Unsupervised Generative Modeling Using Matrix Product States

Abstract

Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence. Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states. Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks. We apply our method to generative modeling of several standard data sets including the Bars and Stripes random binary patterns and the MNIST handwritten digits to illustrate the abilities, features, and drawbacks of our model over popular generative models such as the Hopfield model, Boltzmann machines, and generative adversarial networks. Our work sheds light on many interesting directions of future exploration in the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to realize on quantum devices.

Many-body topological invariants from randomized measurements in synthetic quantum matter

“…1 and 2 of the main text). The estimations for random measurements were obtained using a sampling algorithm of the occupation probabilities P U (s) for Matrix-Product-States [35].…”

Machine learning by unitary tensor network of hierarchical tree structure

“…As one of the most powerful numerical tools for studying quantum manybody systems [6][7][8][9], tensor networks (TNs) have drawn more attention. For instance, TNs have been recently applied to solve machine learning problems such as dimensionality reduction [10,11] and handwriting recognition [12,13]. Just as a TN allows the numerical treatment of difficult physical systems by providing layers of abstraction, deep learning achieved similar striking advances in automated feature extraction and pattern recognition using a hierarchical representation [14].…”

“…A TTN of hierarchical structure [29] suits the two-dimensional (2D) nature of images more than those based on a one-dimensional (1D) TN, e.g. matrix product state (MPS) [12,13,30]. Secondly, we explicitly connects machine learning to quantum quantities, such as fidelity and entanglement.…”

Encoding of matrix product states into quantum circuits of one- and two-qubit gates

“…MPS can efficiently describe the ground states and (purified) thermal states of one-dimensional (1D) gapped systems [8][9][10][11][12]. It has also been widely and successfully applied to other areas including statistic physics [13], non-equilibrium quantum physics [9, 14-17], field theories [18][19][20][21][22][23], machine learning [24][25][26][27][28], and so on.In particular, MPS is an important model in quantum information and computation (see, e.g., [29][30][31][32]). It can represent a large class of states, including GHZ [33] and AKLT states [34,35], which can implement non-trivial quantum computational tasks [36,37].…”

“…MPS can efficiently describe the ground states and (purified) thermal states of one-dimensional (1D) gapped systems [8][9][10][11][12]. It has also been widely and successfully applied to other areas including statistic physics [13], non-equilibrium quantum physics [9,[14][15][16][17], field theories [18][19][20][21][22][23], machine learning [24][25][26][27][28], and so on.…”