Optical holonomic quantum computer

Abstract

In this paper the idea of holonomic quantum computation is realized within quantum optics. In a non-linear Kerr medium the degenerate states of laser beams are interpreted as qubits. Displacing devices, squeezing devices and interferometers provide the classical control parameter space where the adiabatic loops are performed. This results into logical gates acting on the states of the combined degenerate subspaces of the lasers, producing any one qubit rotations and interactions between any two qubits. Issues such as universality, complexity and scalability are addressed and several steps are taken towards the physical implementation of this model.

Mathematical foundations of holonomic quantum computer

“…We apply the results of last section to Quantum Optics and discuss about (optical) Holonomic Quantum Computation proposed by [6] and [11].…”

“…Let H 0 be a Hamiltonian with nonlinear interaction produced by a Kerr medium., that is H 0 =hXN(N − 1), where X is a certain constant, see [11]. The eigenvectors of H 0 corresponding to 0 is {|0 , |1 }, so its eigenspace is Vect {|0 , |1 } ∼ = C 2 .…”

“…We believe that these examples will become important in the near future. But their works ( [6], [7] and [11]) are a bit coarse in the mathematical point of view. Therefore in this paper we give a mathematical reinforcement to them.…”

Geometric quantum computation on solid-state qubits

“…In these schemes, however, geometrically available was only Abelian Berry phase, and additional dynamic manipulations were required for universal quantum computation. A scheme based solely on holonomies has been proposed for quantum optical systems [9]. However, it relies on nonlinear optics, which may make this scheme less practical.…”

Consistency of the Adiabatic Theorem

“…The role of the adiabatic theorem in the study of slowly varying quantum mechanical systems spans a vast array of fields and applications, such as the Landau-Zener theory of energy level crossings in molecules [9,10], quantum field theory [11], and Berry's phase [12]. In recent years geometric phases [13] have been proposed to perform quantum information processing [14,15,16], with adiabaticity assumed in a number of schemes for geometric quantum computation (e.g., [17,18,19,20]). Additional interest in adiabatic processes has arisen in connection with the concept of adiabatic quantum computing, in which slowly varying Hamiltonians appear as a promising mechanism for the design of new quantum algorithms and even as an alternative to the conventional quantum circuit model of quantum computation [21,22,23].…”