Quaternionic Computing

Abstract

We introduce a model of computation based on quaternions, which is inspired on the quantum computing model. Pure states are vectors of a suitable linear space over the quaternions. Other aspects of the theory are the same as in quantum computing: superposition and linearity of the state space, unitarity of the transformations, and projective measurements. However, one notable exception is the fact that quaternionic circuits do not have a uniquely defined behaviour, unless a total ordering of evaluation of the gates is defined. Given such an ordering a unique unitary operator can be associated with the quaternionic circuit and a proper semantics of computation can be associated with it. The main result of this paper consists in showing that this model is no more powerful than quantum computing, as long as such an ordering of gates can be defined. More concretely we show, that for all quaternionic computation using n quaterbits, the behaviour of the circuit for each possible gate ordering can be simulated with n+1 qubits, and this with little or no overhead in circuit size. The proof of this result is inspired of a new simplified and improved proof of the equivalence of a similar model based on real amplitudes to quantum computing, which states that any quantum computation using n qubits can be simulated with n+1 rebits, and in this with no circuit size overhead. Beyond this potential computational equivalence, however, we propose this model as a simpler framework in which to discuss the possibility of a quaternionic quantum mechanics or information theory. In particular, it already allows us to illustrate that the introduction of quaternions might violate some of the ``natural'' properties that we have come to expect from physical models.