We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows: we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra; we extract an equational theory for a quantum programming language from the algebraic theory; we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.
Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are hermitian operators which are idempotents also. We call them quantum idempotents. They are also important for the conceptual understanding of quantum theory because projection operators also represent observation process on quantum system. In this paper we explore the algebra of quantum idempotents and show that they generate Iterant algebra (defined in the paper), Lie algebra, Grassmann algebra and Clifford algebra which is very interesting because these later algebras were introduced for the geometry of spaces and hence are called geometric algebras. Thus the projection operator representation gives a new meaning to these geometric algebras in that they are also underlying algebras of quantum processes and also they bring geometry closer to the quantum theory. It should be noted that projection operators not only make lattices of quantum logic but they also span projective geometry. We will give iterant representations of framed braid group algebras, parafermion algebras and the
su(3) algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).