes have no obvious tractable and systematic way of being improved. A promising alternative is to use Neural-network quantum states (NQS) based on Restricted Boltzmann Machines (RBMs). With binary visible and hidden units NQS have proven to be a highly effective at describing quantum states of interacting spin- 1 2 lattice systems. The application of NQS to bosonic systems has so far been based on one-hot encoding from machine learning where the multi-valued site occupation is distributed across several binary-valued visible units of an RBM. Compared to spin- 1 2 systems one-hot encoding greatly increases the number of variational parameters whilst also making their physical interpretation opaque. Here we revisit the construction of NQS for bosonic systems by reformulating a one-hot encoded RBM into a correlation operator applied to a reference state, analogous to the structure of the projected variational ansatzes. In this form we then propose a number of specialisations of the RBM motivated by the physics of the BHM and the ability to capture exactly the projected variational ansatzes. We analyse in detail the variational performance of these new RBM variants for a 10 × 10 BHM, using both a standard Bose condensate state and a pre-optimised Jastrow + many-body correlator state as the reference state of the calculation. Several of our new ansatzes give robust results as nearly good as one-hot encoding across the regimes of the BHM, but at a substantially reduced cost. Such specialised NQS are thus primed tackle bosonic lattice problems beyond the accuracy of classic variational wavefunctions.
Abstract Projected variational wavefunctions such as the Gutzwiller, many-body correlator and Jastrow ansatzes have provided crucial insight into the nature of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two or more spatial dimensions. However, these ansatzes have no obvious tractable and systematic way of being improved. A promising alternative is to use Neural-network quantum states (NQS) based on Restricted Boltzmann Machines (RBMs). With binary visible and hidden units NQS have proven to be a highly effective at describing quantum states of interacting spin- 1 2 lattice systems. The application of NQS to bosonic systems has so far been based on one-hot encoding from machine learning where the multi-valued site occupation is distributed across several binary-valued visible units of an RBM. Compared to spin- 1 2 systems one-hot encoding greatly increases the number of variational parameters whilst also making their physical interpretation opaque. Here we revisit the construction of NQS for bosonic systems by reformulating a one-hot encoded RBM into a correlation operator applied to a reference state, analogous to the structure of the projected variational ansatzes. In this form we then propose a number of specialisations of the RBM motivated by the physics of the BHM and the ability to capture exactly the projected variational ansatzes. We analyse in detail the variational performance of these new RBM variants for a 10 × 10 BHM, using both a standard Bose condensate state and a pre-optimised Jastrow + many-body correlator state as the reference state of the calculation. Several of our new ansatzes give robust results as nearly good as one-hot encoding across the regimes of the BHM, but at a substantially reduced cost. Such specialised NQS are thus primed tackle bosonic lattice problems beyond the accuracy of classic variational wavefunctions.
Neural network quantum states (NQS) have been widely applied to spin-1/2 systems, where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here, we propose a more direct generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach, we present variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Furthermore, we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the xyz spin-1 basis using only M=2N hidden units, compared to M∼O(N2) required in the Sz basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the xyz basis with only M=N hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.
Recently, the use of neural quantum states for describing the ground state of many- and few-body problems has been gaining popularity because of their high expressivity and ability to handle intractably large Hilbert spaces. In particular, methods based on variational Monte Carlo have proven to be successful in describing the physics of bosonic systems such as the Bose-Hubbard (BH) model. However, this technique has not been systematically tested on the parameter space of the BH model, particularly at the boundary between the Mott insulator and superfluid phases. In this work, we evaluate the capabilities of variational Monte Carlo with a trial wavefunction given by a Restricted Boltzmann Machine to reproduce the quantum ground state of the BH model on several points of its parameter space. To benchmark the technique, we compare its results to the ground state found through exact diagonalization for small one-dimensional chains. In general, we find that the learned ground state correctly estimates many observables, reproducing to a high degree the phase diagram for the first Mott lobe and part of the second one. However, we find that the technique is challenged whenever the system transitions between excitation manifolds, as the ground state is not learned correctly at these boundaries. We improve the quality of the results produced by the technique by proposing a method to discard noisy probabilities learned in the ground state.
In recent years, neural quantum states have emerged as a powerful variational approach, achieving state-of-the-art accuracy when representing the ground-state wave function of a great variety of quantum many-body systems, including spin lattices, interacting fermions or continuous-variable systems. However, accurate neural representations of the ground state of interacting bosons on a lattice have remained elusive. We introduce a neural backflow Jastrow Ansatz, in which occupation factors are dressed with translationally equivariant many-body features generated by a deep neural network. We show that this neural quantum state is able to faithfully represent the ground state of the 2D Bose-Hubbard Hamiltonian across all values of the interaction strength. We scale our simulations to lattices of dimension up to 20 × 20 while achieving the best variational energies reported for this model. This enables us to investigate the scaling of the entanglement entropy across the superfluid-to-Mott quantum phase transition, a quantity hard to extract with non-variational approaches.
Abstract Motivated by recent advances in the representation of ground state wavefunctions of quantum many-body systems using restricted Boltzmann machines as variational ansatz, we utilize an open-source platform for constructing such ansatz called NetKet to explore the extent of applicability of restricted Boltzmann machines to bosonic lattice models. Within NetKet, we design and train these machines for the one-dimensional Bose-Hubbard model through a Monte Carlo sampling of the Fock space. We vary parameters such as the strength of the onsite repulsion, the chemical potential, the system size and the maximum site occupancy and use converged equations of state to identify phase boundaries between the Mott insulating and superfluid phases. We compare the average density and the energy to results from exact diagonalization and map out the ground state phase diagram, which agrees qualitatively with previous finding obtained through conventional means.
Abstract Neural-network quantum states (NQSs) have become a powerful tool in many-body physics. Of the numerous possible architectures in which neural-networks can encode amplitudes of quantum states the simplicity of the complex restricted Boltzmann machine (RBM) has proven especially useful for both numerical and analytical studies. In particular devising exact NQS representations for important classes of states, like Jastrow and stabilizer states, has provided useful clues into the strengths and limitations of the RBM based NQS. However, current constructions for a system of N spins generate NQS with M ∼ O(N 2) hidden units that are very sparsely connected. This makes them rather atypical NQS compared to those commonly generated by numerical optimisation. Here we focus on compact NQS, denoting NQS with a hidden unit density α = M/N ⩽ 1 but with system-extensive hidden-visible unit connectivity. By unifying Jastrow and stabilizer states we introduce a new exact representation that requires at most M = N − 1 hidden units, illustrating how highly expressive α ⩽ 1 can be. Owing to their structural similarity to numerical NQS solutions our result provides useful insights and could pave the way for more families of quantum states to be represented exactly by compact NQS.
Accuracy of restricted Boltzmann machines for the one-dimensional J1−J2 Heisenberg model
Neural networks have been recently proposed as variational wave functions for quantum many-body systems [G. Carleo and M. Troyer, Science 355, 602 (2017)]. In this work, we focus on a specific architecture, known as restricted Boltzmann machine (RBM), and analyse its accuracy for the spin-1/2 J_1-J_2 J 1 − J 2 antiferromagnetic Heisenberg model in one spatial dimension. The ground state of this model has a non-trivial sign structure, especially for J_2/J_1>0.5 J 2 / J 1 > 0.5 , forcing us to work with complex-valued RBMs. Two variational Ansätze are discussed: one defined through a fully complex RBM, and one in which two different real-valued networks are used to approximate modulus and phase of the wave function. In both cases, translational invariance is imposed by considering linear combinations of RBMs, giving access also to the lowest-energy excitations at fixed momentum k. We perform a systematic study on small clusters to evaluate the accuracy of these wave functions in comparison to exact results, providing evidence for the supremacy of the fully complex RBM. Our calculations show that this kind of Ansätze is very flexible and describes both gapless and gapped ground states, also capturing the incommensurate spin-spin correlations and low-energy spectrum for J_2/J_1>0.5 J 2 / J 1 > 0.5 . The RBM results are also compared to the ones obtained with Gutzwiller-projected fermionic states, often employed to describe quantum spin models [F. Ferrari, A. Parola, S. Sorella and F. Becca, Phys. Rev. B 97, 235103 (2018)]. Contrary to the latter class of variational states, the fully-connected structure of RBMs hampers the transferability of the wave function from small to large clusters, implying an increase in the computational cost with the system size.