  Representations of quantum computations are almost always based on a tensor
product $\otimes$-structure. This coincides with what we are able to execute in
our experiments, as well as what we observe in Nature, but it makes certain
familiar quantum primitives convoluted. Reversible classical circuits, diagonal
operations or controlled unitaries all have very elegant and simple matrix
representations that cannot be expressed succinctly as a circuit in a simple
gate set, complicating quantum algorithm design and circuit optimization.
  We propose a new additive presentation of quantum computation to address
this. We show how conventional circuits can be expressed in the additive space
and how they can be recovered. In particular in our formalism we are able to
synthesize high-level multi-controlled primitives from low-level circuit
decompositions, making it an invaluable tool for circuit optimization. Our
formulation also accepts a circuit-like diagrammatic representation and
proposes a novel and simple interpretation of quantum computation.
