Superlattice model for quantum gates

Abstract

Probability electronic current conservation in superlattices leads to a natural connection with the qubit and quantum gate concepts of quantum information and the introduction of Bloch spheres and Hopf bundles. Superlattices or multilayer devices with coupled channels are described by sectionally constant potentials in the longitudinal direction and arbitrary lateral dependency, which allow the explicit analytic calculation of the scattering amplitudes connecting ingoing with outgoing wave functions. A superlattice with n open channels and two terminals can be seen as a quantum gate, since both can be viewed as quantum systems with n components, each of which can have two states given by input and output. Taking into account the dimensionality of the respective Hilbert spaces, it results that the one-channel or mode superlattice corresponds to a single qubit; the two-channels case to two qubits, and the three- and four-channels cases to three qubits. The coupling of modes or channels corresponds to the entanglement of qubits. As shown in this work, superlattices with interacting energy modes constitute physical systems which allow to understand and evaluate explicitly the main features used to describe entangled qubits through the study of Hopf fiber bundles on spheres. Explicit examples for superlattice gate models with coupled channels are provided.