From Geometry to Quantum Computation

The aim of this paper is to introduce our idea of Holonomic Quantum Computation (Computer). Our model is based on both harmonic oscillators and non-linear quantum optics, not on spins of usual quantum computation and our method is moreover completely geometrical. We hope that therefore our model may be strong for decoherence.Comment: Latex File, 12 pages, A talk at the 2nd International Symposium `` Quantum Theory and Symmetries'', Krakow, 18-21, July, 200

Introduction to Grassmann Manifolds and Quantum Computation

Geometrical aspects of quantum computing are reviewed elementarily for non-experts and/or graduate students who are interested in both Geometry and Quantum Computation. In the first half we show how to treat Grassmann manifolds which are very important examples of manifolds in Mathematics and Physics. Some of their applications to Quantum Computation and its efficiency problems are shown in the second half. An interesting current topic of Holonomic Quantum Computation is also covered. In the Appendix some related advanced topics are discussed.Comment: Latex File, 28 pages, corrected considerably in the process of refereeing. to appear in Journal of Applied Mathematic

Exponentiation of certain Matrices related to the Four Level System by use of the Magic Matrix

In this paper we show how to calculate explicitly the exponential of certain matrices, which are evolution operators governing the interaction of the four level system of atoms and the radiation, etc. We present a consistent method in terms of the magic matrix by Makhlin. As a closely related subject, we derive a closed form expression of the Baker-Campbell-Hausdorff formula for a class of matrices in SU(4), by use of the method developed by the present authors in quant-ph/0610009.Comment: Latex ; 13 pages ; 2 figures ; substantial changes (including the title) made. To appear in Yokohama Mathematical Journal (2007