Quantum stochastic modelling and tensor networks

• Main Author: Yang, Chengran
• Other Authors: Gu Mile
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2020
• Subjects: Science::Physics::Atomic physics::Quantum theory; Science::Physics::Atomic physics::Statistical physics
• Online Access: https://hdl.handle.net/10356/144661
• Institution: Nanyang Technological University

Predicting a stochastic process' future lies at the heart of many scientific areas. A predictive model extracts information from a stochastic process' past and uses it to generate future statistics. There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information. By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits. Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems. In this thesis, we explore the connection between tensor networks and quantum predictive models. A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models. First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models. MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory. Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models. Third, we propose a family of divergence measures to quantify the distance between two stochastic processes. This family of divergence measures overcomes certain weaknesses of the existing measures. Moreover, we propose an efficient means of computing the divergence measure using our MPS methods. Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process. We use machine learning methods to propose a systematic algorithm for learning the quantum model. This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.