Fractal quantum many-body scars and Hamiltonian inverse design from ZX-calculus

Diagrammatic languages such as ZX-calculus provide compact and intuitive descriptions of quantum processes and have become established tools for circuit simplification, verification, and compilation. However, their potential as a framework for constructing many-body states and the Hamiltonians that host them remains largely unexplored. Here, we introduce families of fractal many-body states obtained from ZX-diagrams based on the Sierpi\'nski triangle and Sierpi\'nski carpet. By construction, the underlying graph connectivity imposes atypical subvolume-law minimum-cut upper bounds on the entanglement, while the actual states are parametrically less entangled still: the triangle family obeys an area law, whereas the carpet family displays approximately logarithmic scaling across the available system sizes. Additionally, their local observables retain fractal-like spatial structure, identifying these states as natural candidates for atypical eigenstates in otherwise thermalizing systems. For the triangle family, we combine parent-Hamiltonian methods, insights from ZX-calculus, and local ZX identities that certify exact annihilation of the target state, producing frustration-free Hamiltonians whose terms admit simple representations in the same diagrammatic language as the states themselves. We then construct 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. Our results demonstrate, through this explicit construction, that ZX-calculus can serve as a framework for Hamiltonian inverse design, in which quantum many-body scars, their local annihilators, and the chaotic Hamiltonians embedding them can be constructed and related through a set of graphical identities.