  Graphical calculi are vital tools for representing and reasoning about
quantum circuits and processes. Some are not only graphically intuitive but
also logically complete. The best known of these is the ZX-calculus, which is
an industry candidate for an Intermediate Representation; a language that sits
between the algorithm designer's intent and the quantum hardware's gate
instructions. The ZX calculus, built from generalised Z and X rotations, has
difficulty reasoning about arbitrary rotations. This contrasts with the
cross-hardware compiler TriQ which uses these arbitrary rotations to exploit
hardware efficiencies. In this paper we introduce the graphical calculus ZQ,
which uses quaternions to represent these arbitrary rotations, similar to TriQ,
and the phase-free Z spider to represent entanglement, similar to ZX. We show
that this calculus is sound and complete for qubit quantum computing, while
also showing that a fully spider-based representation would have been
impossible. This new calculus extends the zoo of qubit graphical calculi, each
with different strengths, and we hope it will provide a common language for the
optimisation procedures of both ZX and TriQ.
