Completeness and the ZX-calculus

Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not sufficient: the new formalisms also need to have equivalent deductive power. This requirement is captured by the property of completeness, which means that any equality that can be derived using some standard formalism can also be derived graphically. In this thesis, I consider the ZX-calculus, a graphical language for pure state qubit quantum mechanics. I show that it is complete for pure state stabilizer quantum mechanics, so any problem within this fragment of quantum theory can be fully analysed using graphical methods. This includes questions of central importance in areas such as error-correcting codes or measurement-based quantum computation. Furthermore, I show that the ZX-calculus is complete for the single-qubit Clifford+T group, which is approximately universal: any single-qubit unitary can be approximated to arbitrary accuracy using only Clifford gates and the T-gate. In experimental realisations of quantum computers, operations have to be approximated using some such finite gate set. Therefore this result implies that a wide range of realistic scenarios in quantum computation can be analysed graphically without loss of deductive power. Lastly, I extend the use of rigorous graphical languages outside quantum theory to Spekkens' toy theory, a local hidden variable model that nevertheless exhibits some features commonly associated with quantum mechanics. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, which is non-local; it thus offers insights into the similarities and differences between classical and quantum theories. I develop a graphical calculus similar to the ZX-calculus that fully describes Spekkens' toy theory, and show that it is complete. Hence, stabilizer quantum mechanics and Spekkens' toy theory can be fully analysed and compared using graphical formalisms. Intuitive graphical languages can replace conventional formalisms for the analysis of many questions in quantum computation and foundations without loss of mathematical rigour or deductive power.</p

Eine &quot;3He-NMR-Anordnung zur Untersuchung der Dynamik paraelektrischer Defekte in Festkoerpern im Temperaturbereich 0.4-4 K

Available from TIB Hannover: DW 1655 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Holant clones and the approximability of conservative holant problems

We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set F of functions is “universal in the conservative case”, which means that all functions are contained in the holant clone generated by F together with all unary functions. When F is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when F is universal in the conservative case

Picturing Quantum Processes: A First Course on Quantum Theory and Diagrammatic Reasoning

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Imaging of spinal tumors

Spinal tumors are often categorized into extradural, intradural extramedullary, or intramedullary. Although this classification represents somewhat of an overgeneralization as a lesion may reside in two compartments, it still helps to characterize spinal tumors. In the intradural, extramedullary space, primary tumors, such as neurofibroma and meningeoma, are relatively common. Secondary tumors or leptomeningeal enhancement also occur. In the intramedullary space, primary tumors are far more common than secondary tumors or metastases. © 2006 Springer Medizin Verlag

Imaging of spinal tumors

Spinal tumors are often categorized into extradural, intradural extramedullary, or intramedullary. Although this classification represents somewhat of an overgeneralization as a lesion may reside in two compartments, it still helps to characterize spinal tumors. In the intradural, extramedullary space, primary tumors, such as neurofibroma and meningeoma, are relatively common. Secondary tumors or leptomeningeal enhancement also occur. In the intramedullary space, primary tumors are far more common than secondary tumors or metastases. © 2006 Springer Medizin Verlag

Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin

We analyse the complexity of approximate counting constraint satisfactions problems #CSP(F), where F is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known if F contains arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and requiring only those to be in F. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. Furthermore, we also consider what happens if only the pinning functions are assumed to be in F. We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition

Supratentorial tumors

Magnetic resonance imaging is a routine diagnostic measure for a suspected intracerebral mass. Computed tomography is usually also indicated. Further diagnostic procedures as well as the interpretation of the findings vary depending on the tumor location. This contribution discusses the symptoms and diagnostics for supratentorial tumors separated in relation to their intra- or extracranial location. Supratentorial tumors include astrocytoma, differentiated by their circumscribed and diffuse growth, ganglioglioma, ependyoma, neurocytoma, primitive neuroectodermal tumors (PNET), oligodendroglioma, dysembryoplastic neuroepithelial tumors (DNET), meningoangiomatosis, pineal tumors, hamatoma, lymphoma, craniopharyngeoma and metastases. The supratentorial extracranial tumors include the choroid plexus, colloid cysts, meningeoma, infantile myofibromatosis and lipoma. The most common sub-forms, especially of astrocytoma, will also be presented. © 2007 Springer Medizin Verlag

[Infratentorial tumors]

This article gives an overview concerning the typical infratentorial tumors of adults

[Multiple sclerosis].

Multiple sclerosis is the most common chronic inflammatory disease of myelin with interspersed lesions in the white matter of the central nervous system. Magnetic resonance imaging (MRI) plays a key role in the diagnosis and monitoring of white matter diseases. This article focuses on key findings in multiple sclerosis as detected by MRI