Topological Quantum Computing: Foundations, Research Trends, and Long-Term Impact

Abstract

Topological quantum computation relies on the concepts of non-Abelian anyons and topological superconductors, holding great promise for noise-protected computation scalable since it stores quantum information coherently, irrespective of position. Included in this review are fundamentals like Majorana fermion zero modes, current innovations at Microsoft regarding nanowires, along with challenges and plans for scaling quantum computation at the level of million-qubits by the year 2030. It employs the use of anyons, a kind of quasi particle that exists in a two-dimensional setting. The world lines of the anyons cross each other in a braid fashion in the three-dimensional spacetime (one time-dimensional and two space-dimensional). The braids correspond to the logic gates in a computer. The one big advantage that the use of the quantum braid has over the trapped quantum particles has to do with stability. A small disturbance in a system may cause the decoherence of the quantum states and errors in calculation in a regular quantum computer, but in the braids, the disturbance does not affect the topological properties because it is like the difference between cutting and reconnecting a string to create a new braid compared to a ball (standing in place for a regular trapped quantum particle in four-dimensional spacetime) hitting a wall. Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum Turing machine model. That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.