# 2.9. Flying Qubit¶

In this tutorial, we discuss multi-terminal electron transport in various nanodevices. As an example, we focus on so-called Electron flying qubits, which are solid-state counterparts of the quantum optics devices. Basic building blocks of these qubits are the folowing semiconductor bases nanodevices:

• Tunneling-coupled wires, TCW - the electronic counterpart of the optical beam splitter, see Figure 2.9.2;

• Aharonov-Bohm rings, ABR - the electronic counterpart of the optical interferometer, or their analogy realized in asymetrically gated regions, see the central region of Figure 2.9.3;

• Circuits containing these elements connected in a series, see Figure 2.9.3.

Left rectangular regions with numbers 1 and 2 are incoming leads, where the electron can be injected into the nanodevice. We will assume that it is injected into the first lead. Right rectangular regions with numbers 3 and 4 are outgoing leads, where the electron can be detected after propagating through the entire nanodevice. The functionality of the electron flying qubits request a reflectionless propagation of the electron. If the electron is reflected and returs to one of the incoming leads, a part of the quantum information is lost. The important task of numerical simulations is to identify regimes where reflection is reduced as much as possible.

The interior part of the nanodevices is assumed to be made from 2D GaAs-based semiconductor and includes regions with different electrostatic potentials and applied gate voltages. Those govern the energy profie through which the electron propagates. Colors reflect the strength of the electrostatic potential in different parts of the device, ranging from $$0$$ eV (dark blue color) up to $$\gg 1$$ eV (dark red color).

Figure 2.9.2 An example of the geometry of TCW connected to four terminals.

Figure 2.9.3 An example of the geometry of a circuit containin two TCWs and one ABR. The cicuit is also connected to four terminals.

Let us first discuss transport in TCW. The horisontal line in Figure 2.9.2 shows a potential barrier separating two paths, along which the electron can move towards the outgoing leads. Red parts of the barrier are impenetrable for the electron while the electron can tunnel through the light blue segment. The latter is precisely the region where the quantum interference btween the upper- and lower- parth takes place. Having experienced the inteference, the electron wave-function is split between the upper- and lower- path. As a result, there is a finite probability to detect the electron in one of the outgoing leads, either 3 or 4. This probability depends both on the electron energy and on the parameters of the nanodevice, including the height of the tunneling barrier. The nextnano software allows one to calculate the transmission from the lead. No.1 to the leads 3 and 4. The theoretical backgroung involves the numerical solution of the Schroedinger equation and the Contact Block Reduction method

Figure 2.9.4 Transmission

\begin{align}\begin{aligned}E_z &= \frac{\hbar^2 k_z^2}{2 m_e^*}\\\psi(z) &= \exp (\pm i k_z z)\end{aligned}\end{align}