Representation Theorem for Matrix Product States

Abstract

In this work, we investigate the universal representation capacity of the Matrix Product States (MPS) from the perspective of boolean functions and continuous functions. We show that MPS can accurately realize arbitrary boolean functions by providing a construction method of the corresponding MPS structure for an arbitrarily given boolean gate. Moreover, we prove that the function space of MPS with the scale-invariant sigmoidal activation is dense in the space of continuous functions defined on a compact subspace of the $n$-dimensional real coordinate space ${\mathbb{R}}^n$. We study the relation between MPS and neural networks and show that the MPS with a scale-invariant sigmoidal function is equivalent to a one-hidden-layer neural network equipped with a kernel function. We construct the equivalent neural networks for several specific MPS models and show that non-linear kernels such as the polynomial kernel which introduces the couplings between different components of the input into the model appear naturally in the equivalent neural networks. At last, we discuss the realization of the Gaussian Process (GP) with infinitely wide MPS by studying their equivalent neural networks.

arXiv comment19 pages

References in corpus (19)

• The density-matrix renormalization group in the age of matrix product states
• Efficient classical simulation of slightly entangled quantum computations
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• DMRG and periodic boundary conditions: a quantum information perspective
• Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
• Renormalization and tensor product states in spin chains and lattices
• Tensor network states and geometry
• Equivalence of restricted Boltzmann machines and tensor network states
• Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
• Approximating quantum many-body wave-functions using artificial neural networks
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Neural network representation of tensor network and chiral states
• Categorical Tensor Network States
• Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
• Language Design as Information Renormalization
• Accelerating Block Coordinate Descent for Nonnegative Tensor Factorization

Cited by in corpus (1)

• The Bayesian Method of Tensor Networks