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REVIEW 2 major objections 5 minor 85 references

Sierpiński-triangle ZX-diagram states are exact quantum many-body scars of a local chaotic Hamiltonian, with the parent Hamiltonian, annihilators, and scar-preserving deformation all expressed in ZX-calculus.

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2026-08-01 04:32 UTC pith:GTDWTSJ3

load-bearing objection Real new method—ZX inverse design for scars—but the exactness proof is a set of asserted diagrams, so the referee should check those identities before signing off. the 2 major comments →

arxiv 2607.22495 v1 pith:GTDWTSJ3 submitted 2026-07-24 quant-ph cond-mat.stat-mechcond-mat.str-el

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

Marcin Szyniszewski This is my paper
classification quant-ph cond-mat.stat-mechcond-mat.str-el
keywords quantum many-body scarsZX-calculusSierpiński triangleSierpiński carpetparent Hamiltoniansfrustration-freequantum chaosHamiltonian inverse design
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that ZX-calculus, a graphical language for qubit operations, can be a construction tool for quantum many-body physics, not just a circuit-simplification toolkit. It introduces many-body states whose ZX-diagrams are Sierpiński triangles and carpets, shows these states have anomalously low entanglement (area law for the triangle, near-logarithmic for the carpet) and self-similar local observables, and then builds local Hamiltonians, expressed as ZX-diagrams, for which the triangle state is an exact zero-energy eigenstate. A local deformation makes the surrounding spectrum chaotic, with level statistics matching Gaussian unitary ensemble, while the fractal state remains an exact eigenstate in the spectral bulk, the defining signature of a quantum many-body scar. A sympathetic reader would care because this supplies an inverse-design method: state, annihilators, parent Hamiltonian, and chaotic embedding all live in one diagrammatic language, with exact annihilation certified by local graphical rewrite rules.

Core claim

The central claim is that a Sierpiński-triangle ZX-diagram defines a many-body state that is an exact zero-energy eigenstate of a frustration-free local Hamiltonian, Hpar, whose terms are simple ZX diagrams built from parity projectors, a singlet projector, and phase-cancelling operators. The annihilation is certified locally by ZX identities: inserting a Hamiltonian term into the boundary of the fractal diagram and applying rewrite rules yields exactly zero, exposing the phase-cancellation mechanism. Adding a local deformation V = −Σ H^b_i, also written as a ZX diagram, preserves the exact eigenstate but destroys positivity, pushes the state into the bulk of the spectrum, and produces chaot

What carries the argument

The central object is the fractal ZX-diagram: a graph-like ZX diagram (Z-spiders joined by Hadamard edges) laid out as a Sierpiński triangle or carpet, with generation-dependent phases and open legs defining the qubits of the state. The carrying mechanism for the Hamiltonian construction is a set of local annihilators, H^a_i(k), H^b_i(k), and H^c_i(k), formed by sandwiching a phase-cancelling rank-one operator Ξ_i(k) between parity and singlet projectors, together with the local ZX identities (Eqs. 15-17, 19) showing each term annihilates the fractal state after diagrammatic rewriting. The same language then supplies the scar-preserving deformation: subtracting H^b terms swaps the outer wire

Load-bearing premise

The construction collapses if the local ZX annihilation identities, the equalities-to-zero that certify H^a_i|ψ⟩ = H^b_i|ψ⟩ = H^c_i|ψ⟩ = 0, are not valid for every generation and phase assignment, or if the small-system null-space search picked annihilator supports that do not generalize.

What would settle it

Verify by exact diagonalization on the next computed generation (L = 33 or L = 65) that the fractal ZX state has zero energy under Hpar and under Hpar + V, and that each local identity (15)-(17), (19) reduces to exact zero when simplified symbolically; a single nonzero norm or a non-GUE level statistic after symmetry resolution would falsify the scar claim. A simpler check is to contract identity (15) for a single site with the phase assignment of generation 5 and confirm that the resulting diagram vanishes exactly.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For the triangle family, the parent Hamiltonian is frustration-free, positive semidefinite, and expressed entirely as ZX diagrams; the fractal state is an exact zero-energy ground state, yielding a diagrammatically certified local Hamiltonian with a highly atypical eigenstate.
  • The annihilation identities depend only on local phase cancellation, so the same Hamiltonian forms work for arbitrary generation-dependent phase functions, not just the numerically studied π(1−1/g) family.
  • The deformed Hamiltonian Hpar + λV has GUE level statistics ⟨r⟩ = 0.601(4) in the resolved symmetry sector, placing the area-law fractal state in the bulk of a chaotic spectrum as an exact quantum many-body scar.
  • The Sierpiński-carpet states show approximately logarithmic entanglement and self-similar profiles, but need annihilators with larger support, making them less immediately suited to the same Hamiltonian engineering.
  • All steps, from state preparation to local constraints to chaotic embedding, are representable in ZX-calculus, establishing it as a Hamiltonian inverse-design framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the diagrammatic certification holds beyond the computed sizes, it provides an exact, phase-independent proof of scarred eigenstates at system sizes far beyond exact diagonalization, since the identities are local and symbolic rather than numeric.
  • The triangle/carpet contrast suggests a tunable hierarchy: by choosing other self-similar graphs or phase assignments, one may interpolate between area-law and logarithmic or power-law entanglement, giving a testable family of fractal scar candidates.
  • Combining this construction with a graphical calculus that handles linear combinations, such as ZXW, could yield one-parameter families of scar-preserving deformations with arbitrary real coefficients, and potentially diagrammatic scar towers through spectrum-generating algebras.
  • Because the resulting terms are three- and five-site local interactions with compact diagrammatic forms, the construction is a plausible starting point for compiling scar Hamiltonians onto programmable quantum simulators, where native multibody interactions are becoming available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper introduces two families of many-body quantum states obtained by promoting Sierpiński-triangle and Sierpiński-carpet graphs to ZX-diagrams. For the triangle family, MPS calculations show area-law entanglement with self-similar spatial profiles of local observables; for the carpet family, the entanglement grows approximately logarithmically over accessible sizes, far below the min-cut upper bound. For the triangle states, the author constructs a local frustration-free parent Hamiltonian from null spaces of reduced density matrices, expresses the local terms as compact ZX-diagrams, and asserts diagrammatic identities (Eqs. (15)–(17) and (19)) certifying exact annihilation. A local deformation H_par + λV is then introduced that preserves the exact eigenstate and, for λ=1, produces level statistics consistent with GUE, leading to the claim that the fractal ZX state is a quantum many-body scar of a chaotic Hamiltonian.

Significance. If the asserted identities are proved, the result is significant: it demonstrates that ZX-calculus can act as a design language in which the target state, its local annihilators, the parent Hamiltonian, and a scar-preserving chaotic deformation are all represented diagrammatically. The construction is phase-generic and gives compact local Hamiltonian terms that would be dense in a standard tensor-network parent construction. The numerical work is carefully parameterized (χ_max=1024, ε=10^-20, symmetry-resolved spectral statistics), and the explicit embedding of a zero-energy area-law state in a GUE-spectrum model is an instructive example. The MPS results and min-cut bounds are reproducible from the described procedure. The main gap is the absence of derivations or machine-checked certificates for the exact annihilation identities, on which the central claim rests.

major comments (2)
  1. [Sec. IV A, Eqs. (15)–(17) and (19)] The exact-scar construction hinges on the local annihilation identities (15)–(17) and (19). The paper presents these as final diagram equalities to zero with no step-by-step derivation, algebraic expansion, or machine-checked certificate. Since these identities are the only support for the claims H_a_i(k)|ψ⟩=0, H_b_i(k)|ψ⟩=0, H_c_i(k)|ψ⟩=0, and the subsequent deformation property, they are load-bearing. A single miscopied phase or an incorrect Hadamard edge would invalidate the parent Hamiltonian. I request a supplementary derivation (e.g., a sequence of ZX rewrites, or an explicit Pauli-algebra expansion for each identity) or a reproducible PyZX/ZXLive script that verifies them. This is necessary before the exactness of the scar construction can be accepted.
  2. [Sec. IV A, paragraph 'Next, we will assume...' and Eq. (14)] The construction relies on the assumption that 'other sites can be annihilated by phase-independent annihilators composed of Pauli operators.' This assumption is not made precise, and the paper does not verify that the compact terms (14a)–(14c) annihilate the fractal state for every position in the periodic pattern {a,b,a,c,...} and for every generation g. The identities (15)–(17) contain generic phases α,β,γ,δ, but no argument is given that the boundary phase pattern of the Sierpiński triangle always matches one of these configurations. Please supply a systematic enumeration of the local boundary subdiagrams (or an induction on g) showing that each term position is covered. Without this, the parent Hamiltonian may contain terms that do not annihilate the target state.
minor comments (5)
  1. [Sec. II A, Eq. (4)] The rewrite rules in Eq. (4) are terse; for readers not familiar with ZX-calculus, a pointer to the complete rule set with illustrative diagrams would help. In particular, the bialgebra rule (b) in Eq. (4a) is easy to confuse with the spider fusion (f).
  2. [Sec. III B, Fig. 3] The color map for generations is hard to distinguish, especially for generations 6–9. Using line styles or markers would improve readability.
  3. [Sec. III C] The MPS convergence checks are described as 'stable on the scale of the plotted data' but no quantitative data are given. Please report the discarded weight or fidelity for the largest system sizes, or an example of convergence.
  4. [Sec. IV A, after Eq. (18)] The periodic pattern {a,b,a,c,...} is shown for g=4; please give the rule for arbitrary g. It seems to follow the boundary recursion of the Sierpiński triangle, but the formal definition would help.
  5. [Sec. IV B, Eq. (21)] The diagram for H_a'^i(k) is described as 'swapping the outer wires' of H_a^i(k). This is only obvious for λ=1; for general λ the diagram is not given. The text already mentions the need for ZX-related calculi for real coefficients, but a brief explanation of why λ=1 is special would be useful.

Circularity Check

0 steps flagged

No circularity: inverse design is explicit and diagnostics are independent

full rationale

The paper's central maneuver is self-avowed inverse design: a fractal ZX-diagram state is chosen first, and local annihilators, a parent Hamiltonian, and a scar-preserving deformation are constructed so that the state is an exact zero-energy eigenstate. This is not a hidden fit or a renamed prediction; it is the stated goal ("Hamiltonian inverse design" in the title and abstract). The level statistics and the entanglement-vs-energy plot are independent diagnostics of the constructed Hamiltonian: ⟨r⟩=0.601(4) is an emergent numerical property of H_par+λV, not an input used to set λ. The exactness of the eigenstate under the deformation is by construction from annihilation identities, but the paper does not present this as a prediction. The ZX identities in Eqs. (15)-(17) and (19) are the load-bearing mathematical core and are asserted as final equalities rather than derived step-by-step or machine-certified; this is an evidence-strength / omitted-proof concern, not circularity, because the identities are checkable by the cited sound and complete rewrite rules and are independent of the fitted numerical null-space search. The numerical search constrains the operator supports, but the final identities are stated for arbitrary phase functions, so they are not merely a refit of the small-g data. Self-citations [33,77,78] are peripheral remarks about scar degeneracies and are not load-bearing for the derivation. No exhibited equation reduces a claimed output to an input by construction in a way the paper conceals.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The construction is explicit and self-contained: it introduces no new physical entities, but it does rely on standard diagrammatic and MPS tools plus one ad hoc ansatz about Pauli annihilators. The chosen phase families and truncation parameters are free choices of the demonstration, not fitted to an external dataset.

free parameters (4)
  • spider phase family φ_g = π/g; π/2^{g-1}; π(1−1/g); π(1−1/2^{g-1})
    Four hand-chosen phase schedules used to define the fractal states; the Hamiltonian scar construction uses π(1−1/g). The entanglement scalings are reported across these families, so the conclusions are tied to these choices, though the authors argue the diagrammatic annihilation is phase-agnostic.
  • deformation strength λ = λ=1 (positive and negative values also tested)
    Coefficient of the scar-preserving perturbation V in Hpar+λV; λ=1 is chosen because it gives a compact ZX form; chaos is shown for several nonzero λ.
  • MPS bond dimension χmax / truncation ϵ = χmax=1024, ϵSVD=10^-20
    Computational truncation parameters; convergence to the plotted entanglement data is checked, but the area-law conclusion is only as solid as this convergence.
  • local term range k = 3 and 5 site terms
    Smallest intervals with nontrivial null spaces in the reduced density matrices; used to define Hpar. This is determined by the state, not fitted to a target observable, but it is an input to the Hamiltonian design.
axioms (4)
  • standard math ZX-calculus rewrite rules are sound and complete
    Used to certify the annihilation identities (Eqs. 15–17, 19); soundness/completeness cited to Refs [57–61].
  • domain assumption Parent Hamiltonian from kernels of reduced density matrices is frustration-free and has the target as a zero-energy eigenstate
    Standard MPS parent-Hamiltonian theorem invoked in Sec. II C.
  • standard math Min-cut of the tensor-network graph bounds the entanglement entropy
    Each cut wire carries bond dimension 2, so entropy ≤ number of cut wires; used in Sec. III B.
  • ad hoc to paper Non-central sites of the fractal state can be annihilated by phase-independent Pauli operators
    Stated explicitly in Sec. IV A: 'we will assume that other sites can be annihilated by phase-independent annihilators composed of Pauli operators'; validated by the subsequent construction, but it is an ansatz that guides the analytic form of Hpar.

pith-pipeline@v1.3.0-alltime-deepseek · 20795 in / 14150 out tokens · 140962 ms · 2026-08-01T04:32:52.417345+00:00 · methodology

0 comments
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2607.22495 by Marcin Szyniszewski.

Figure 1
Figure 1. Figure 1: FIG. 1. Summary of the methodology: a fractal ZX-diagram [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Successive generations [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The von Neumann entanglement entropy as a function [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Entanglement entropy divided by [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Local expectation value [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Symmetry-resolved distributions of the ratio [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Entanglement entropy vs energy for all nondegenerate [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The von Neumann entanglement entropy as a function [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Local expectation value [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Entanglement entropy divided by [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗

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