A geometrical approach to quantum holonomic computing algorithms
- 作者：Samoilenko ; A.M. ; Prykarpatsky ; Y.A. ; Taneri ; Ufuk ; Prykarpatsky ; A.K. ; Blackmore ; D.L.
- 关键词：Quantum computers ; Quantum algorithms ; Dynamical systems ; Grassmann manifolds ; Symplectic structures ; Connections ; Holonomy groups ; Lax type integrable flows
- 刊名：Mathematics and Computers in Simulation
- 出版年：2004
- 期刊代码：88_03784754
- 类别：cp
- 出版时间：June 4, 2004
- 卷：66
- 期：1
- 页码：1-20
- 文件大小：209 K
摘要
The article continues a presentation of modern quantum mathematics backgrounds started in [Quantum Mathematics and its Applications. Part 1. Automatyka, vol. 6, AGH Publisher, Krakow, 2002, No. 1, pp. 234–2412; Quantum Mathematics: Holonomic Computing Algorithms and Their Applications. Part 2. Automatyka, vol. 7, No. 1, 2004]. A general approach to quantum holonomic computing based on geometric Lie-algebraic structures on Grassmann manifolds and related with them Lax type flows is proposed. Making use of the differential geometric techniques like momentum mapping reduction, central extension and connection theory on Stiefel bundles it is shown that the associated holonomy groups properly realizing quantum computations can be effectively found concerning diverse practical problems. Two examples demonstrating two-form curvature calculations important for describing the corresponding holonomy Lie algebra are presented in detail.