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Title: The abstract structure of quantum algorithms
Author: Zeng, William J.
ISNI:       0000 0004 6061 8638
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2015
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Quantum information brings together theories of physics and computer science. This synthesis challenges the basic intuitions of both fields. In this thesis, we show that adopting a unified and general language for process theories advances foundations and practical applications of quantum information. Our first set of results analyze quantum algorithms with a process theoretic structure. We contribute new constructions of the Fourier transform and Pontryagin duality in dagger symmetric monoidal categories. We then use this setting to study generalized unitary oracles and give a new quantum blackbox algorithm for the identification of group homomorphisms, solving the GROUPHOMID problem. In the remaining section, we construct a novel model of quantum blackbox algorithms in non-deterministic classical computation. Our second set of results concerns quantum foundations. We complete work begun by Coecke et al., definitively connecting the Mermin non-locality of a process theory with a simple algebraic condition on that theory's phase groups. This result allows us to offer new experimental tests for Mermin non-locality and new protocols for quantum secret sharing. In our final chapter, we exploit the shared process theoretic structure of quantum information and distributional compositional linguistics. We propose a quantum algorithm adapted from Weibe et al. to classify sentences by meaning. The clarity of the process theoretic setting allows us to recover a speedup that is lost in the naive application of the algorithm. The main mathematical tools used in this thesis are group theory (esp. Fourier theory on finite groups), monoidal category theory, and categorical algebra.
Supervisor: Coecke, Bob ; Olivervicary, Jamie Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Categories (Mathematics) ; Quantum Algorithms ; Quantum computing ; Monoidal Categories ; Frobenius algebras ; Group homomorphisms