Engineering the Kitaev chain

Nanotechnology enables the study of various quantum phenomena on real hardware. For instance, semiconducting and superconducting nanostructures can define single-electron transistors, quantum dots, Josephson junctions, and many other examples of quantum devices. It's a wonderful sandbox.
In this thesis, we exploit such a technology to bring the Kitaev chain model to life. The Kitaev Hamiltonian, discussed in the second chapter of this dissertation, describes a chain of N fermionic sites coupled by a standard tunneling and a more exotic superconducting pairing. It is one of the simplest models able to bring the concept of topology into condensed matter physics. Proposed more than twenty years ago, it attracted many experimental groups around the world, due to the promise of realizing a topologically protected qubit. This would be encoded into the Majorana bound states predicted to appear at the ends of the chain. However, such a qubit was never made, due to the difficulty of reproducing the Kitaev model with realistic, hence imperfect, materials.
Here, we demonstrate that engineering Kitaev chains with state-of-the-art materials is possible, by compensating imperfections with fine tuning. As opposed to top-down approaches, this requires building the chain site-by-site and tuning carefully each of them. In this work, each site is represented by a semiconducting quantum dot, while short semiconducting-superconducting hybrids mediate the inter-dot couplings. First, we describe minimal arrays of two quantum dots, show how to control every term of the Kitaev Hamiltonian, and detect the appearance of Majorana bound states. Then, we generalize the tuning procedure to three-site Kitaev chain devices. We also study the additional complications caused by multiple superconductors on the same device.
The main downside of a few-site Kitaev chain is the lack of topological protection. Nevertheless, we demonstrate that its Majorana bound states already exhibit partial protection (against some parameter perturbations), which increases substantially from two- to three-site chains. In the outlook, we propose to generalize the techniques described here to realize a rudimentary Majorana qubit and scale up to even longer Kitaev chains, whose partial protection evolves into topological as N grows.