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Quantum Machine Learning Matrix Product States

Jacob Biamonte

Published Apr 6, 2018• 1 citation
OpenAlex

Abstract

Matrix product states minimize bipartite correlations to compress the classical data representing quantum states. Matrix product state algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics. Matrix product states have a further range of applications in machine learning. Finding matrix product states is in general a computationally challenging task, a computational task which we show quantum computers can accelerate. We present a quantum algorithm which returns a classical description of a $k$-rank matrix product state approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires $O(n\cdot k^2)$ quantum gates, yielding sufficient conditions for our quantum variational algorithm to terminate in polynomial-time.

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Matrix product states minimize bipartite correlations to compress the classical data representing quantum states. Matrix product state algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics. Matrix product states have a further range of applications in machine learning. Finding matrix product states is in general a computationally challenging task, a computational task which we show quantum computers can accelerate. We present a quantum algorithm which returns a classical description of a kkk-rank matrix product state approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires O(n⋅k2)O(n\cdot k^2)O(n⋅k2) quantum gates, yielding sufficient conditions for our quantum variational algorithm to terminate in polynomial-time.
Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.
Matrix Product States have been extensively explored as a powerful tool for simulating quantum states in image classification task. However, most research has focused on classical simulations or computations involving high-dimensional unitaries, and significant challenges still exist in the practical preparation of Matrix Product States on quantum computers. This paper proposes a novel and practically feasible quantum variational algorithm based on Matrix Product States for image classification task. We design a hardware-efficient quantum circuit with several adjustable entangling operators to prepare the local tensors in Matrix Product States and integrate minimal residuals to ensure computational stability. We demonstrate that our algorithm can reduce the parameter complexity from growing exponentially with the system size to a linear scale. To validate the effectiveness of this quantum variational algorithm, we conducted experiments on the MNIST dataset, achieving accuracies of 99.95% and 95.96% for binary and ten-class classification tasks, which outperforms other related quantum algorithms. This work advances the practical application of quantum machine learning in resource-constrained environments of quantum computing.
Abstract Hybrid quantum–classical algorithms are a promising candidate for developing uses for NISQ devices. In particular, parametrised quantum circuits (PQCs) paired with classical optimizers have been used as a basis for quantum chemistry and quantum optimization problems. Tensor network methods are being increasingly used as a classical machine learning tool, as well as a tool for studying quantum systems. We introduce a circuit pre-training method based on matrix product state machine learning methods, and demonstrate that it accelerates training of PQCs for both supervised learning, energy minimization, and combinatorial optimization.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
Abstract Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Originally developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them suitable for various machine learning tasks within the quantum realm. Here, we present a matrix product state (MPS) model, where the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. Additionally, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Initially developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Drawing inspiration from these quantum methods, specifically the Matrix Product States (MPS), we apply them in a classical machine learning setting. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them effective in a supervised learning framework. Here, we present an MPS model, in which the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. In addition, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.