Quantum Statistical Inference

TL;DR

This thesis explores quantum statistical inference, demonstrating quantum algorithms for Gaussian processes and addressing quantum causality, with applications in machine learning, quantum correlations, and communication channels.

Contribution

It introduces a novel quantum algorithm for solving linear systems in Gaussian processes and applies quantum causality concepts to quantum correlations and communication capacity.

Findings

Quantum algorithms improve Gaussian process training.

Experimental demonstration on quantum hardware.

Analytical tools for quantum causal inference.

Abstract

In this thesis, I present several results on quantum statistical inference in the following two directions. Firstly, I demonstrate that quantum algorithms can be applied to enhance the computing and training of Gaussian processes (GPs), a powerful model widely used in classical statistical inference and supervised machine learning. A crucial component of the quantum GP algorithm is solving linear systems with quantum computers, for which I present a novel algorithm that achieves a provable advantage over previously known methods. I will also explicitly address the task of encoding the classical data into a quantum state for machine learning applications. I then apply the quantum enhanced GPs to Bayesian deep learning and present an experimental demonstration on contemporary hardware and simulators. Secondly, I look into the notion of quantum causality and apply it to inferring spatial and temporal quantum correlations, and present an analytical toolkit for causal inference in quantum data. I will also make the connection between causality and quantum communications, and present a general bound for the quantum capacity of noisy communication channels.