  In 2008 Coecke and Duncan proposed the graphical ZX-calculus rewrite system
which came to formalize reasoning with quantum circuits, measurements and
quantum states. The ZX-calculus is sound for qubit quantum mechanics. Hence,
equality of diagrams under ZX-equivalent transformations lifts to an equality
of corresponding equations over matrices. Conversely, in 2014 Backens proved
completeness, establishing that any derivation done in stabilizer quantum
mechanics with matrices can be derived graphically using the ZX-calculus. A
graphical rewrite system that is both confluent and also terminates uniquely is
called canonical: Applying alternate sequences of rewrites to the same initial
diagram, a rewrite system is confluent whenever all resulting diagrams can be
manipulated to establish graphical equivalence. Here we show that a reduced
ZX-rewrite system is already confluent in the Heisenberg picture for stabilizer
quantum mechanics. Moreover, any application of a subset of ZX-rewrites
terminates uniquely and irrespective of the order of term rewrites in the
Heisenberg picture for stabilizer quantum mechanics. The ZX-system is hence
Heisenberg-canonical for stabiliser quantum mechanics. For a stabilizer circuit
on $n$-qubits with $l$ single-qubit gates and $g$ two-qubit gates, the circuit
output can be derived graphically in the Heisenberg picture using no more than
$(\frac{1}{2}\cdot g+l)\cdot n$ graphical rewrites, thereby providing a
graphical proof of the Gottesman-Knill theorem. Finally, we establish that each
stabilizer state described by a Clifford circuit gives rise to a non-negative
parent Hamiltonian with $n+1$ terms and a one-dimensional kernel spanned by the
corresponding stabilizer state. Such parent Hamiltonians can be derived with
$\mathcal{O}(t\cdot n)$ graphical rewrites for a low energy state prepared by a
$t$-gate Clifford circuit.
