Abstract
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.
Highlights
Representing involved mathematical formulae with simple diagrams has been a common strategy to drive progress in physics
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
We find that symmetric diagrams represent symmetric states, offering a straightforward, diagrammatic way to implement crystal symmetries, not available to other tensor networks
Summary
Representing involved mathematical formulae with simple diagrams has been a common strategy to drive progress in physics. To exemplify the power of the ZXH representation further, we prove entirely diagrammatically that the 2D AKLT-state reduces to a graph state under a suitable set of measurements To use ZXHdiagrams to represent quantum states that live in larger Hilbert spaces (such as the spin-1 states in a 1D AKLTstate) we need to encode these larger Hilbert spaces into tensor products of two-dimensional Hilbert spaces We solve this problem by resorting to the representation theory of SU(2), which tells us there is a unique N -dimensional representation given by the symmetric subspace of N − 1 copies of C2.
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