Mathematical foundations of holonomic quantum computer

Abstract

We make a brief review of (optical) Holonomic Quantum Computer (or Computation) proposed by Zanardi and Rasetti (quant-ph 9904011) and Pachos and Chountasis (quant-ph 9912093), and give a mathematical reinforcement to their works. *

Introduction to Grassmann manifolds and quantum computation

“…The full story would require detailed knowledge of quantum mechanics, quantum optics, and global analysis, which are not discussed in this paper. This model was proposed by Zanardi and Rasetti [22,31], and it has been developed by Fujii [5,6,7,8] and Pachos [20,21].…”

Basic Properties of Coherent and Generalized Coherent Operators Revisited

“…We make a brief review of some basic properties of coherent operators (10). For the elegant proofs see the book [2], or the paper [6] and its references.…”

“…By the way we are very interested in Geometric Quantum Computer, in particular, Holonomic Quantum Computer which has been proposed by Zanardi and Rassetti [7], [8] and developed by Fujii [9], [10], [11] and Pachos and Chountasis [12]. As a general introduction to Quantum Computer (Computation) [13] or [14] are recommended.…”

Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

“…We have that the mapping P : Gr m (H) −→ P m (H) is a bijection (see [4], [5], [6]) and defines an injection of the set Gr m (H) in the vector space of Hermitian operators on the Hilbert space H. This injection induces a topology and a differential structure on the set Gr m (H). The set Gr m (H), together with this differential structure is known as the Grassmann manifold of mdimensional complex subspaces of the Hilbert space H. As it follows from the above discussion, further we can identify the following two objects: P m (H) and Gr m (H).…”

“…This action is transitive and for any point X ∈ Gr m (H), the corresponding stabilizer subgroup is U(X)×U(X ⊥ ). Therefore, the manifold Gr m (H) can be considered as the homogeneous space (see, for example, [4], [5], [6]) where Y = u(X) (and therefore Y ⊥ = u(X ⊥ )). This transformation is defined by the condition: Γ( u(f )) = u(Γ(f )).…”