Quantum computing with Octonions

There are two schools of "measurement-only quantum computation". The first ([11]) using prepared entanglement (cluster states) and the second ([4]) using collections of anyons, which according to how they were produced, also have an entanglement pattern. We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections. This notion is the leading character in the paper. The largest continuous family, in a sense made precise in Corollary 4.2, is associated with the octonions and this example leads to a universal computational scheme. Adiabatic quantum computation also fits into this rubric as a limiting case: nearby projections are nearly equiangular, so as a gapped ground state space is slowly varied the corrections to unitarity are small.