Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation.
Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical.
In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
