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Title: Computational power of quantum many-body states and some results on discrete phase spaces
Author: Gross, David
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2008
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This thesis consists of two parts. The main part is concerned with new schemes for measurement-based quantum computation. Computers utilizing the laws of quantum mechanics promise an exponential speed-up over purely classical devices. Recently, considerable attention has been paid to the measurement-based paradigm of quantum computers. It has been realized that local measurements on certain highly entangled quantum states are computationally as powerful as the well-established model for quantum computation based on controlled unitary evolution. Prior to this thesis, only one family of quantum states was known to possess this computational power: the so-called cluster state and some very close relatives. Questions posed and answered in this thesis include: Can one find families of states different from the cluster, which constitute universal resources for measurement-based computation? Can the highly singular properties of the cluster state be relaxed while retaining universality? Is the quality of being a computational resource common or rare among pure states? We start by establishing a new mathematical tool for understanding the connection between local measurements on an entangled quantum state and a quantum computation. This framework - based on finitely correlated states (or matrix product states) common in many-body physics - is the first such tool general enough to apply to a wide range of quantum states beyond the family of graph states. We employ it to construct a variety of new universal resource states and schemes for measurement-based computation. It is found that many entanglement properties of universal states may be radically different from those of the cluster: we identify states which are locally arbitrarily close to a pure state, exhibit long-ranged correlations or cannot be converted into cluster states by means of stochastic local operations and classical communication. Flexible schemes for the compensation of the inherent randomness of quantum measurements are introduced. We proceed to provide a complete classification of a natural class of states which can take the role of a single logical qubit in a measurement-based quantum computer. Lastly, it is demonstrated that states can be too entangled to be useful for any computational purpose. Concentration of measure arguments show that this problem occurs for the dramatic majority of all pure states. The second part of the thesis is concerned with discrete quantum phase spaces. We prove that the only pure states to possess a non-negative Wigner function are stabilizer states. The result can be seen as a finite-dimensional analogue of a classic theorem due to Hudson, who showed that Gaussian states play the same role in the setting of continuous variable systems. The quantum phase space techniques developed for this argument are subsequently used to quantize a well-known structure from classical computer science: the Margulis expander.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available