Seeding neural network quantum states with tensor network states

TL;DR

This work solves the challenge of efficiently seeding neural-network quantum states (NNQS) by converting matrix product states (MPS) into restricted Boltzmann machine (RBM) wave functions through a canonical polyadic (CP) decomposition. The CP-decomposed MPS yields an RBM with a single multinomial hidden unit, enabling initial states that are close to ground states and can be refined via variational Monte Carlo (VMC) with stochastic reconfiguration. The method scales polynomially with system size and variational parameters, with initial-state errors dropping roughly as 1/R^2 as CP rank R increases, and empirical evidence shows R = O(n) suffices for size-independent accuracy. The approach is demonstrated on the one-dimensional transverse-field Ising model, where open-boundary CP-derived states effectively seed periodic-boundary VMC and the final energies approach the true ground state upon optimization, highlighting potential applicability to more complex systems with intricate nodal structures. The work also discusses extensions to higher dimensions, translationally invariant tensor networks, and combinations with binomial RBMs to enhance expressivity.

Abstract

We find an efficient approach to approximately convert matrix product states (MPSs) into restricted Boltzmann machine wave functions consisting of a multinomial hidden unit through a canonical polyadic (CP) decomposition of the MPSs. This method allows us to generate well-behaved initial neural network quantum states for quantum many-body ground-state calculations in polynomial time of the number of variational parameters and systematically shorten the distance between the initial states and the ground states while increasing the rank of the CP decomposition. We demonstrate the efficiency of our method by taking the transverse-field Ising model as an example and discuss possible applications of our method to more general quantum many-body systems in which the ground-state wave functions possess complex nodal structures.

Figures (15)

• Figure 1: Schematic figure of the conversion from MPSs to the RBM wave function through the CP decomposition. (a) MPS representation of tensor $T$. (b) CP decomposition of tensor $T$, which is efficiently computed from the MPS representation. (c) RBM wave function with a multinomial hidden unit, which is equivalent to the CP decomposed tensor.
• Figure 2: (a) CP decomposition of a tensor. A black dot represent the Kronecker delta tensor $\delta$. Each factor $A^{(i)}$ is a $d_i \times R$ matrix with $R$ being the rank of the CP decomposition. (b) MPS representation of the CP decomposed tensor. The black dot in (a) can be rewritten as a product of Kronecker delta tensors with smaller orders. The product of the Kronecker delta tensor and factor $A^{(i)}$ can be rewritten as MPS $F^{(i)}$.
• Figure 3: Tensors needed for the ALS method for the CP decomposition of MPSs. We show an example of a tensor with the order $n=5$ when updating the $j=3$ element. (a) MPS representation of tensor $X$. (b) Matrix $Y^{(j)}$ obtained by the product between the matricized tensor, written in MPS form $G^{(i)}$, and the matrix obtained by the Khatri-Rao product, written by the Kronecker delta tensors $\delta$ and matrix $A^{(i\not=j)}$. Matrix $Y^{(j)}$ is used to update matrix $A^{(j)}$ during the ALS method. (c) Left matrix $L^{(j)}$ at the $j$th step of the ALS method. (d) Right matrix $R^{(j)}$ at the $j$th step of the ALS method. Matrices $L^{(j)}$ and $R^{(j)}$ are used to construct matrix $Y^{(j)}$.
• Figure 4: Infidelity as a function of iterations for $h/J = 2$ and $n=16$. We show the results for ranks ranging from $R=8$ to $R=64$. Different colors represent different ranks of the CP decomposition, whereas different symbols represent different initial factors for the CP decomposition.
• Figure 5: (a) Infidelity and energy difference as a function of the rank $R$ of the CP decomposition for $h/J = 2$ and $n=16$. Different symbols represent different initial factors for the CP decomposition. Open symbols are for the infidelity and filled symbols are for the energy difference. (b) Same as panel (a) but shown on a logarithmic scale. We plot the line proportional to $R^{-2}$ as a reference. Note that, hereafter, all energy values are expressed in the units of $J$.