Quantum Computation by Geometrical Means

A geometrical approach to quantum computation is presented, where a non-abelian connection is introduced in order to rewrite the evolution operator of an energy degenerate system as a holonomic unitary. For a simple geometrical model we present an explicit construction of a universal set of gates, represented by holonomies acting on degenerate states. 1. Prologue Abelian [Sha] and non-abelian [Wil] geometrical phases in quantum theory have been considered as a deep and fascinating subject. They provide a natural connection between the evolution of a physical system with degenerate structure and differential geometry. Here we shall present a model where these concepts can be explicitly applied for quantum computation [Zan]. The physical setup consists of an energy degenerate quantum system on which we perform an adiabatic isospectral evolution described by closed paths in the parametric space of external variables. The corresponding evolution operators acting on the code-state in the degenerate eigenspace are given in terms of holonomies and we can use them as quantum logical gates. This is a generalization of the Berry phase or geometrical phase, to the non-abelian case, where a non-abelian adiabatic connection, A, is produced from the geometrical structure of the degenerate spaces. In particular, on each point of the manifold of the external parameters there is a code-state attached and a transformation between these bundles of codes is dictated by the connection A. In order to apply this theoretical construction to a concrete example we employ a model with CP geometry, that is a complex projective manifold with two complex coordinates. This is interpreted as a qubit [Pac]. A further generalization with the tensor product of m CP models and additional interaction terms parametrized by the Grassmannian manifold, G(4, 2), is interpreted as a model of quantum computer. The initial code-state is written on the degenerate eigenspace of the system. The geometrical evolution operator is a unitary acting on it and it is interpreted as 2000 Mathematics Subject Classification. Primary 81P68. The author was supported in part by TMR Network under the contract no. ERBFMRXCT96 0087. c ©2002 American Mathematical Society