Researchers have introduced a new family of fractal quantum many-body states, constructed from ZX-diagrams based on the Sierpiński triangle and Sierpiński carpet. These states exhibit atypically low entanglement, with the triangle family obeying an area law and the carpet family displaying approximately logarithmic scaling. Furthermore, their local observables retain fractal-like spatial structure, identifying them as natural candidates for atypical eigenstates in otherwise thermalizing systems.

The study explores the potential of ZX-calculus, a diagrammatic language for quantum processes, as a framework for constructing many-body states and the Hamiltonians that host them. By combining parent-Hamiltonian methods, ZX-calculus insights, and local ZX identities, the scientists have designed frustration-free Hamiltonians that exactly annihilate the target state. These Hamiltonians admit simple representations in the same diagrammatic language as the states themselves.

A key advancement is the construction of a local deformation that produces chaotic level statistics while embedding the fractal ZX state in the bulk of the energy spectrum as an exact quantum many-body scar. This result demonstrates that ZX-calculus can serve as a framework for Hamiltonian inverse design, allowing for the construction and relation of quantum many-body scars, their local annihilators, and the chaotic Hamiltonians embedding them, all through a set of graphical identities.