Creating subgroups of U(2w) for quantum-minus computers
Alexis De Vos, Michiel Boes · Journal of Physics Conference Series · 2011
Classical reversible computers on w bits are isomorphic to the (finite) symmetric group S 2 w ; quantum computers on w qubits are isomorphic to the (Lie) unitary group U(2 w ). We investigate and classify groups X which represent computers intermediate between classical reversible computers and quantum computers. Such intermediate groups X may exist in three flavours: finite groups of order larger than (2 w )!, infinite but discrete groups, and Lie groups of dimension smaller than (2 w ). For our purpose, we start from 1-qubit transformations, represented by 2 × 2 unitary matrices. We call this group the creator. Its members are called gates and act on one qubit. Controlled gates are quantum circuits acting on w qubits, such that the 1-qubit transformation (applied to a particular qubit) depends on the state of the w − 1 other qubits. The controlled gates generate the group X of 2 w × 2 w matrices, called the creation. We discuss all creators of order up to 8. Additionally a creator of order 16 and one of order 192 are discussed.