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Quantum Information Processing using the Power-of-SWAP
This project is a comprehensive investigation into the application of the exchange interaction,
particularly with the realization of the SWAP^1/n quantum operator, in quantum information
processing. We study the generation, characterization and application of entanglement in such
systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical
study of combinations of these gates is an interesting avenue of research that we have
explored, though due to the exponential scaling of the complexity of the problem with the
number of qubits in the system, numerical techniques, though good for few-qubit systems, are
found to be inefficient for this research problem when we look at systems with higher number
of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric
group Sn, we employ group-theoretic methods to find the relevant invariant subspaces
and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving
symmetry of the vectors spanning the various invariant subspaces. We also devise new
ways of characterizing entanglement and approach the separability problem by looking at
permutation symmetries of subsystems of quantum states. This idea is found to form a
bridge with the entanglement characterization tool of Peres-Horodecki’s Partial Positive
Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented ‘distance’ measures of entanglement, besides devising a new entanglement witness
in the ‘engle’. In terms of applications, we define five different formalisms for quantum
computing: the circuit-based model, the encoded qubit model, the cluster-state model,
functional quantum computation and the qudit-based model. Later in the thesis, we explore
the idea of quantum computing based on decoherence-free subspaces. We also investigate
ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum
communication protocols and quantum memory.Trinity Barlow Scholarship by Trinity College (University of Cambridge), Nehru Bursary by Nehru Trust for Cambridge University, Hitachi CASE Grant by Hitachi-Cavendish Laboratory, Grants from Semiconductor Physics (SP) and Thin Film Magnetism (TFM) Groups, Cavendish Laboratory, University of Cambridg