Electronic Thesis and Dissertation Repository

Thesis Format

Monograph

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Barron, Tatyana

Abstract

The principal objective of this study is to investigate how the Kahler geometry of a classical phase space influences the quantum information aspects of the quantum Hilbert space obtained from geometric quantization and vice versa. We associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the states associated this way are separable when the subset is a finite union of products. We presented an asymptotic result for the average entropy over all the pure states on the Hilbert space H0(M1,L1⊗N) ⊗ H0(M2,L2⊗N), where H0(Mj,Lj⊗N) is the space of holomorphic sections of the N-th tensor powers of hermitian ample line bundle Lj on compact complex manifolds Mj. The coefficients of this asymptotic expression capture certain topological and geometric properties of the manifold.

In another project related to quantum computing, we constructed an exact synthesis algorithm for quantum gates in the groups U3n(Z[1/3,e2πi/3]) and U3n(Z[1/3,e2πi/3]) over the multi-qutrit Clifford+T gate set with the help of ancilla.

Summary for Lay Audience

We live in a three-dimensional world where our everyday experiences and physical phenomena occur in geometric space. To understand these phenomena, physicists and mathematicians create mathematical frameworks. For example, Newtonian Mechanics describes how a particle's motion relates to the forces acting on it. A key focus in many scientific theories is understanding how changes in one quantity affect others. In this thesis, we explore how changes in a geometric space relate to concepts from quantum information theory.

The quantum world behaves in strange ways that classical physics can't explain. For instance, quantum particles can exist in multiple places at once, and particles can become entangled so that a change in one instantly affects the other, even across vast distances.

The location and properties of a quantum particle, or quantum state, are described by matrices, which are arrays of numbers. These matrices help us understand phenomena like entanglement. In our study, we take geometric objects known as K\"ahler manifolds and associate quantum states with them. We study properties related to the entanglement of these states versus the properties of the geometric objects.

In another project of a slightly different flavour, we construct circuits that can be used for computers that make use of quantum mechanics. The classical computers (the common computers that we see every day) use gates that manipulate the bits (0s and 1s) to execute an algorithm. However, the set of gates is very big. So, we select a small number of convenient gates that can be used (in a circuit) to generate any desirable gate. The quantum version of this process is even more complicated owing to the set of gates being infinite. To this end, we contributed by adding one more algorithm to tackle this process.

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