Tree tensor network classifiers for machine learning: from quantum-inspired to quantum-assisted

Abstract

We describe a quantum-assisted machine learning (QAML) method in which multivariate data is encoded into quantum states in a Hilbert space whose dimension is exponentially large in the length of the data vector. Learning in this space occurs through applying a low-depth quantum circuit with a tree tensor network (TTN) topology, which acts as an unsupervised feature extractor to identify the most relevant quantum states in a data-driven fashion, analogous to coarse-graining strategies used in renormalization group methodologies from statistical physics. This unsupervised feature extractor then feeds a supervised linear classifier together with a set of truth labels for the data type, and encodes the output in a small-dimensional quantum register. In contrast to previous work on quantum-inspired TTN classifiers, in which the embedding map and class decision weights did not map the data to well-defined quantum states, we present an approach that can be implemented on gate-based quantum computing devices. In particular, we identify an embedding map with accuracy similar to the recently defined exponential machines (Novikov et al., arXiv:1605.03795), but which produces valid quantum state embeddings of classical data vectors, and utilize manifoldbased gradient optimization schemes to produce isometric operations mapping quantum states to a register of qubits defining a class decision. We detail methods for efficiently obtaining one-point and two-point correlation functions of the vectors defining the decision boundary of the quantum model, which can be used for model interpretability, as well as methods for obtaining classification decisions from partial data vectors. Further, we show that the use of isometric tensors can significantly aid in the human interpretability of the correlation functions extracted from the decision weights, and may produce models that are less susceptible to adversarial perturbations. We demonstrate our methodologies in applications utilizing the MNIST handwritten digit dataset and a multivariate timeseries dataset of human activity recognition.