Quantum programming and synthesis: Internalizing Clifford operations and beyond

Abstract

Clifford operations are a subset of quantum operations used extensively in quantum error correction and classical simulation of quantum circuits.

The first part of the thesis is motivated by the problem of programming with generalized Clifford operations, such as the quantum Fourier transform. We delve into the algebraic complications that arise for systems of even dimension $d$, and we are particularly interested in the case when $d$ is a power of $2$. We apply our results in the design of a quantum functional programming language, in which the user does not have to worry about these irrelevant complications.

Later, we consider the problem of compiling circuits over universal gate sets. In particular, we study the problem of multi-qutrit exact synthesis over a variety of gate sets including Clifford gates.

Lastly, we present a framework for defining a symplectic form on an object in a sufficiently structured category, and lay out the theory, generalizing the theory of symplectic forms on a finite dimensional vector space or locally compact abelian group. In the process, we develop new results and perspectives on operations defined on categories.