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SIAM Journal on Mathematics of Data Science

Representation Theorems for Matrix Product States

Erdong Guo, David Draper

Published Oct 13, 2025• 1 citation
Publisher

Abstract

Abstract. In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of [Formula: see text]. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model’s representational capacity.

DOI
10.1137/24m172010x

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Abstract. In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of [Formula: see text]. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model’s representational capacity.
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 nnn-dimensional real coordinate space Rn\mathbb{R^{n}}Rn. 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.
Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.
Tensor network, which originates from quantum physics, is emerging as an efficient tool for classical and quantum machine learning. Nevertheless, there still exists a considerable accuracy gap between tensor network and the sophisticated neural network models for classical machine learning. In this work, we combine the ideas of matrix product state (MPS), the simplest tensor network structure, and residual neural network and propose the residual matrix product state (ResMPS). The ResMPS can be treated as a network where its layers map the "hidden" features to the outputs (e.g., classifications), and the variational parameters of the layers are the functions of the features of the samples (e.g., pixels of images). This is different from neural network, where the layers map feed-forwardly the features to the output. The ResMPS can equip with the non-linear activations and dropout layers, and outperforms the state-of-the-art tensor network models in terms of efficiency, stability, and expression power. Besides, ResMPS is interpretable from the perspective of polynomial expansion, where the factorization and exponential machines naturally emerge. Our work contributes to connecting and hybridizing neural and tensor networks, which is crucial to further enhance our understanding of the working mechanisms and improve the performance of both models.
We introduce a novel normal form representation of Boolean functions in terms of products of binary matrices, hereafter referred to as the Binary Matrix Product (BMP) representation. BMPs are analogous to the Tensor-Trains (TT) and Matrix Product States (MPS) used, respectively, in applied mathematics and in quantum many-body physics to accelerate computations that are usually inaccessible by more traditional approaches. BMPs turn out to be closely related to Binary Decision Diagrams (BDDs), a powerful compressed representation of Boolean functions invented in the late 80s by Bryant that has found a broad range of applications in many areas of computer science and engineering. We present a direct and natural translation of BMPs into Binary Decision Diagrams (BDDs), and derive an elementary set of operations used to manipulate and combine BMPs that are analogous to those introduced by Bryant for BDDs. Both BDDs and BMPs are practical tools when the complexity of these representations, as measured by the maximum bond dimension of a BMP (or the accumulated bond dimension across the BMP matrix train) and the number of nodes of a BDD, remains polynomial in the number of bits, nnn. In both cases, controlling the complexity hinges on optimizing the order of the Boolean variables. BMPs offer the advantage that their construction and manipulation rely on simple linear algebra -- a compelling feature that can facilitate the development of open-source libraries that are both more flexible and easier to use than those currently available for BDDs. An initial implementation of a BMP library is available on GitHub, with the expectation that the close conceptual connection to TT and MPS techniques will motivate further development of BMP methods by researchers in these fields, potentially enabling novel applications to classical and quantum computing.
Data representation in quantum state space offers an alternative function space for machine learning tasks. However, benchmarking these algorithms at a practical scale has been limited by ineffective simulation methods. We develop a quantum kernel framework using a Matrix Product State (MPS) simulator and employ it to perform a classification task with 165 features and 6400 training data points, well beyond the scale of any prior work. We make use of a circuit ansatz on a linear chain of qubits with increasing interaction distance between qubits. We assess the MPS simulator performance on CPUs and GPUs and, by systematically increasing the qubit interaction distance, we identify a crossover point beyond which the GPU implementation runs faster. We show that quantum kernel model performance improves as the feature dimension and training data increases, which is the first evidence of quantum model performance at scale.
The matrix product state (MPS) belongs to the most important models in, for example, quantum information sciences and condensed-matter physics. However, realizing an NNN-qubit MPS with large NNN and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or nnn-qubit gates with $n\ensuremath{\gg}2$ to carry or produce the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. Essentially different from the existing compiling methods, our idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPSs of the strongly correlated spin models show that the constructed quantum circuits can simulate the MPSs with much fewer qubits than the sizes of the MPSs themselves. This method paves a feasible and efficient path to realizing useful and/or exotic quantum states and MPS-based models as quantum circuits on near-term quantum platforms.