Oxford Scientist Builds Many-Body States Using ZX-Calculus Designs

Marcin Szyniszewski has developed a method for building quantum many-body states directly from the geometry of fractals, using a diagrammatic language called ZX-calculus. The work introduces families of states derived from the Sierpiński triangle and Sierpiński carpet, exhibiting surprising entanglement behaviors; the triangle family follows an area law, while the carpet family scales approximately logarithmically. Researchers found that this approach not only describes quantum states but actively constructs them, a previously underexplored application of ZX-calculus beyond circuit manipulation. “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,” Szyniszewski writes. The research demonstrates a way to guarantee the existence of quantum many-body scars within chaotic energy spectra, embedding fractal states through frustration-free Hamiltonians.

ZX-calculus, a diagrammatic language for quantum processes, is now being applied to the construction of quantum many-body states, extending its traditional role in circuit manipulation. While typical quantum states adhere to subvolume-law expectations for entanglement, the triangle family exhibits an area law, meaning entanglement growth scales with the surface area of the system. In contrast, the carpet family demonstrates approximately logarithmic scaling, a significantly slower increase in entanglement with system size. These differing behaviors arise from the unique connectivity imposed by the fractal structures, which create atypical subvolume-law minimum-cut upper bounds on entanglement. Szyniszewski combined parent-Hamiltonian methods with ZX-calculus insights to create frustration-free Hamiltonians for the triangle family, certifying exact annihilation of the target state with local ZX identities.

This difference in entanglement arises from the unique connectivity inherent in each fractal structure, creating a specific arrangement of quantum bits, and this process embeds the fractal state as a precise quantum many-body scar within a chaotic energy spectrum.