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REVIEW 2 major objections 5 minor 71 references

Tower of Structured Excited States from Measurements

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Pith's one-line read A global measurement can prepare towers of structured excited states in logarithmic circuit depth.

desk verdict A genuinely new measurement-based route to scar towers and Dicke states that is correct in its core math, but the log-depth claim leans on an unverified MPS preparation assumption and a confusing ancilla count. read the letter →

arxiv 2411.17020 v2 pith:SA7P2J5O submitted 2024-11-26 quant-ph cond-mat.other

Yuxuan Guo , Yuto Ashida This is my paper · ORCID
classification quant-phcond-mat.other
keywords measurement-basedstatepreparationquantummany-bodyscarsphaseestimationmatrixproductstatesDickeHeisenberglimitmetrologyglobalchargemeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Preparing highly entangled states usually requires circuits whose depth grows with system size, but measurements can shortcut that. The paper claims that a global measurement—realized by quantum phase estimation—can project a simple, low-entanglement resource state onto a tower of structured excited states, including quantum many-body scars in the AKLT, constrained domain-wall, and XX spin chains, as well as high-weight Dicke states. If correct, this gives a logarithmic-depth recipe for states that reach the Heisenberg limit in quantum metrology, with a failure probability that decays exponentially in system size. The central mechanism is that the resource state is a weighted superposition of the target states labelled by the eigenvalue of a global charge, so measuring that charge picks out one excitation-number sector.

What carries the argument

The mechanism is a resource state (Eq. (4)) that is a product of local operators applied to a low-bond-dimension MPS $|\Psi_0\rangle$, giving a coherent superposition of the target states $|E_n\rangle$ with approximately binomial weights; combined with quantum phase estimation on $\hat{U} = e^{i2\pi\hat{Q}/2^m}$ using $m = O(\log L)$ ancilla qubits, which resolves the global charge $\hat{Q}$ modulo $2^m$ and projects onto the sector with the measured value. For non-onsite observables such as total momentum, the controlled unitary is implemented by a measurement-feedback circuit that performs a controlled translation in $O(1)$ depth using shared Bell pairs and local Pauli corrections, a construction that generalizes to any matrix product unitary.

What would settle it

Run the full protocol on a noiseless simulator for the spin-$\frac{1}{2}$ XX chain at $L = 64, 128, 256$ with a fixed relative tolerance $\delta$, and record (i) the probability of obtaining an outcome $n$ inside the window $(1\pm\delta)n_0$ and (ii) the actual circuit depth required to prepare the site-dependent MPS of Eq. (5). If the success probability decays faster than the predicted $e^{-O(\delta^2 L)}$, or if the preparation depth can be shown to grow faster than polylogarithmically in $L$ under any known construction, the central efficiency claim fails.

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Extended reading notes

Core claim

The paper establishes that the resource state $|\Psi\rangle = \prod_j(\sqrt{1-w}\,\hat{I} + e^{ikj}\sqrt{w}\,\hat{O}^\dagger_j)\,|\Psi_0\rangle$, which is an area-law-entangled matrix product state, decomposes into a superposition of the excited states $|E_n\rangle = (\hat{J}^\dagger)^n|\Psi_0\rangle/\sqrt{N_n}$ that carry distinct eigenvalues of a global $U(1)$ charge $\hat{Q}$. A phase-estimation measurement of $\hat{Q}$ therefore projects $|\Psi\rangle$ onto a definite excitation-number sector $n$, and because the outcome distribution is approximately Gaussian with width $\Delta \propto \sqrt{n_0}$, a state with $O(L)$ excitations is produced with probability $1 - e^{-O(\delta^2 L)}$. The protocol runs in $O(\log L)$ circuit depth with $O(\log L)$ ancilla qubits per site, provided the unitaries $\hat{U} = e^{i2\pi\hat{Q}/2^m}$ can be applied as products of commuting local gates; for momentum, which is not an on-site observable, an $O(1)$-depth measurement-feedback construction realizes a controlled translation, extending the scheme to non-onsite symmetries expressible as matrix product unitaries.

Load-bearing premise

The load-bearing premise is that the site-dependent matrix-product resource state of Eq. (4)—including the case of $m$-site operators with bond dimension $\chi d^{m-1}$—can be prepared in polylogarithmic (or doubly logarithmic) circuit depth by the cited general MPS algorithms and then directly combined with the global phase-estimation measurement, a step for which the paper supplies citations but no explicit construction.

Editorial extensions

If this is right

  • The same protocol—prepare the appropriate MPS and measure the global charge—produces AKLT scars, Onsager scars in the spin-$\frac{1}{2}$ XX chain, scars of the constrained domain-wall model, spin-1 XX scars, and high-weight Dicke states.
  • The success probability for $n$ excitations is given by Eq. (7) and is approximately Gaussian; tuning $w$ controls the mean $n_0$, so a state with $O(L)$ excitations is obtained with failure probability $e^{-O(\delta^2 L)}$ for any fixed relative tolerance $\delta$.
  • Measuring total momentum via the measurement-feedback controlled translation prepares states that on-site charges cannot distinguish, such as the Arovas $A$ state of the AKLT Hamiltonian.
  • The resulting scar states carry genuine multipartite entanglement with superextensive quantum Fisher information $F \propto L^2$, so they can saturate the Heisenberg limit in parameter estimation.
  • If an exact excitation number is required, repeating the protocol $O(\sqrt{n})$ times suffices; for Lie-algebra scars, repeated feedback reduces this to $O(\log n)$ repetitions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The projection-by-global-charge logic should also work for any tower generated by a ladder operator that raises a $U(1)$ charge, not only the Lie-algebra examples treated here; the essential ingredient is only that the resource state have nonzero, controlled overlap with each $|E_n\rangle$.
  • The controlled-translation feedback construction suggests that other non-onsite symmetries with matrix-product-unitary form—such as anomalous boundary symmetries of symmetry-protected topological phases—could be measured in $O(1)$ depth, giving a route to preparing the corresponding boundary states without preparing the full bulk.
  • A practical upshot not stressed in the paper is that the Gaussian tolerance window removes the need for exact postselection in metrology: the experimenter can certify success from the measured charge sector while accepting any outcome in the window.
  • A quantitative feature worth testing on hardware is the predicted width $\Delta \propto \sqrt{n_0}$ of the outcome distribution; it is a fingerprint of the scheme that distinguishes it from other preparation methods.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a measurement-based preparation protocol for towers of structured excited states. It starts from an area-law resource state |Ψ⟩ = ∏_j (√(1−w) I + e^{ikj}√w O^†_j)|Ψ0⟩, which is argued to be an MPS with constant bond dimension, and uses quantum phase estimation to measure a global charge Q, projecting onto the states |E_n⟩ = (J^†)^n / √N_n |Ψ0⟩ with probability p_n = N_n w^n (1−w)^{L−n} / (n!)². The authors claim that states with O(L) excitations can be prepared with O(log L) circuit depth, O(log L) ancillas, and failure probability e^{−O(δ²L)}. Applications include AKLT scars, Onsager scars, constrained domain-wall scars, Dicke states, spin-1 XX scars, and, via a controlled-translation gate, the Arovas A state; an extension to non-onsite symmetries represented by matrix product unitaries is also sketched.

Significance. The framework is elegant and potentially useful: it unifies several known scar towers under a single measurement-based narrative and gives explicit analytic probability distributions with numerical verification (Fig. 2 and Fig. SM3). The controlled-translation construction is a concrete O(1)-depth feedback implementation of a non-onsite global gate, and the MPU generalization is conceptually appealing. If the resource-state preparation step can be justified, the log-depth and exponential-tail claims would be a meaningful advance for preparing metrologically relevant multipartite entangled states. The paper is an analytic construction rather than a fit; model-specific normalization factors are derived and then checked numerically, which is a strength.

major comments (2)
  1. [Eqs. (4)-(5), paragraph beginning 'Accordingly, the area-law-entangled...'] The advertised O(log L)-depth claim rests on the cited preparation algorithms [43–49], but the paper does not verify their hypotheses for the specific resource state. For k=π the tensors C_j in Eq. (5) are period-2 and are formed from the non-unitary R_j; after blocking into a translation-invariant MPS, no proof is given that the blocked tensor is normal/injective or that a gapped parent Hamiltonian exists at the w values used to reach n=O(L). If the authors instead rely on the approximate measurement-feedback protocol of Ref. [49], they need to account for the preparation error in the projection identity Eq. (6) and in the failure bound Eq. (9). The remark in SM2.A (Fig. SM2 caption) asserts an injectivity claim but does not carry out the check for the AKLT, Onsager, or domain-wall examples. Without this, the central O(log L) efficiency claim is not established for the headline scar examples, although product-state Dicke preparation is not affected.
  2. [Eqs. (6)-(7)] The identity |E_n⟩ = Π_n |Ψ⟩ / √p_n with p_n = N_n w^n (1−w)^{L−n} / (n!)² is exact only under implicit conditions: the local operators O^†_j must be nilpotent on the relevant local space, repeated or overlapping applications must vanish, and every ordered n-tuple contributing to (J^†)^n must generate each n-excitation configuration with the same multiplicity n!. These conditions hold in the listed examples, where N_n is computed combinatorially, but they are not stated. For a generic m-site O^†_j with overlapping supports, the expansion of ∏_j R_j is not proportional to ∑_n (J^†)^n / n!, so Eq. (7) should be presented as applying to a class of models satisfying stated conditions rather than as a universal formula. The subsequent Gaussian approximation and the failure estimate Eq. (9) inherit this issue; they are verified per example in the SM, but not proven for the general construction.
minor comments (5)
  1. [Discussion, resource-count sentence] The sentence 'O(log L) circuit depth, O(log L) ancilla qubits per site' appears inconsistent with Eq. (10), where m = O(log L) ancilla qubits are used for the whole system; if a different resource count is intended (for instance, for the GHZ fan-out construction), it should be defined explicitly.
  2. [Circuit implementation, Eq. (10)] The phase-estimation readout Q_{s_l} is a non-negative integer modulo 2^m, but several examples (Onsager scar, domain-wall, spin-1/2 XX) start from Q0 = −L/2 and target negative charges; the paper should specify how signed charges are encoded in the phase-estimation measurement.
  3. [SM4, Arovas A state] The Arovas A preparation uses a Trotterized global unitary U(α) whose circuit depth is not analyzed; the paper should state explicitly that this application is not covered by the O(log L) resource claim made for the charge-measurement examples.
  4. [References] Refs. [45] and [51] are the same paper (Piroli, Styliaris, and Cirac, PRL 127, 220503), and Refs. [33] and [69] are also duplicates; these should be consolidated.
  5. [After Eq. (5)] The phrase 'finite-depth circuits' is imprecise for the cited algorithms [43–48], which give polynomial-logarithmic depth rather than constant depth; 'log-depth' or 'poly(log L)-depth' would be more accurate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central projection identity and probability formulas are derived algebraically from the definitions, and the cited MPS preparation results are independent external inputs.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. The central identity Eq. (6), |E_n> = Pi_n / sqrt(p_n) |Psi>, is not an input renamed as a prediction: it follows from expanding the product in Eq. (4) and comparing with (J^dagger)^n in Eq. (2); the same expansion gives p_n in Eq. (7), with N_n fixed by the normalization of (J^dagger)^n |Psi_0>. The Gaussian form and the failure bound Eq. (9) are obtained by saddle-point/Stirling analysis in the Supplemental Material (e.g., Eqs. (SM19)-(SM20) and (SM27)-(SM29)), with numerical fitting used only as a check of the analytically derived expressions. The only load-bearing use of prior work is the assumption that normal (and, via Ref. [49], nonnormal) MPS resource states can be prepared by cited O(poly(log L)) or O(log log L) protocols; these are external results, not self-citations, and no step reduces to the paper's own target. The single self-citation, Ref. [13] in the introduction, is background on measurement-induced entanglement and is not used to justify the protocol's claims. The protocol's efficiency claim does depend on the unverified applicability of the cited MPS-preparation algorithms to the period-2, site-dependent tensors of Eq. (5); that is a correctness or rigor concern, not circularity, because the cited algorithms are independent of and prior to this paper's conclusions.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The protocol uses standard quantum phase estimation and matrix product state formalism. The main axioms are the known scar eigenstate results, the efficient preparability of MPS resource states, the exactness of the controlled translation gadget, and the saddle-point approximation. The free parameters are user-controlled (w, alpha), not fitted to data.

free parameters (2)
  • w
    Weight parameter 0<w<1 in the resource state (Eq. 4). Chosen by the user to set the expected excitation number n0. Not fitted to data.
  • alpha
    Trotter angle in the Arovas A preparation (Eq. SM48). Trade-off between approximation error O(alpha^2) and repetition overhead O(1/(alpha^2 epsilon)).
assumptions (5)
  • domain assumption Tower states |E_n⟩ = (J†)^n/√N_n |Ψ0⟩ are exact eigenstates of the respective Hamiltonians (AKLT, XX, domain-wall).
    The protocol projects onto these states, so their eigenstate property is inherited from prior literature (Refs [56-58, 28, 60, 61]). The paper does not re-derive these results.
  • domain assumption The resource state |Ψ⟩ in Eq. (4) is preparable by finite-depth or O(poly(log L))-depth circuits using known MPS preparation protocols.
    Invoked in the main text after Eq. (5) with citations [43-49] to MPS preparation algorithms. The specific non-translation-invariant state is not explicitly constructed.
  • standard math Phase estimation with m = O(log L) ancilla qubits resolves the global charge Q unambiguously.
    For integer Q with range smaller than 2^m, the QFT-based phase estimation is exact. The paper assumes this without discussing signed charge offsets.
  • standard math The Gaussian/saddle-point approximation of the excitation-number distribution p_n is valid for L ≫ 1 and n0 = O(L).
    Derived via Stirling's formula in the supplementary (SM2). Numerical fits in Fig. SM3 agree, but the approximation is asymptotic.
  • domain assumption The controlled translation gate CT can be realized by an O(1)-depth measurement-feedback circuit with Bell pairs and Pauli corrections.
    Constructed in the main text (Eqs. 14-16) and validated in SM3. The gadget assumes the ability to prepare Bell pairs, perform Bell measurements, and apply feedback in constant depth.

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Pith. "Pith review of Tower of Structured Excited States from Measurements." pith.science (2026). https://pith.science/paper/SA7P2J5O

@misc{pith2026241117020,
  author       = {Pith},
  title        = {Pith review of: Tower of Structured Excited States from Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SA7P2J5O}},
  note         = {Machine review of arXiv:2411.17020}
}
abstract

Preparing highly entangled quantum states is a key challenge in quantum metrology and quantum information science. Measurements, especially those of global observables, offer a simple and efficient way to generate entanglement between subsystems when they are measured as a whole. We introduce a log-depth protocol leveraging quantum phase estimation to measure a global observable, such as total magnetization and momentum. We demonstrate its capability to prepare towers of structured excited states that are useful in quantum metrology; examples include quantum many-body scars in various models, including the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, the constrained domain-wall model, and the spin-$\frac{1}{2}$ and spin-$1$ XX chains. The same method is also applicable to preparing the Dicke states of high weight. In addition, we propose a protocol for momentum measurement that avoids disturbing the system, facilitating the preparation of states beyond the above construction, such as the Arovas $A$ state of the AKLT Hamiltonian. Our results expand the utility of measurement-based approaches to accessing highly entangled states in quantum many-body systems.

Figures

Figures reproduced from arXiv: 2411.17020 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the proposed preparation protocol. (a) We first [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Probabilities of getting [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Measurement-feedback protocol to realize the C [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.