Robust Hadamard gate for optical and ion trap holonomic quantum computers

We consider one possible implementation of Hadamard gate for optical and ion trap holonomic quantum computers. The expression for its fidelity determining the gate stability with respect to the errors in the single-mode squeezing parameter control is analytically derived. We demonstrate by means of this expression the cancellation of the squeezing control errors up to the fourth order on their magnitude. Holonomic quantum computation exploiting non-abelian geometrical phases (holonomies) was primarily proposed in Ref. [1] and was further worked out in Ref. [2]. Various implementations of holonomic quantum computer (HQC) have been proposed recently. Namely, it was suggested to realize the HQC within quantum optics (optical HQC) [3]. Laser beams in a non-linear Kerr medium were used for this purpose. Two different sets of control devices could be used in this case. The first one consists of one and two mode displacing and squeezing devices. The second one includes SU(2) interferometers. Squeezing and displacement of the vibrational modes of the trapped ions were suggested to use for realization of HQC in Ref. [4]. This implementation of HQC is mathematically similar to the first embodiment of the optical HQC offered in Ref. [3]. Particularly, expressions for the adiabatic connection and holonomies are the same. Another proposed implementation of HQC was HQC with neutral atoms in cavity QED [5]. The coding space was spanned by the dark states of the atom trapped in a cavity. Dynamics of the system was governed by the generalized Λ-system Hamiltonian. Mathematically similar semiconductor-based implementation of HQC was proposed in Ref. [6]. One-qubit gates were realized in the framework of the same generalized Λ-system as in Ref. [5]. However its physical implementation exploits semiconductor excitons driven by sequences of laser pulses [6]. For the two-qubit gate the bi-excitonic shift was used. The generalized Λ-system with different Rabi frequencies parametrization was exploited recently for HQC implemented by Rf-SQUIDs coupled through a microwave cavity [7]. One more solid state implementation of HQC based on Stark effect was proposed in Ref. [8]. Quantum computers including HQC are analog-type devices. Thus unavoidable errors in the assignment of the classical control parameters lead to an errors in quantum gates and in the case when the tolerance of quantum computation is not large enough the computation fails. This obstacle (inaccuracy) is also related to the decoherence problem [9]. The effect of the 1