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Description
We define a free space qubit for free electron to be a system where an electron wavepacket located on the left side of some finite space represents $|1\rangle$ and the wavepacket on the right side represents $|0\rangle$. As the system propagates in time through a given potential set by a laser, the electron moves rapidly between 3 states: $|0\rangle$, $|1\rangle$ and $\frac{1}{\sqrt{2}}(|0\rangle -|1\rangle)$. Such a system represents an electron in an infinite well or a harmonic oscillator, and the rapid transition between the states is can be understood as a Talbot effect for electrons in a shaped potential.
The potential created by the laser can capture the electron in the transverse direction, while it propagates in z axis with time. In particular, if the electron is initiated in a gaussian wavepacket in space, we the spatial form of the wavefunction will remain the same as in a guided mode.
When the electrons are captured in the transverse direction, we can create a the two qubit qubit phase shift can be achieved through coulomb interaction of the two electrons, that allows for a CNOT gate and therefore completes a full set of quantum operations.