Measurement-based quantum computation on cluster states

We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size

Measurement-based quantum computation with cluster states

In this thesis we describe the one-way quantum computer (QCc), a scheme of universal quantum computation that consists entirely of one-qubit measurements on a highly entangled multi-particle state, the cluster state. We prove universality of the QCc, describe the underlying computational model and demonstrate that the QCc can be operated fault-tolerantly. In Chapter 2 we show that the QCc can be regarded as a simulator of quantum logic networks. In this way, we give the universality proof and establish the link to the network model, the common model of quantum computation. We also indicate that the description of the QCc as a network simulator is not adequate in every respect. In Chapter 3 we derive the computational model underlying the one-way quantum computer, which is very different from the quantum logic network model. The QCc has no quantum input, no quantum output and no quantum register, and the unitary gates from some universal set are not the elementary building blocks of QCc-quantum algorithms. Further, all information that is processed with the QCc are the outcomes of one-qubit measurements and thus processing of information exists only at the classical level. The QCc is nevertheless quantum mechanical as it uses a highly entangled cluster state as the central physical resource. In Chapter 4 we show that there exist nonzero error thresholds for fault-tolerant quantum computation with the QCc. Further, we outline the concept of checksums in the context of the QCc which may become an element in future practicable and adequate methods for fault-tolerant QCc-computation.In dieser Dissertation beschreiben wir den Einweg-Quantenrechner (QCc), ein Schema zum universellen Quantenrechnen, das allein aus Einteilchenmessungen an einem hochgradig verschraenkten Vielteilchenzustand, dem Clusterzustand, besteht. Wir beweisen die Universalitaet des QCc, beschreiben das zugrunde liegende Rechnermodell und zeigen, dass der QCc fehlertolerantes Quantenrechnen erlaubt. In Kapitel 2 zeigen wir, dass der QCc als ein Simulator quantenlogischer Netzwerke aufgefasst werden kann. Damit beweisen wir dessen Universalitaet und stellen den Zusammenhang zum Netzwerkmodel her, welches das verbreitete Model eines Quantenrechners darstellt. Wir weisen auch darauf hin, dass die Beschreibung des QCc als Netzwerksimulator nicht in jeder Hinsicht passend ist. In Kapitel 3 leiten wir das dem Einweg-Quantenrechner zugrunde liegende Rechnermodell her. Es ist sehr verschieden vom Netzwerkmodell des Quantenrechners. Der QCc besitzt keinen Quanten-Input, keinen Quanten-Output und kein Quantenregister. Unitaere Quantengatter aus einem universellen Satz sind nicht die elementaren Bestandteile von QCc-Quantenalgorithmen. Darueber hinaus sind die Messergebnisse aus den Einteilchenmessungen die einzige Information, die vom QCc verarbeitet wird, und somit existiert Informationsverarbeitung beim QCc nur auf klassischem Niveau. Dennoch arbeitet der QCc fundamental quantenmechanisch, da er den hochverschraenkten Clusterzustand als zentrale physikalische Resource nutzt. In Kapitel 4 zeigen wir, dass positive Fehlerschranken fuer das fehlertolerante Quantenrechnen mit dem QCc existieren. Desweiteren skizzieren wir das Konzept der Pr{"u}fsummen im Zusammenhang mit dem QC, das ein Element zukuenfitiger praktikabler und zweckmaessiger Methoden fuer fehlertolerantes QCc-Quantenrechnen werden kann

A Computational Model for Quantum Measurement

Is the dynamical evolution of physical systems objectively a manifestation of information processing by the universe? We find that an affirmative answer has important consequences for the measurement problem. In particular, we calculate the amount of quantum information processing involved in the evolution of physical systems, assuming a finite degree of fine-graining of Hilbert space. This assumption is shown to imply that there is a finite capacity to sustain the immense entanglement that measurement entails. When this capacity is overwhelmed, the system's unitary evolution becomes computationally unstable and the system suffers an information transition (`collapse'). Classical behaviour arises from the rapid cycles of unitary evolution and information transitions. Thus, the fine-graining of Hilbert space determines the location of the `Heisenberg cut', the mesoscopic threshold separating the microscopic, quantum system from the macroscopic, classical environment. The model can be viewed as a probablistic complement to decoherence, that completes the measurement process by turning decohered improper mixtures of states into proper mixtures. It is shown to provide a natural resolution to the measurement problem and the basis problem.Comment: 24 pages; REVTeX4; published versio

What Makes a Computation Unconventional?

A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.Comment: Based on an invited lecture for the 'Symposium on Natural/Unconventional Computing and Its Philosophical Significance' at the AISB/IACAP World Congress 2012, University of Birmingham, July 2-6, 201