Computational power of quantum many-body states and some results on discrete phase spaces

This thesis consists of two parts. The main part is concerned with new schemes for measurementbased quantum computation. Computers utilizing the laws of quantum me chanics promise an exponential speed-up over purely classical devi ces. Recently, considerable attention has been paid to the measurement-ba sed p radigm of quantum computers. It has been realized that local measur ments on certain highly entangled quantum states are computational ly as powerful as the well-established model for quantum computation based o n controlled unitary evolution. Prior to this thesis, only one family of quantum states was kn own to possess this computational power: the so-called cluster state and s ome very close relatives. Questions posed and answered in this thesis incl ude: Can one find families of states different from the cluster, which con stitute universal resources for measurement-based computation? Can the high ly s ngular properties of the cluster state be relaxed while retaining u iversality? Is the quality of being a computational resource common or rare among pure states? We start by establishing a new mathematical tool for underst anding the connection between local measurements on an entangled quantum state and a quantum computation. This framework – based on finitely corr elated states (or matrix product states) common in many-body physics – is t he 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 c omputation. It is found that many entanglement properties of universal sta te may be radically different from those of the cluster: we identify state s which are locally arbitrarily close to a pure state, exhibit long-ranged corr elations or cannot be converted into cluster states by means of stochastic loca l operations and classical communication. Flexible schemes for the compens ation of the inherent randomness of quantum measurements are introduce d. W proceed to provide a complete classification of a natural class o f tates which can take the role of a single logical qubit in a measurement-b ased quantum computer. Lastly, it is demonstrated that states can be too e ntangled to be useful for any computational purpose. Concentration of mea sure arguments show that this problem occurs for the dramatic majority of al l pure states. The second part of the thesis is concerned with discrete quan tum phase spaces. We prove that the only pure states to possess a non-ne gative Wigner function are stabilizer states. The result can be seen as a fin ite-dimensional analogue of a classic theorem due to Hudson, who showed that G aussian states play the same role in the setting of continuous variab le systems. The quantum phase space techniques developed for this argument are subsequently used to quantize a well-known structure from classi cal computer science: the Margulis expander.