Advantages and learning for quantum modelling

Abstract

Simulating stochastic processes using less resources is a key pursuit in many sciences. This involves identifying and extracting the past information relevant to the process' future behavior and formulating `predictive models' for inferring the latter from the former. Significant efforts have been made towards finding the optimal predictive models that require the least amount of information. Quantum technologies offer a promising means to this end, allowing equally accurate future predictions whilst storing less past information. In this thesis, I explore three aspects of research related to this idea. First, we demonstrate quantum models can have unbounded memory advantage by studying a family of stochastic processes where a random walk on real numbers is modelled with progressively greater precision. Next, we document the optimal quantum models that generate predictions using unitary circuits. We use these to document `ambiguity of optimality', a uniquely quantum phenomena where the optimal model depends on whether it is used in i.i.d versus single-shot settings. Finally, we look at a related problem of learning how a desired unitary can be synthesized using unknown pulses. Together, these development help to further understand the properties and power of quantum models, as well as build towards tools for their synthesis.