  From Feynman diagrams to tensor networks, diagrammatic representations of
computations in quantum mechanics have catalysed progress in physics. These
diagrams represent the underlying mathematical operations and aid physical
interpretation, but cannot generally be computed with directly. In this paper
we introduce the ZXH-calculus, a graphical language based on the ZX-calculus,
that we use to represent and reason about many-body states entirely
graphically. As a demonstration, we express the 1D AKLT state, a symmetry
protected topological state, in the ZXH-calculus by developing a representation
of spins higher than 1/2 within the calculus. By exploiting the simplifying
power of the ZXH-calculus rules we show how this representation
straightforwardly recovers the AKLT matrix-product state representation, the
existence of topologically protected edge states, and the non-vanishing of a
string order parameter. Extending beyond these known properties, our
diagrammatic approach also allows us to analytically derive that the Berry
phase of any finite-length 1D AKLT chain is $\pi$. In addition, we provide an
alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced
to a graph state, demonstrating that it is a universal quantum computing
resource. Lastly, we build 2D higher-order topological phases diagrammatically,
which we use to illustrate a symmetry-breaking phase transition. Our results
show that the ZXH-calculus is a powerful language for representing and
computing with physical states entirely graphically, paving the way to develop
more efficient many-body algorithms and giving a novel diagrammatic perspective
on quantum phase transitions.
