The ZX-Calculus is Canonical in the Heisenberg Picture for Stabilizer Quantum Mechanics

Abstract: 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.