Neural-Network Quantum States, String-Bond States, and Chiral Topological States

Neural-network quantum states have recently been 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, because of their nonlocal geometry, neural-network quantum states and their string-bond-state 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.

Popular Summary

Studying low-temperature physics, where quantum effects are prominent, is one of the central problems in condensed-matter physics. Theoretical works often rely on numerical simulations to describe matter in that regime. The complexity of these systems, however, makes them very difficult to solve, and one needs to engineer suitable approximations to the wave function of quantum systems. Mathematical tools known as tensor networks—which compress information of multivariable functions in a controlled way—have been extremely successful in this respect, bringing both insights over the data structure of quantum states and powerful algorithms to study them. Recently, artificial neural networks also have been suggested as a way to approximate quantum states. In this work, we build a connection between these neural-network quantum states and tensor networks, allowing us to introduce new, powerful methods to study challenging problems in condensed-matter physics.

We focus on a machine-learning model known as the restricted Boltzmann machine and provide an exact mapping to a class of tensor networks called string-bond states. This sheds light on the underlying architecture of restricted Boltzmann machines and allows us to generalize it while combining the advantages of both tensor networks and neural-network quantum states. We successfully apply these techniques to the problem of representing chiral topological states, peculiar states of matter that have eluded traditional tensor-network approaches.

These discoveries introduce an important bridge between tensor networks and artificial neural networks, highlighting both their descriptive power and limitations. We expect that these tools will have applications in both quantum many-body physics and machine learning.

Figure 1

Geometry of Ansatz wave functions: (a) Jastrow wave functions include correlations within all pairs of spins. (b) MPS in 2D cover the lattice with one snake. (c) EPS include all spin correlations within each plaquette ($2\times 2$ on the figure) and mediate correlations between distant spins through overlapping plaquettes. (d) SBS cover the lattice with many 1D strings on which the interactions within spins are captured by a MPS.

Figure 2

(a) Boltzmann machines approximate a probability distribution by the Boltzmann weights of an Ising Hamiltonian on a graph including visible units (corresponding to the spins $s_j$) and hidden units $h_i$, which are summed over. (b) Restricted Boltzmann machines (here in 2D) only include interactions between the visible and the hidden units.

Figure 3

(a) A locally connected RBM is an EPS where each plaquette encodes the local connections to a hidden unit. (b) Once expressed as a SBS, a fully connected RBM can be represented by many strings on top of each other. Enlarging the RBM by using noncommuting matrices to nonlocal SBS induces a geometry in each string.

Figure 4

Energy difference with the exact ground-state energy of a spin-1 extension of a RBM [Eq. (27)] with $D=2$ and different number of strings for the AKLT model on a spin-1 chain with eight spins. A nonlocal SBS with noncommuting matrices and one string is exact within numerical accuracy.

Figure 5

Energy of $H_l$ per site for different optimized Ansatz wave functions on a square lattice. The number of parameters ($N_p$) is modified by increasing the bond dimension $D$ (local SBS, $N_p\propto D^2$), the size of the plaquettes (EPS, $N_p\propto M_P{2}^P$, where $M_P$ is the number of plaquettes and $P$ is the number of spins in one plaquette), the number of strings $M_S$ (nonlocal SBS and dSBS, $N_p\propto M_S$), or the number of hidden units $M_h$ (RBM, $N_p\propto M_h$). (a) The $4\times 4$ lattice for which the energy difference with the exact ground-state energy is plotted. (b) The $10\times 10$ lattice for which the exact ground-state energy is unknown and the reference energy of the Laughlin state is indicated as a black line. (c) Optimization of wave functions that have been multiplied by the Laughlin wave function on a $10\times 10$ lattice. The original RBM results are indicated for reference as grey crosses.

Figure 6

Partition of the lattice used to compute the topological entanglement entropy.

Figure 7

(a) The spin-spin correlation function between one lattice site (in red) and all other spins on the lattice measured on the optimized l-RBM with lowest energy reveals the antiferromagnetic behavior of the correlations. (b) Decay of the correlations with the distance across the direction indicated in (a) as a white solid line. The error bars are within dot size, and finite-size effects can already be seen for the last point.