Introduction to Grassmann manifolds and quantum computation.

*(English)*Zbl 1037.81014The article is a review based on a series of the author’s lectures. It is written in a clear way, with many illustrative examples. The results are not new but presented from an original perspective. Several open problems are formulated. The main subject is Grassmann geometry and related topics (homogeneous spaces, Stiefel manifolds, flag manifolds) treated in an algebraic way, with explicit use of natural coordinates and bases. The physical context of quantum computations is rather omitted.

The first part of the paper is fully devoted to the problems of computing the volumes of Grassmann manifolds and unitary groups (i.e, the otherwise known volumes are expressed by some complicated integrals in special coordinates). The physical motivation (“the understanding of entanglements or entangled measures”) is just briefly mentioned.

In the second part of the paper the author considers the construction, efficiency and geometric interpretation of some unitary operations (on spaces of high dimension which are tensor products of many copies of \({\mathbb C}^2\)). In particular, tensor products of the so called “controlled-NOT gate” operations are considered. In the last section a simplified version of holonomic quantum computation is shortly presented. The starting point is an infinite-dimensional separable Hilbert space, the associated Stiefel and Grassmann manifolds, and the appropriate principal \(U(m)\) bundle and the associated vector bundles. The information encoded as a point in the fiber of the vector bundle is processed by performing some path integration (a holonomy operation).

The first part of the paper is fully devoted to the problems of computing the volumes of Grassmann manifolds and unitary groups (i.e, the otherwise known volumes are expressed by some complicated integrals in special coordinates). The physical motivation (“the understanding of entanglements or entangled measures”) is just briefly mentioned.

In the second part of the paper the author considers the construction, efficiency and geometric interpretation of some unitary operations (on spaces of high dimension which are tensor products of many copies of \({\mathbb C}^2\)). In particular, tensor products of the so called “controlled-NOT gate” operations are considered. In the last section a simplified version of holonomic quantum computation is shortly presented. The starting point is an infinite-dimensional separable Hilbert space, the associated Stiefel and Grassmann manifolds, and the appropriate principal \(U(m)\) bundle and the associated vector bundles. The information encoded as a point in the fiber of the vector bundle is processed by performing some path integration (a holonomy operation).

Reviewer: Jan L. Cieśliński (Białystok)

##### MSC:

81P68 | Quantum computation |

81Q70 | Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory |

15A69 | Multilinear algebra, tensor calculus |

15A90 | Applications of matrix theory to physics (MSC2000) |

53C30 | Differential geometry of homogeneous manifolds |

53C80 | Applications of global differential geometry to the sciences |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |