AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States

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