Abstract
In this paper, I study the invariant subspaces of quantum states under SWAPα gates arising from the exchange interaction and their use in quantum computation. I investigate the generation and characterization of invariant-subspace vector-states that arise from such gates. I also state a condition for the locus of states that are accessible using the SWAPα gates, given an initial input state.
Highlights
Open AccessThe non-local physical phenomena related to quantum entanglement [1] [2] [3]have played a pivotal role in the development of quantum mechanics during the last century
Since local single-qubit operations are difficult to implement in hardware, Divincenzo et al subsequently looked at how universal quantum computation can be achieved by using only SWAPα gates if one encodes pseudo-spin qubits using three physical qubits [33]
We find that a combination of all SWAPα gates between a finite number of qubits comprise a group that is isomorphic to the Symmetric Group SSnn, which is the group of all permutations or self-bijections of a set of nnelements with the operation of composition, and that only certain points in the Hilbert Space are accessible using the SWAPα gates, given a particular input state
Summary
Have played a pivotal role in the development of quantum mechanics during the last century. The SWAPα gate, together with single-qubit operations, is found to produce a universal set of quantum gates (enabling any quantum computation) [33] These two-qubit gates can be realised in several physical systems. Since local single-qubit operations are difficult to implement in hardware, Divincenzo et al subsequently looked at how universal quantum computation can be achieved by using only SWAPα gates if one encodes pseudo-spin qubits using three physical qubits [33]. This naturally leads to the question of whether universal quantum computation could be implemented purely using SWAPα with a specific value of α.
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