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arXiv:quant-ph/0111075 · Optical Fiber Communication Conference

Optical Holonomic Quantum Computation

Demosthenes Ellinas, Jiannis K. Pachos

Published Jan 1, 2001• 0 citations
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Abstract

Holonomic quantum computation (HQC) is materialized here with quantum optics components. Holonomies are the generalization of the Berry phases to unitary matrices. In a nonlinear Kerr medium the degenerate states of a laser beams are interpreted as qubits. Control manipulations with displacers, squeezers and two-mode interfering devices in a cyclic, adiabatic fashion produces holonomies. Hence, they are employed as logical gates for our HQC proposal. The effects of errors from imperfect control of classical parameters, the looping variation of which builds up holonomic gates, are also investigated. Additionally, the influence of quantum decoherence on holonomic tele-portation used as a computational primitive is studied, and particular features of control error resilience are discussed.

DOI
10.1364/icqi.2001.pb15

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Holonomic quantum computation (HQC) is materialized here with quantum optics components. Holonomies are the generalization of the Berry phases to unitary matrices. In a nonlinear Kerr medium the degenerate states of a laser beams are interpreted as qubits. Control manipulations with displacers, squeezers and two-mode interfering devices in a cyclic, adiabatic fashion produces holonomies. Hence, they are employed as logical gates for our HQC proposal. The effects of errors from imperfect control of classical parameters, the looping variation of which builds up holonomic gates, are also investigated. Additionally, the influence of quantum decoherence on holonomic tele-portation used as a computational primitive is studied, and particular features of control error resilience are discussed.
In this paper the idea of holonomic quantum computation is realized within quantum optics. In a nonlinear 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 in 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, scalability, and reversibility are addressed and several steps are taken towards the physical implementation of this model.
Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.
We show that the notion of generalized Berry phase i.e., non-abelian holonomy, can be used for enabling quantum computation. The computational space is realized by a nnn-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold calM\\cal McalM. The point of calM\\cal McalM represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold calM\\cal McalM induce non trivial unitary transformations on the computational space. For a generic system it is shown that this mechanism allows for universal quantum computation by composing a generic pair of loops in calM.\\cal M.calM.
Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek–Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed. The conception of quantum computer (QC) and quantum calculation developed in 80-th [1]–[2] were found to be fruitful both for computer science and mathematics as well as for physics [3]. Although a device being able to perform quantum calculations is far away from practical realization, there is a number of theoretical proposals of such a construct [4]–[16]. Recently holonomic realization of QC was proposed [17],[18]. It is based on the notion of non-Abelian Berry phase[19].To perform quantum computations in this approach one needs a parametrically driven quantum system described by the Hamiltonian H(λ) with adiabatically evolving parameters λ(t) = (λ1(t),..., λN(t)). Adiabaticity means that Ω ≪ ωmin where Ω is the characteristic frequency in the Fourier spectrum of λ(t) and ωmin is the minimal transition frequency in the spectrum