{"id":"00a64578-e6f6-460b-bad1-e3fd0ff8a7dc","arxiv_id":"2411.17020","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A phase-estimation measurement of a global charge or momentum projects an easy-to-prepare matrix product state onto towers of quantum many-body scar states and Dicke states in logarithmic circuit depth.","lead":"This paper introduces a logarithmic-depth circuit that measures a global property, like total magnetization, to prepare highly entangled 'scar' states and Dicke states. It matters because it offers a shallow, measurement-based recipe for states that are useful in quantum sensing and many-body physics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central O(log L) claim assumes the period-2 resource MPS of Eq. (5) is preparable by the cited algorithms, but the paper gives no verification of injectivity, transfer-matrix gap, or error accounting for the approximate protocol.","rationale":"I read the central claim as: for the resource state of Eq. (4), a log-depth phase-estimation measurement of a global charge projects it onto the tower |E_n⟩ of Eq. (2), with Gaussian probability and O(log L) total depth. The projection identity and probability formulas are plausible; the examples' combinatorial normalizations check out. The weakest point is not the phase-estimation circuit itself — with an integer-shifted charge, m=O(log L) ancillas suffice and the controlled unitaries are products of commuting local gates — but the preparation of the non-product resource states. The cited literature is strong evidence, but it is not a substitute for verifying that the specific site-dependent, period-2 MPS satisfies the algorithms' hypotheses. Because the current verdict is CONDITIONAL, this concern reinforces rather than changes it: no reason to reject, but the missing construction should be supplied or the claim restricted. The concrete test would settle the question by checking the transfer matrix and compiling the circuit.","tokens_in":17329,"tokens_out":32886,"duration_ms":337743,"concrete_test":"Block two sites of the AKLT resource tensors C_j of Eq. (5) with k=π and a chosen w (e.g., w=1/4) into a uniform MPS of bond dimension χ=4; compute the transfer matrix for L=64,128,256 and check (i) injectivity of the blocked tensor and (ii) the spectral gap below the largest eigenvalue. Separately, compile the blocked MPS with the algorithm of Ref. [49] and estimate its depth and fidelity. If the gap is O(1) and the prepared state has fidelity 1−ε with ε = o(e^{-δ²L}) for the chosen δ, the central claim holds; if the gap closes as L grows or the compilation requires a uniform parent Hamiltonian, the O(log L) claim should be downgraded to conditional on an explicit preparation construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The protocol's logarithmic-depth claim depends on two linked steps: the resource state |Ψ⟩ of Eq. (4) must be preparable in O(poly(log L)) depth, and the phase-estimation projection must then act on the exact |Ψ⟩. The paper asserts preparation by citing [43–49] (main text after Eq. (5)), but those algorithms have applicability conditions (normal/injective MPS, gapped parent Hamiltonian; Ref. [49] is an approximate measurement-feedback protocol). The tensors C_j in Eq. (5) are site-dependent (period-2 for k=π) and are built with the non-unitary operators R_j = √(1−w)I + e^{ikj}√w O_j; they are not obtained from |Ψ0⟩ by a constant-depth unitary circuit. The authors do not block the period-2 MPS into a uniform injective MPS, do not show the parent Hamiltonian remains gapped for the w used to reach n=O(L), and do not analyze how the approximate preparation error of [49] propagates through Eq. (6) and the failure bound Eq. (9). If any cited preparation protocol is inapplicable, the advertised O(log L) depth is unsupported for the headline examples beyond product-state Dicke preparation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17533,"tokens_out":15762,"duration_ms":156648,"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":[{"comment":"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.","section":"Eqs. (4)-(5), paragraph beginning 'Accordingly, the area-law-entangled...'"},{"comment":"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.","section":"Eqs. (6)-(7)"}],"minor_comments":[{"comment":"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.","section":"Discussion, resource-count sentence"},{"comment":"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.","section":"Circuit implementation, Eq. (10)"},{"comment":"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.","section":"SM4, Arovas A state"},{"comment":"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.","section":"References"},{"comment":"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.","section":"After Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the resource-state preparation step: if the cited MPS preparation protocols do not cover the period-2, non-unitary resource states, the central efficiency claim is unsupported. This is fixable in revision by adding explicit blocked tensors, injectivity/normality checks, gapped parent-Hamiltonian verification, or an explicit argument that Ref. [49] applies exactly or with controlled error. I do not see grounds for rejection; the examples are concrete and the probability analysis is largely sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis paper is worth a serious look. The core idea is genuinely new: take an area-law MPS resource state built as a product of local raising operators on a simple initial state, then use quantum phase estimation to project onto a tower of excited states with a fixed global charge. That combination, plus the O(1)-depth measurement-feedback controlled-translation gate for momentum measurement, goes beyond earlier measurement-based scar preparation, which needed O(L) depth and all-to-all connectivity. The concrete examples—AKLT scars, Onsager scars, the constrained domain-wall model, Dicke states, and the Arovas A state—are worked out in enough detail that the probability distributions are derived analytically and checked numerically. The central projection identity (Eq. 6) and the probability analysis are correct for these models.\n\nThe soft spots are real but not fatal. The main one is that the advertised O(log L) depth depends on the resource MPS, which has site-dependent (period-2) tensors, being preparable by the cited algorithms [43-49]. The paper simply asserts this after Eq. (5) and does not verify injectivity, transfer-matrix gap, or account for the error of the approximate protocol [49]. For a Letter this might be acceptable as a pointer to known methods, but a referee should ask the authors to block the period-2 MPS into a uniform injective form and state explicitly that the cited preparation algorithms apply. If they cannot, the log-depth claim for the non-product initial states (e.g., AKLT) would be unsupported.\n\nThe other thing I'd flag is the resource count. The text says 'O(log L) ancilla qubits per site,' which is confusing. Phase estimation uses O(log L) ancillas total; the GHZ-state fan-out uses O(L) ancillas if that is the implementation. The paper should clarify what exactly is being counted.\n\nThe ancillary issues—signed charges in phase estimation, the details of the m-site operator case, and the approximate Arovas A preparation—are minor and can be handled in revision. The citation pattern looks fine; the prior work on MPS preparation and on O(L)-depth scar preparation is properly cited.\n\nBottom line: this is a solid, inventive protocol paper. It deserves peer review. I'd read it again after the authors tighten the resource-state preparation argument and fix the ancilla-count wording. Recommend sending to a serious referee, not desk reject.","headline":"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.","tokens_in":33,"tokens_out":5861,"would_cite":true,"duration_ms":111348,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A global measurement can prepare towers of structured excited states in logarithmic circuit depth.","keywords":["measurement-based state preparation","quantum many-body scars","quantum phase estimation","matrix product states","Dicke states","Heisenberg limit","quantum metrology","global charge measurement"],"falsifier":"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.","tokens_in":63,"feed_emoji":"⚛","tokens_out":9317,"duration_ms":130712,"temperature":0.7,"pith_summary":"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.","feed_headline":"Global measurement prepares quantum many-body scars in log depth","feed_subtitle":"Prepare a low-entanglement MPS, measure total magnetization, and project onto metrologically useful excited states.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the phase-estimation algorithm used to realize the global charge measurement.","marker":"[36]"},{"why":"Cited as the low-depth MPS preparation methods that produce the resource state $|\\Psi\\rangle$; the logarithmic-depth claim depends on them.","marker":"[43–49]"},{"why":"Prepares the AKLT ground state, the starting point $|\\Psi_0\\rangle$ for the AKLT scar tower, in constant depth.","marker":"[16–18]"},{"why":"Defines Onsager's scars and the raising-operator construction used in the spin-$\\frac{1}{2}$ XX example.","marker":"[28]"},{"why":"Define the constrained domain-wall model whose scar tower is prepared in Example 3.","marker":"[60, 61]"},{"why":"Provide the O(1)-depth measurement-feedback GHZ preparation that enables simultaneous local controlled gates.","marker":"[51, 52]"},{"why":"Introduces the Arovas A state, the target of the momentum-measurement protocol.","marker":"[67]"},{"why":"Provides the CZX boundary-symmetry example showing the controlled-translation feedback construction extends to general matrix product unitaries.","marker":"[68]"},{"why":"Define matrix product unitaries, the class of non-onsite symmetries to which the momentum-measurement scheme generalizes.","marker":"[40, 41]"}],"fun_headline_variants":["Global measurement yields tower of excited states in log depth","Log-depth global measurement prepares metrological excitations","Log-depth global charge measurement builds many-body scars","Quantum phase estimation on global charge creates excited towers"],"cache_read_input_tokens":20224,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Global measurement yields tower of excited states in log depth","Log-depth global measurement prepares metrological excitations","Log-depth global charge measurement builds many-body scars","Quantum phase estimation on global charge creates excited towers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3400,"prompt_tokens":1007,"completion_tokens":2393,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":2333}},"tokens_in":623,"tokens_out":2393,"duration_ms":16497,"temperature":1.0,"reasoning_tokens":2333,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:37:27.438525+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Shibata, N","cited_arxiv_id":null,"evidence_quote":"Defines Onsager's scars and the raising-operator construction used in the spin-$\\frac{1}{2}$ XX example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Arovas A state, the target of the momentum-measurement protocol."},{"cited_title":"Chen, Z.-X","cited_arxiv_id":null,"evidence_quote":"Provides the CZX boundary-symmetry example showing the controlled-translation feedback construction extends to general matrix product unitaries."}],"review_version":1}