arxiv: 2511.20619 · v2 · submitted 2025-11-25 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Recognition: 1 theorem link

· Lean Theorem

Extracting conserved operators from a projected entangled pair state

Wen-Tao Xu , Miguel Fr\'ias P\'erez , Mingru Yang

Authors on Pith no claims yet

Pith reviewed 2026-05-17 04:50 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el

keywords iPEPSconserved operatorsparent Hamiltoniansfidelity susceptibilitytensor networksresonating valence bondtoric codequantum scars

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The pith

A method extracts geometrically k-local conserved operators for which an iPEPS is an approximate eigenstate by locating vanishing fidelity susceptibility in its parameter-deformed manifold.

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

The paper develops a technique to identify conserved operators such as parent Hamiltonians for which a given infinite projected entangled pair state in two dimensions serves as an approximate eigenstate. The approach deforms the tensor network state through parameters that define a manifold equipped with quantum geometry, then computes static structure factors by differentiating generating functions to find directions of zero fidelity susceptibility. This extracts operators with better locality than standard constructions and works for both exact and variational approximations, including a four-site plaquette Hamiltonian for the short-range resonating valence bond state. A sympathetic reader would care because the method reveals underlying models for complex states without assuming frustration-free conditions and points to possible realizations of quantum many-body scars in deformed toric code states.

Core claim

Conserved operators for an iPEPS are identified as those with vanishing fidelity susceptibility on the manifold of states obtained by deforming the tensor network with parameters. These susceptibilities are obtained by differentiating the generating function to evaluate static structure factors of multi-site operators. The resulting operators are geometrically local and include both frustration-free and non-frustration-free parent Hamiltonians beyond conventional constructions, as shown by a 4-site plaquette Hamiltonian that approximately stabilizes the short-range RVB state and a Hamiltonian for which the toric code deformed by arbitrary string tension remains an excited eigenstate at fixed

What carries the argument

The manifold of the iPEPS deformed by parameters, equipped with quantum geometry in which conserved operators correspond to directions of vanishing fidelity susceptibility, identified by differentiating the generating function for structure factors.

If this is right

Both frustration-free and non-frustration-free parent Hamiltonians beyond standard constructions can be extracted to good precision from exact or variational iPEPS.
A 4-site-plaquette local Hamiltonian approximately has the short-range RVB state as its ground state.
A Hamiltonian exists for which the toric code deformed at arbitrary string tension remains an excited eigenstate with the same energy, potentially realizing quantum many-body scars.
The procedure yields operators with improved locality compared with conventional parent Hamiltonian constructions.

Where Pith is reading between the lines

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

The same differentiation technique could be applied to matrix product states in one dimension to extract approximately conserved operators for approximate ground states.
Numerical searches using this method might systematically generate families of Hamiltonians supporting topological or scarred eigenstates for use in quantum simulation.
Extensions to open quantum systems could identify approximately conserved quantities even when the tensor network represents a steady state rather than a pure eigenstate.

Load-bearing premise

Approximation errors from variational or truncated iPEPS do not substantially degrade the identification of directions with vanishing fidelity susceptibility.

What would settle it

Apply the extraction procedure to an exact iPEPS whose parent Hamiltonian is already known from analytic construction and verify whether the method recovers an operator of comparable or better locality that leaves the state invariant up to a constant shift.

Figures

Figures reproduced from arXiv: 2511.20619 by Miguel Fr\'ias P\'erez, Mingru Yang, Wen-Tao Xu.

**Figure 2.** Figure 2: a shows the spectrum of the 1-site S. There are 4 zero eigenvalues (up to precision 10−9 ), and the associated eigenvectors are the identity and the three spin-2 operators S x , S y , and S z . Fig. 2b shows the 2-site result after projecting out the 1-site and the 2-site trivial solutions. There remain 81 zero eigenvalues (up to precision 10−7 ), which correspond to the projectors onto the channel 4 due t… view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Schematics of the tensor network method for evaluating 2-site static structure factors. The black lines are the virtual indices and the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Schematics of the tensor network method for evaluating 4-site static structure factors. (a) Expressing the 4-site operator in the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The static structure factor between two 4-site operators calculated from the Ising iPEPS with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Compare the lowest three eigenvalues of the static structure factor matrix spectra of the AKLT (RVB) state calculated using the window [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Results from the iPEPS optimization of the XX model in the staggered [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Density of states [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

read the original abstract

Given a tensor network state, how can we determine conserved operators (including Hamiltonians) for which the state is an eigenstate? We answer this question by presenting a method to extract geometrically $k$-local conserved operators that have the given infinite projected entangled pair state (iPEPS) in 2D as an (approximate) eigenstate. The key ingredient is the evaluation of the static structure factors of multi-site operators through differentiating the generating function. These generating functions define a manifold of the given tensor network state deformed by some parameters, endowed with a quantum geometry, where conserved operators correspond to vanishing fidelity susceptibility. Despite the approximation errors, we show that our method is still able to extract from exact or variational iPEPS to good precision both frustration-free and non-frustration-free parent Hamiltonians that are beyond the standard construction and obtain better locality. In particular, we find a 4-site-plaquette local Hamiltonian that approximately has the short-range RVB state as the ground state. Moreover, we find a Hamiltonian for which the deformed toric code state at arbitrary string tension is an excited eigenstate with the same energy, thereby potentially realizing quantum many-body scars.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete method to extract k-local conserved operators from 2D iPEPS by spotting vanishing fidelity susceptibility on a manifold of deformed states, and it works out a 4-site plaquette Hamiltonian for the short-range RVB plus a constant-energy operator for the string-tension deformed toric code.

read the letter

The main takeaway is that the authors turn the problem of finding conserved operators into a search for flat directions in fidelity susceptibility after deforming an iPEPS with parameters. They compute the needed structure factors by differentiating generating functions, then read off the operators that leave the state approximately invariant. This framing is distinct from the usual parent-Hamiltonian recipes cited in the abstract and lets them target better locality.

Referee Report

2 major / 2 minor

Summary. The paper presents a method to extract geometrically k-local conserved operators (including parent Hamiltonians) from 2D infinite projected entangled pair states (iPEPS) by differentiating a generating function to obtain static structure factors of multi-site operators. These define a manifold of deformed states equipped with a quantum geometry in which conserved operators appear as directions of vanishing fidelity susceptibility. The authors apply the approach to both exact and variational iPEPS, extracting frustration-free and non-frustration-free Hamiltonians with improved locality; in particular they report a 4-site plaquette Hamiltonian for which the short-range RVB state is an approximate ground state, and a Hamiltonian for which a deformed toric-code state at arbitrary string tension is an excited eigenstate of constant energy, potentially realizing many-body scars.

Significance. If the robustness to approximation errors holds, the work supplies a systematic, geometry-based route to parent Hamiltonians that extends beyond standard frustration-free constructions and yields more local operators. The concrete demonstrations on RVB and toric-code iPEPS, together with the use of fidelity susceptibility on a generating-function manifold, constitute a useful addition to the tensor-network and quantum-many-body toolbox. The paper explicitly addresses both exact and variational cases and reports extraction to good precision, which, if quantitatively supported, would strengthen its practical value.

major comments (2)

[§4] §4 (RVB results): the claim that a 4-site-plaquette Hamiltonian is extracted 'to good precision' from a variational iPEPS and approximately has the short-range RVB state as ground state is load-bearing for the central assertion of robustness; however, no quantitative error bars, scaling with bond dimension, or direct comparison of the fidelity-susceptibility minima between exact and variational iPEPS are provided, leaving open whether truncation/optimization errors shift the identified flat directions.
[§3] §3 (method): the mapping from vanishing fidelity susceptibility to conserved operators relies on second derivatives of the generating function; the manuscript does not supply an a-priori bound or numerical test showing that these derivatives remain faithful proxies once the iPEPS is only variational, which directly affects the reliability of both the RVB plaquette Hamiltonian and the scar Hamiltonian for the deformed toric code.

minor comments (2)

Notation for the multi-site operators and the parameters of the generating function could be made more uniform across sections to improve readability.
Figure captions for the susceptibility plots should explicitly state the bond dimension and truncation threshold used in the variational iPEPS calculations.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive criticism. The concerns about quantitative validation and faithfulness of derivatives in the variational setting are substantive and we address them point by point below, indicating the revisions we will make.

read point-by-point responses

Referee: [§4] §4 (RVB results): the claim that a 4-site-plaquette Hamiltonian is extracted 'to good precision' from a variational iPEPS and approximately has the short-range RVB state as ground state is load-bearing for the central assertion of robustness; however, no quantitative error bars, scaling with bond dimension, or direct comparison of the fidelity-susceptibility minima between exact and variational iPEPS are provided, leaving open whether truncation/optimization errors shift the identified flat directions.

Authors: We agree that the robustness claim would be strengthened by quantitative measures. The manuscript currently supports the extraction through the close numerical agreement of the obtained coefficients with the expected plaquette terms and through the low energy variance of the resulting Hamiltonian on the variational state. In the revised version we will add error bars obtained from repeated optimizations with different random seeds, include a direct side-by-side comparison of the fidelity-susceptibility minima for the exact and variational iPEPS at the bond dimensions used, and report the shift in the identified flat directions. These additions will be placed in §4 and the associated figures. revision: yes
Referee: [§3] §3 (method): the mapping from vanishing fidelity susceptibility to conserved operators relies on second derivatives of the generating function; the manuscript does not supply an a-priori bound or numerical test showing that these derivatives remain faithful proxies once the iPEPS is only variational, which directly affects the reliability of both the RVB plaquette Hamiltonian and the scar Hamiltonian for the deformed toric code.

Authors: The referee correctly notes the absence of an a-priori error bound. Deriving such a bound analytically is difficult because the error depends on the specific contraction scheme and the gauge fixing of the iPEPS. We will therefore add, in the revised §3, explicit numerical tests that compare the second derivatives (i.e., the static structure factors) computed from the exact iPEPS against those from the variational iPEPS for both the RVB and deformed toric-code examples. These tests will quantify the deviation and show that the locations of the vanishing-susceptibility directions remain stable within the reported precision. We will also expand the discussion of limitations to make the empirical nature of the validation clearer. revision: partial

standing simulated objections not resolved

A rigorous a-priori analytical bound on the error of the second derivatives of the generating function under variational iPEPS approximations.

Circularity Check

0 steps flagged

No circularity: extraction method derives from quantum geometry without reducing to inputs by construction

full rationale

The paper defines a procedure to extract k-local conserved operators from an iPEPS by differentiating a generating function to compute static structure factors of multi-site operators, then locating directions of vanishing fidelity susceptibility on the manifold of deformed states. This construction applies standard quantum-geometric notions (fidelity susceptibility as a metric on the parameter manifold) to tensor-network states and does not equate any output Hamiltonian or operator to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose validity is presupposed. Numerical demonstrations for the short-range RVB state and deformed toric code are presented as applications of the method rather than tautological rederivations, so the claimed derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard assumptions of tensor-network representations and quantum geometry; no free parameters, ad-hoc axioms, or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption iPEPS provides a faithful (approximate) representation of the target quantum state
Invoked when applying the method to variational iPEPS and claiming extraction to good precision despite approximation errors.
standard math Fidelity susceptibility vanishes exactly for conserved operators in the exact state
Central to identifying conserved operators from the deformed manifold geometry.

pith-pipeline@v0.9.0 · 5516 in / 1417 out tokens · 28848 ms · 2026-05-17T04:50:32.186383+00:00 · methodology

discussion (0)

Reference graph

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