NMR control techniques have made many important contributions in developing set of tools for optimal quantum control, ranging from spin echo to dynamical decoupling. Using such optimal quantum control techniques, the emulation of what are known as anyonic statistics was demonstrated previously. This demonstration showed the ability of NMR QIP for experimental exploration of topological quantum computing where anyonic statistics play a significant role as a means to implement the quantum gates by braiding anyons. In this thesis, going one step further than the previous demonstration, we experimentally emulated the anyonic statistics and demonstrated the path independence of anyonic braiding operations manifested in a 7-qubit Kitaev's lattice model. In our experiments, the anyons are braided along two different loops in which the system's wave function gains a π phase in theory. We experimentally measured the anyonic phases of (153.9 ± 3.8)° and (151.4 ± 3.8)° for the two different braiding paths, demonstrating that experimental anyonic phases are path independent. However, the values differ from the theoretical value due to decoherence and gate imperfections.
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