Quantum computing based on low-dimensional systems in germanium and silicon

Condensed matter physics revolutionized our lives in the first half of the 20th century by providing the tools for the development of the transistor which is the building block of any modern classical computer. The working principle of transistors is based on quantum mechanics indirectly since it relies on the electronic band structures of the used semiconductors. However, the computations performed on a computer built of transistors are purely classical.
In the 1980s, Richard Feynman proposed a groundbreaking idea for the simulation of physics: the use of quantum computers that obey the rules of quantum mechanics rather than classical mechanics to unleash full computational power. Since then, physicists have been trying to build a quantum computer that is predicted to be capable of solving complex problems like prime factorization of large numbers, the search in large unsorted databases, and solving linear systems of equations much faster than a classical computer.
Though a universal quantum computer with millions of quantum bits (qubits) remains a distant goal, significant progress has been made in the past four decades. Demonstrations of quantum advantage over a classical computer for specific tasks have been achieved with superconducting quantum chips hosting ∼ 10 − 100 qubits. However, scaling up to millions of qubits with superconducting devices faces challenges due to the relatively large size of individual superconducting qubits, approximately 10 μm.
A promising alternative realization comes from condensed matter physics as well: the spin qubit. In the original proposal of the so-called Loss-DiVincenzo qubit it was suggested to utilize the spin-1/2 degree of freedom of an electron confined in a quantum dot (QD) as a quantum mechanical two-level system, forming the qubit. It is about two orders of magnitude smaller than superconducting qubits and thus raises hope for better scalability. The number of realized spin qubits stands behind superconducting systems, but the wealth of experience from the advanced semiconductor industry for Si promises fast development in spin qubit research. The integration of industrial processes in spin qubit fabrication has become possible, but electrons in Si come with a disadvantage that could possibly spoil the advantages for scalability: they exhibit only weak intrinsic spin-orbit interaction (SOI). Consequently, fast, all electrical qubit control relies on stray fields from micromagnets on the quantum device. This encouraged research on holes in Si and Ge with very strong SOI enabling ultrafast qubit control without large additional elements on the chip.
The small size of spin qubits comes at the cost of lower connectivity between qubits compared to superconducting systems. While exchange interaction allows for two-qubit gates, its short-range nature makes it incapable of coupling far-distant qubits which is desirable, e.g., for multi-qubit entanglement or quantum error correction. One solution for long-range coupling involves the use of circuit quantum electrodynamics, where the spin qubits in semiconductor QDs interact strongly with the photons in a superconducting cavity.
Furthermore, coupling semiconductors and superconductors (SCs) opens up a whole field of interesting physical phenomena among which are Andreev spin qubits (ASQs) and topological superconductivity with associated Majorana bound states (MBSs). ASQs have the potential to combine the advantages of superconducting and spin qubits in QDs and MBSs promise a new approach towards quantum computing relying on topological protection.
Motivated by these developments, this thesis delves into theoretical studies of low-dimensional hole and electron systems in Ge and Si. Chapter 1 introduces the field of quantum computing with spin qubits. After introducing some basic concepts of quantum computing in general, we discuss the properties of group-IV semiconductors, in particular Si and Ge. We explain what a spin qubit is, how it is implemented in a semiconductor QD, how it is controlled via single- and two-qubit operations, how it can be initialized and read out, and how it interacts with the environment. In the final part of this chapter, we pick two phenomena arising in hybrid SC-semiconductor devices, namely the Andreev bound state which can be harnessed to implement ASQs, and the MBS as an example of a topological state.
In Chapter 2, we analyze in detail one-dimensional Ge hole device designs that optimize SOI. We provide a new analytical approach for the estimation of SOI in the presence of transversal electric and magnetic fields where we treat orbital magnetic fields exactly. Assisted by numerical calculations we analyze the electric and magnetic field dependence as well as the dependence on strain and anisotropies of SOI, g factor, and effective masses. These quantities enter the one-dimensional, low-energy effective model that we derive. In particular, we stress the importance of orbital magnetic fields for the g factor. Considering electrostatic confinement of a QD in the onedimensional system we predict the existence of a g-factor sweet spot that is tunable by strain and confinement potential. We expect highly coherent qubits and fast Rabi frequencies at low power for realistic device parameters. Eventually, we identify a regime of flat bands with promising potential for the simulation of strongly correlated matter.
Whereas in the previously discussed model the magnetic field was pointing in perpendicular direction to the nanowire (NW) axis, in Chapter 3 we analyze similar one-dimensional Ge hole structures with a magnetic field aligned along the NW axis. Also for this case we provide a analytical solution and derive an effective one-dimensional model. Among our core results is the strong renormalization of the effective g factor due to orbital magnetic fields even at weak magnetic fields. Furthermore, we provide a detailed discussion of strain, growth direction, energetically higher-lying valence bands, and different designs of one-dimensional systems. We raise special attention to the large effective g factor and SOI in curved quantum wells at weak electric fields suggesting the applicability of such devices as ideal hosts for MBSs. The same device design exhibits a g factor and SOI independent of the electric field at stronger electric fields, ideal for the optimization of the spin qubit coherence time.
In Chapter 4, we investigate the proximity-induced superconductivity and metallization effects in SC-Ge hole NWs. Taking into account the three-dimensional nature of the NW we predict a strong dependence of the induced gap and metallization effects on the proximity of the hole wavefunction in the Ge part to the SC. Most interestingly, by employing an external electric field the SOI and the proximity-induced pairing potential can be tuned to large values at the same time, making SC-Ge hole NWs a promising platform for quantum information processing via, e.g., ASQs or MBSs.
Finally, in Chapter 5, we propose a novel approach to lifting the valley degeneracy of the Si conduction band edge, which is a major hurdle on the way to large-scale Si-based quantum processors. In experimentally relevant fin field-effect transistor devices shear strain enhances the gap to non-computational valley states to values ∼ 1 meV − 10 meV. This proposal does not rely on atomic-size interface details as predicted for planar Si/SiGe heterostructures in previous works and is robust against interface disorder. We show that the effect remains large for realistic values of applied electric fields and is largely independent of the fin shape.