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arXiv:1910.13454

Multi-Layer Restricted Boltzmann Machine Representation of 1D Quantum Many-Body Wave Functions

Huan He, Yunqin Zheng, Bogdan Andrei Bernevig, Germán Sierra

Published Oct 29, 2019• 2 citations
arXivHTMLPDF

Abstract

We consider representing two classes of 1D quantum wave functions of spin systems, including the AKLT and CFT correlator wave functions, in terms of multi-layer restricted Boltzmann machines. In our prescription, the AKLT wave function can be exactly represented by a 2-layer restricted Boltzmann machine with five hidden spins per visible spin. The construction can be generalized to prove that any MPS wave function on $N$ unit cells with finite bond dimension can be approximated by a 2-layer restricted Boltzmann machine with $\mathcal{O}(N)$ hidden spins within an error which scales linearly with $N$. The Haldane-Shastry wave function or a chiral boson CFT correlator wave function, as any Jastrow type of wave functions, can be exactly written as a 1-layer Boltzmann machine with $\mathcal{O}(N^2)$ hidden spins and $N$ visible spins. Applying the cumulant expansion, we further find that the chiral boson CFT correlator wave function (with small vertex operator conformal dimension $α$, i.e., $α<0.1$) can be approximated, within 99.9\% accuracy up to 22 visible spins, by a 1-layer RBM with $\mathcal{O}(N)$ hidden spins. The cumulant expansion also leads us to a physically inspiring result in which the hidden spins of the restricted Boltzmann machine can be interpreted as the conformal tower of the chiral boson CFT on the cylinder.

DOI
10.48550/arxiv.1910.13454

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We consider representing two classes of 1D quantum wave functions of spin systems, including the AKLT and CFT correlator wave functions, in terms of multi-layer restricted Boltzmann machines. In our prescription, the AKLT wave function can be exactly represented by a 2-layer restricted Boltzmann machine with five hidden spins per visible spin. The construction can be generalized to prove that any MPS wave function on NNN unit cells with finite bond dimension can be approximated by a 2-layer restricted Boltzmann machine with O(N)\mathcal{O}(N)O(N) hidden spins within an error which scales linearly with NNN. The Haldane-Shastry wave function or a chiral boson CFT correlator wave function, as any Jastrow type of wave functions, can be exactly written as a 1-layer Boltzmann machine with O(N2)\mathcal{O}(N^2)O(N2) hidden spins and NNN visible spins. Applying the cumulant expansion, we further find that the chiral boson CFT correlator wave function (with small vertex operator conformal dimension ααα, i.e., α<0.1α<0.1α<0.1) can be approximated, within 99.9% accuracy up to 22 visible spins, by a 1-layer RBM with O(N)\mathcal{O}(N)O(N) hidden spins. The cumulant expansion also leads us to a physically inspiring result in which the hidden spins of the restricted Boltzmann machine can be interpreted as the conformal tower of the chiral boson CFT on the cylinder.
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.
Accuracy of restricted Boltzmann machines for the one-dimensional J1−J2J_1-J_2J1​−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 J1−J2J_1-J_2J1​−J2​ antiferromagnetic Heisenberg model in one spatial dimension. The ground state of this model has a non-trivial sign structure, especially for J2/J1>0.5J_2/J_1>0.5J2​/J1​>0.5, forcing us to work with complex-valued RBMs. Two variational Ans\"atze 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 kkk. 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 Ans\"atze is very flexible and describes both gapless and gapped ground states, also capturing the incommensurate spin-spin correlations and low-energy spectrum for J2/J1>0.5J_2/J_1>0.5J2​/J1​>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 of the computational cost with the system size.
Accuracy of restricted Boltzmann machines for the one-dimensional J1−J2J_1-J_2J1​−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.
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.
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.
Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.