Date of Award

8-2023

Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics, Applied

First Advisor

Alioscia Hamma

Second Advisor

Maxim Olchanyi

Third Advisor

Christopher Fuchs

Abstract

In this thesis, we explore the mathematical properties of the Clifford group, a special subgroup of the unitary group, as well as the additional resources necessary to go beyond the Clifford group. Quantum computation restricted to Clifford circuits and stabilizer states -- i.e. those obtained by Clifford circuits on computational basis states -- can be entirely simulated classically. In fact, quantum advantage lies beyond the Clifford group: the injection of non-Clifford gates into Clifford circuits introduces a quantum resource called magic. Magic is crucial for the unique and distinctive properties of quantum states and serve as a fuel for quantum computation: without magic, a quantum computer cannot do anything that a classical computer cannot do. Beside providing a overview on the stabilizer formalism and the resource theory of magic, this thesis is mostly devoted to a detailed exploration of the mathematical tools behind the stabilizer world and aims to be a practical guide for readers seeking answers about the effectiveness of states in producing resources beyond the stabilizer resources. In particular, it offers a detailed review on the Haar measure over the Clifford group as well as its extension for t-doped Clifford circuits --- Clifford circuit doped with t non-Clifford gates. The thesis delves into the properties and characteristics of these measures, shedding light on their significance, implications and possible applications. This thesis includes both previously known results from existing literature and new, unpublished findings. The main novel contributions of this thesis are as follows. First, we largely improve the lower bound for \epsilon-approximate projective k-design obtained by adding non-Clifford gates to Clifford circuits. The best known bound scales as O(k^4\log^2k\log\epsilon^{-1}), while this thesis achieves O(k^2+\log\epsilon^{-1}) by utilizing Markov process tools and techniques. The second main result of this thesis is the proof of a Fannes-Audenaert type inequality for the resource theory of magic. Specifically, we demonstrate that the difference between stabilizer entropies cannot exceed the trace distance between states, up to a factor scaling only linearly with the number of qubits. This inequality has diverse applications in the field of quantum information, connecting stabilizer entropies and the distance between quantum states.

Comments

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