Quantum many-body scars with tunable entanglement and Hamiltonian inverse design from ZX-calculus

Fractal quantum many-body scars and Hamiltonian inverse design from ZX-calculus introduce a novel framework for constructing atypical eigenstates in thermalizing systems by combining fractal many-body states with diagrammatic representations. The central finding is that ZX-calculus can serve as a diagrammatic route to finding exact quantum many-body scars and exact forms of the corresponding chaotic Hamiltonians, allowing for the construction and relation of these elements through graphical identities.

The Construction of Fractal Many-Body States

The paper introduces families of fractal many-body states obtained from ZX-diagrams based on the Sierpiński triangle and Sierpiński carpet. The fractality is encoded in the diagrammatic structure rather than a real-space lattice geometry. By construction, the underlying graph connectivity imposes atypical subvolume-law minimum-cut upper bounds on the entanglement. Specifically, for the triangle family, these states obey an area law for bipartite entanglement entropy. For the carpet family, they display approximately logarithmic scaling across the available system sizes. Furthermore, their local observables retain fractal-like spatial structure, making them natural candidates for quantum many-body scars (QMBS).

Hamiltonian Inverse Design via Local Annihilators

The construction of a parent Hamiltonian is achieved by combining parent-Hamiltonian methods with local ZX identities that certify exact annihilation of the target state. This process involves several key steps:

1. Constructing reduced density matrices for adjacent sites to find local annihilators, where any positive semidefinite operator whose support lies inside the kernel of these matrices annihilates the target state locally.

2. Using insights from ZX-calculus to search for analytically simple operators that admit compact ZX representations within the local annihilator subspaces, rather than relying solely on numerical kernels.

3. Constructing local Hamiltonian terms, such as those involving projectors like ΠZ i,j = I − ZiZj2 = π, and rank-one projectors like Ξi(k) = 1 exp (−iϕk) exp (iϕk).

4. Verifying annihilation locally by inserting the Hamiltonian term diagram into the corresponding boundary subdiagram of the state and applying ZX rewrite rules, which yields specific identities such as Eq. (15), (16), and (17).

Embedding into a Chaotic Spectrum

The method proceeds in two stages to embed the fractal state as a QMBS:

1. First, constructing a parent Hamiltonian Hpar where the target state is an exact zero-energy ground state, which is shown to be frustration-free because all local terms are positive semidefinite and commute. Spectral diagnostics show that this parent Hamiltonian exhibits near-Poissonian spectral statistics.

2. Second, constructing a perturbed local Hamiltonian by adding a deformation term, such as V = −L X−3 i=4, i even Hb i(gi), which is compactly representable in ZX-diagrams. This deformation is designed to produce chaotic level statistics while embedding the fractal ZX state in the bulk of the energy spectrum as an exact quantum many-body scar. The resulting deformed Hamiltonian preserves most symmetries of the parent model, allowing for a transition from near-Poissonian statistics to Wigner-Dyson statistics (GUE) upon deformation.

Diagrammatic Verification and Conclusion

The paper demonstrates that ZX-calculus can serve as a framework for Hamiltonian inverse design, as the same graphical language describes the entire process: state construction, identification of local operators, parent Hamiltonian construction, and chaotic deformation. The ZX rewriting exposes the local annihilation mechanism that preserves the exact eigenstate. The results confirm that these fractal ZX-diagrams possess subextensive entanglement entropy (area law for triangles) and self-similar profiles of local expectation values, establishing them as promising candidates for atypical eigenstates in otherwise thermalizing systems. This work suggests ZX-calculus can be used as a systematic tool for engineering physically relevant interacting Hamiltonians with prescribed nonthermal eigenstates. The paper concludes by opening avenues to extend this construction to other exact scar states and the study of real-time dynamics using ZXW formalisms.

The gist: Fractal many-body states constructed from Sierpiński triangle and carpet ZX-diagrams act as quantum many-body scars for chaotic Hamiltonians, with the entire construction—from state preparation to Hamiltonian engineering—being described by graphical identities within the ZX-calculus framework.

Key Enumerated Elements:

(The paper enumerates several key concepts and steps, including:)

1. Construction of fractal ZX-diagrams based on Sierpiński triangle and carpet graphs.

2. Investigation of entanglement bounds using minimum-cut to show atypical scaling (area law for triangle, logarithmic for carpet).

3. Construction of a local parent Hamiltonian Hpar where the target state is an exact zero-energy ground state, utilizing kernels of reduced density matrices or analytic ZX representations of local annihilators (e.g.

Based on the provided scientific paper, here are specific improvements for AI systems derived from its core concepts:

)AI System Improvement 1: Hamiltonian Inverse Design Framework using ZX-Calculus

The paper establishes a framework where a target quantum many-body state (a fractal ZX-diagram) can be systematically related to an exact ground state of a local, frustration-free parent Hamiltonian, and subsequently embedded as an atypical eigenstate (Quantum Many-Body Scar) within a chaotic Hamiltonian.

The AI system improvement involves integrating this framework into a generative or design module for quantum Hamiltonians.

Specific Capabilities:

1. Genetically Design Atypical Hamiltonians: The AI can be trained to take the diagrammatic representation of a desired atypical state (encoded via ZX-diagrams) and automatically generate the corresponding frustration-free parent Hamiltonian using local ZX identities (Eqs. 15, 16, 17).

2. Chaotic Embedding Module: The system can then apply a learned or parameterized scar-preserving deformation (like Eq. 20) to this parent Hamiltonian to generate a chaotic Hamiltonian whose bulk spectrum hosts the target state as an exact quantum many-body scar.

3. Symmetry Preservation: Because the construction relies on local ZX identities, the AI can engineer Hamiltonians that are not only chaotic but also preserve significant symmetry structures (e.g., reflection symmetry R, edge symmetries Z1/ZL) of the original parent model, which is critical for targeted physics studies.

How this improves AI systems:

This moves AI beyond simple black-box optimization or state generation toward a principled Inverse Design capability for quantum hardware and simulation targets. It allows researchers to design Hamiltonians with specific, desired non-thermal features (like localized scars) rather than relying on brute-force search or approximations.

)AI System Improvement 2: Automated State/Hamiltonian Verification via Diagrammatic Proofs

The paper demonstrates that the relationship between the state, the parent Hamiltonian, and the chaotic Hamiltonian can be proven diagrammatically using ZX rewrite rules (Appendix B). This provides a rigorous, symbolic verification method.

Specific Capabilities:

1. Symbolic Consistency Checker: An AI module can be tasked with verifying if a proposed local Hamiltonian term (represented as a ZX-diagram) actually annihilates the target fractal state, by checking if the diagrammatic identity (Eqs. 15-17) holds true under ZX rewrite rules.

2. Hamiltonian Simplification and Canonicalization: The system can use PyZX/ZXLive tools to simplify complex, numerically derived local annihilators into their minimal, algebraically simplest ZX forms, ensuring that the resulting Hamiltonian terms are as compact as possible for subsequent simulation or hardware mapping.

How this improves AI systems:

This significantly reduces the risk of introducing errors in complex quantum design pipelines. Instead of relying solely on numerical convergence checks (which can be fooled), the AI uses exact diagrammatic algebra to certify physical constraints, ensuring that the designed Hamiltonian truly hosts the desired scar state.

)AI System Improvement 3: Entanglement Structure Predictor and Classifier

The paper links graph-theoretic properties of ZX-diagrams (minimum cut bounds) to actual entanglement scaling (area law vs. logarithmic/power-law). It also shows how local observables exhibit fractal spatial structures.

Specific Capabilities:

1. Entanglement Signature Classifier: An AI can analyze a generated or simulated quantum state's underlying diagrammatic structure and predict its expected entanglement scaling (e.g., Area Law, Logarithmic Scaling, or Power-Law Scaling) based on the topology of its ZX-diagram (Triangle vs. Carpet connectivity).

2. Atypicality Detector: The system can analyze local observable expectation values (like Fig. 6) to detect self-similar, fractal spatial modulations, providing a direct diagnostic for identifying atypical eigenstates in thermalizing systems.

How this improves AI systems:

This allows the AI to rapidly screen and categorize quantum states based on their physical behavior without running expensive full MPS simulations immediately. It acts as a fast filter for identifying potential Quantum Many-Body Scar candidates among millions of possible states, focusing computational resources only on those exhibiting the desired atypical signatures.