Using Abstract Harmonic Analysis and the Lie Group Theory for the Study of Parameterized Quantum Circuits

Abstract

Quantum computers are believed to be able to significantly outperform classical ones in terms of running time required to solve various problems. Near-term quantum computers, that can already be available in the nearest future, will have fairly limited resources, thus implying additional limitations and challenges. Near-term quantum algorithms are primarily based on parameterized quantum circuits. A parameterized quantum circuit is a quantum circuit, which is run repeatedly, while changing the numerical parameters some of the quantum operations in response to previous measurement results. Parameterized quantum circuits, however, need to be optimized, which can be simplified by endowing them with some mathematical structure, e.g., the ability to take derivatives or compute Fourier transforms. Here we study the possibility of using non-commutative Fourier transforms as a tool to find useful mathematical structure in parameterized quantum circuits.To our knowledge this thesis is the first work, where non-commutative Fourier transforms have been applied to parameterized quantum circuits. Our results include computations and theorems about non-commutative Fourier spectrum on parameterized quantum circuits. The results of this thesis provide a foundation, that opens the door for further study into derivatives and gradients of expectation functions on parameterized quantum circuits via the means of abstract harmonic analysis.