Quantum Computing with Octonions

There are two schools of “measurement-only quantum computation”. The first (Phys. Rev. Lett. 86(22), 5188–5191 (2001)) using prepared entanglement (cluster states) and the second (Phys. Rev. Lett. 101(1), 010501 (2008)) 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.