In a quantum ring connected with two external leads the spin properties of an incoming electron are modified by the spin-orbit interaction resulting in a transformation of the qubit state carried by the spin. The ring acts as a one qubit spintronic quantum gate whose properties can be varied by tuning the Rashba parameter of the spin-orbit interaction, by changing the relative position of the junctions, as well as by the size of the ring. We show that a large class of unitary transformations can be attained with already one ring -or a few rings in series -including the important cases of the Z, X, and Hadamard gates. By choosing appropriate parameters the spin transformations can be made unitary, which corresponds to lossless gates. The electron spin degree of freedom is one of the prospective carriers [1, 2] of qubits, the fundamental units in quantum information processing. In order to implement quantum operations on electron spins, appropriate gates are necessary that operate on this type of qubits. We note that in the present context the word 'gate' stands for an elementary logical operation [3]. In this paper we show that a one dimensional ring [4] connected with two external leads made of a semiconductor structure [5], such as InGaAs in which Rashba-type [6] spin-orbit interaction is the dominant spin-flipping mechanism, can render such a gate. Conductance properties of this kind of rings have been discussed earlier in the case of diametrically connected leads [7,8,9].By taking here a new point of view, we focus explicitly on the spin transformation characteristics of this device, and show that those can be appropriately controlled by varying its geometrical and physical parameters in the experimentally feasible range [7,8]. We shall determine the effects of changing the radius and the relative positions of the junctions, as well as the influence of varying the strength of the spin-orbit interaction via an external electric field. The conditions under which the incoming and transmitted spinors are connected unitarily will be determined, leading in principle, to a lossless single qubit gate. By connecting a few such rings in an appropriate manner, one can achieve practically all the important one qubit gates [3].We consider a ring of radius a in the x − y plane and assume a tunable static electric field [7] in the z direction characterized by the parameter α. Then the spin dependent Hamiltonian [9, 10] of a charged particle of effective mass m * iswhere ϕ is the azimuthal angle of a point on the ring, hΩ =h 2 /2m * a 2 is the parameter characterizing the kinetic energy of the charge and ω =α/ha is the frequency associated with the spin-orbit interaction. Apart from constants, the Hamiltonian (1) is the square of the sum of the z component of the orbital angular momentum operator L z = −i ∂ ∂ϕ , and of ω Ω S r , where S r = σ r /2 is the radial component of the spin (both measured in units ofh). H commutes in a nontrivial way with K = L z + S z , the z component of the total angular momentum. H also commutes...
