Creating subgroups of U(2^w) for quantum-minus computers

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). Although S_{2^w} is a subgroup of U(2^w), the step from S_{2^w} to U(2^w) is huge. We investigate and classify groups X which are simultaneously supergroup of S_{2^w} and subgroup of U(2^w), such that they represent computers which are 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)^2. The larger the group, the more powerful the computer may be, but the smaller the group, the easier it can be to build the computer hardware. In the present paper, we investigate the first two flavours only. For our purpose, we start from 1-qubit transformations, represented by 2 * 2 unitary matrices, forming a group which is simultaneously a subgroup of U(2) and a supergroup of the group (isomorphic to S_2) consisting of the 2 * 2 IDENTITY matrix and the 2 * 2 NOT matrix. We call this group the creator X_2. 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 a group of 2^w * 2^w matrices, called the creation X. We discuss all creators of order up to 8. Additionally a creator of order 16 and one of order 192 are discussed.

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