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

Abstract

Classical reversible computers on w bits are isomorphic to the (finite) symmetric group S2w; quantum computers on w qubits are isomorphic to the (Lie) unitary group U(2w). 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 (2w)!,infinite but discrete groups, andLie groups of dimension smaller than (2w).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. 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 2w × 2w 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.