# From topology in condensed matter to quantum information and spintronics

**Quantum many-body systems in condensed matter physics can realize boundary states or bulk excitations that are protected by topology. Such states or excitations are endowed with a robustness to local perturbations by topology. This topological robustness renders such boundary states or bulk excitations promising candidates as physical platforms for spintronics or quantum information. This project aims at constructing local quantum many-body Hamiltonians in three-dimensional space that support non-Abelian topological ordered ground states.**

The classical bit takes two values, say zero or one. A quantum bit takes any value on a circle, i.e., it can be thought of as a phase. As such a quantum bit is prone to decoherence, i.e., the dynamical scrambling of the phase of an isolated quantum state when the isolation is removed by coupling it to the environment. Decoherence times of the order of minutes have been achieved for few quantum bits, but the decoherence time usually decays exponentially fast with the number of quantum bits. This is typically so when quantum bits are stored locally in space. If the probability for decoherence per unit of time for a local quantum bit is 0 < p < 1, and if decoherence events for two local quantum bits are uncorrelated (owing to locality), it follows that the probability for decoherence per unit of time for n local quantum bits is 0 < p^n << 1.

However, the decoherence time does not need to decrease exponentially fast with the number of quantum bits as a matter of principle when the quantum information is not stored locally. This is so when the quantum bit is carried by a fractionalized excitation of a quantum many-body system, say the fractional quantum Hall effect (FQHE) at the filling fraction 5/2. It hosts fractional excitations that are a signature of the phenomenon of topological order. Hence, whenever topological order is the rule, then there are fractional excitations that can be used to encode non-local quantum bits.

This project aims at establishing the conditions to support non-Abelian topological order (like in the FQHE at the filling fraction 5/2) for strongly interacting electrons hosted in a three-dimensional material (unlike in the FQHE which lives in 2d).