AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states

TL;DR

This paper introduces the ZXH-calculus, a graphical language for representing and reasoning about many-body quantum states, demonstrating its effectiveness on AKLT states and topological phases, and enabling new analytical insights.

Contribution

The paper develops the ZXH-calculus for graphically representing many-body states, extending it to higher spins, and applying it to AKLT states and topological phases, providing new analytical tools.

Findings

Recovered AKLT matrix-product state representation

Proved the Berry phase of AKLT chain is π

Demonstrated 2D AKLT state as a universal resource

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.