Effective Hamiltonian for two interacting double-dot exchange-only qubits and their controlled-NOT operations

Double dot exhange only qubit represents a promising compromise between high speed and simple fabrication in solid state implementations. A couple of interacting double dot exhange only qubits, each composed by three electrons distributed in a double quantum dot, is exploited to realize Controlled-NOT (CNOT) operations. The effective Hamiltonian model of the composite system is expressed by only exchange interactions between pairs of spins. Consequently the evolution operator has a simple form and represents the starting point for the research of sequences of operations that realize CNOT gates. Two different geometrical configurations of the pair are considered and a numerical mixed simplex and genetic algorithm is used. We compare the non physical case in which all the interactions are controllable from the external and the realistic condition in which intra-dot interactions are fixed by the geometry of the system. In the latter case, we find the CNOT sequences for both the geometrical configurations and we considered a qubit system where electrons are electrostatically confined in two quantum dots in a silicon nanowire. The effects of the geometrical sizes of the nanowire and of the gates on the fundamental parameters controlling the qubit are studied by 2 E. Ferraro* et al. exploiting a Spin Density Functional Theory based simulator. Consequently, CNOT gate performances are evaluated.