Geometric Representation of Infinite Arrays of Qubits: Theory and Modelling
Date
2024-09-27
Authors
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Journal ISSN
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ORCID
Type
Thesis
Degree Level
Doctoral
Abstract
In this thesis, we present a multiresolution approach to the theory of quantum information. The motivation behind this work was to provide a methodical mathematical framework for studying an infinite number of qubits, i.e., a structure that may be interpreted as a quantum metamaterial. The core of our model is based on two mathematical constructions with classical roots: the Borel isomorphism and the Haar basis. Here, these constructions are intertwined to establish an identification between $L_2(0,1]$ and the Hilbert space of an infinite array of qubits and to enable analysis of operators that act on arrays of qubits (either finite or infinite). The fusion of these two concepts empowers us to represent quantum operations and observables through geometric operators. As an unexpected upshot, we observe that the fundamental concept of calculus is inherent in an infinite array of qubits; indeed, the antiderivative arises as a natural and indispensable operator in this context.
To achieve this, we embarked upon exploring a generalization of a previously published solvable quantum metamaterial type model interacting with an electromagnetic field \cite{Sowa_2019}. The continuum limit of the interaction Hamiltonian is the essential insight that facilitates solvability in this study. It produces a scale-wise self-similar non-local operator in $L_2(0, 1]$. Therefore, we look at a similar strategy by focusing on the impact of replacing the Pauli matrix $\hat{\sigma}_x$ with $\hat{\sigma}_y$ in the interaction Hamiltonian.
We generalize the results we have found to the entire Bloch sphere by introducing a total operator that describes a continuous movement on the Bloch sphere. The reason is that, through this generalization, one can describe any continuous movement on the Bloch sphere using the operators defined in this framework. Following that we introduce noise to the coupling constant in the original Jaynes-Cummings model. By doing so, we find that the scaled qubit array is harder to engineer than a more regular, uniform one. This is remarkable since the dyadically scaled array has some special properties, such as implicitly containing the antiderivative.
Description
Keywords
Infinite qubit array, Haar transform
Citation
Degree
Doctor of Philosophy (Ph.D.)
Department
Mathematics and Statistics
Program
Mathematics