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REVIEW 3 major objections 3 minor 87 references

Deterministic carving of quantum states with Grover's algorithm

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A few $O(N^{1/4})$ Grover steps, implemented by global rotations and single-photon cavity reflections, deterministically prepare Dicke, GHZ, and Cat states without individual addressing.

desk verdict Clean Grover-based Dicke preparation with a real physical proposal, but the headline resource scaling hides that the needed cooperativity grows with N. read the letter →

arxiv 2501.18884 v4 pith:WK3K2JZT submitted 2025-01-31 quant-ph physics.atom-ph

Omar Nagib , M. Saffman , K. M{o}lmer This is my paper · ORCID
classification quant-phphysics.atom-ph
keywords Grover'salgorithmDickestatescavityQEDdeterministicstatepreparationGHZcatheraldingcooperativity
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

The paper aims to show that the photon-induced phase shift used in probabilistic cavity carving can be turned into a deterministic Grover iteration. It claims that any Dicke state of N atoms — the permutation-symmetric state with exactly m excitations — can be prepared perfectly in $O(m^{1/4})$ Grover steps, and that GHZ and Cat states can be prepared with a few more steps, with each Grover step costing only two or three photon reflections plus global rotations independent of N. If correct, this removes the need for individual addressing, ancillas, or mid-circuit measurements that earlier efficient schemes require, and it matches the scaling of circuit-assisted state preparation in a direct physical setting. The paper also provides a quantitative error analysis showing infidelity scaling as $O(C^{-1/2})$ without heralding and $O(C^{-2/3})$ with heralding, where C is the atom-cavity cooperativity.

What carries the argument

The load-bearing object is the Grover iteration on the Dicke manifold, $G = R(\phi)^{\otimes N}\chi_0 R(-\phi)^{\otimes N}\chi_m$, where $\chi_m$ is phase inversion of the Dicke state $|m\rangle$ (the permutation-symmetric superposition of all N-qubit basis states with exactly m excitations) and $R(\phi)^{\otimes N}$ is a global single-qubit rotation applied to every atom. The key identity is the overlap condition $\langle m|\psi_i\rangle = \sin(\pi/[2(2k+1)])$, which guarantees that exactly k applications of G produce $|m\rangle$ perfectly. The physical mechanism that realizes $\chi_m$ without individual addressing is the dispersive cavity-QED Hamiltonian $H = \hbar\Omega\,\hat{m}\hat{n}_c$, which shifts the cavity resonance by $m\Omega$; reflecting a photon at that shifted frequency gives an ideal reflection amplitude $r_m = -1$ only for the component with m excitations, and $r_n \approx 1$ for all other Dicke components. The global rotation is implemented by driving all atoms identically, so the entire iteration costs two photon reflections for Dicke states and three for GHZ/Cat states, independent of N.

What would settle it

Measure the reflected-photon spectrum from a cavity containing N atoms: the model requires a sharp reflection dip at the m-excitation-shifted frequency $\omega_0 + m\Omega$ with reflection amplitude near $-1$ and near-unity reflection at neighboring Dicke resonances. Alternatively, measure the infidelity of preparing a small-m Dicke state as a function of cooperativity C at the optimal detuning: it should fall as $C^{-1/2}$ unheralded and as $C^{-2/3}$ heralded, with the optimal $d$ moving from $(C/m)^{1/4}$ to $(C/m)^{1/3}$; a different power law or the absence of the detuning trade-off would falsify the central error-model claim.

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Extended reading notes

Core claim

The central claim is that the Grover iteration $G = R(\phi)^{\otimes N} \chi_0 R(-\phi)^{\otimes N} \chi_m$, acting on the Dicke manifold, prepares the Dicke state $|m\rangle$ with unit fidelity in exactly $k$ steps whenever the global rotation angle $\phi$ is chosen so that the initial product state has overlap $\langle m|\psi_i\rangle = \sin(\pi/[2(2k+1)])$. For large N this gives $k \sim 1.24\, m^{1/4}$ for $m \ll N/2$ and $k \sim 0.88\, N^{1/4}$ for $m = N/2$; numerically, every Dicke state with $3\le N\le 500$ can be prepared in four or fewer steps, and the W state $(m=1)$ in a single step. A two-step variant starting from the Dicke state $|N/2\rangle$ in the x-basis prepares the GHZ state in $\sim 0.62\, N^{1/4}$ steps, and an analogous construction prepares Cat states. The physical realization uses the dispersive cavity Hamiltonian $H = \hbar\Omega\,\hat{m}\hat{n}_c$, which shifts the cavity resonance by $m\Omega$ when $m$ atoms are excited, so that a photon tuned to $\omega_0 + m\Omega$ acquires the phase $-1$ only on the component $|m\rangle$; together with global rotations this implements one Grover step with constant resources. The error analysis, based on input–output theory and Kraus operators, predicts the stated cooperativity scalings and quantifies the sensitivity to spatial mode mismatch, giving a linear penalty $(2k+1)(1-\zeta)/2$ in infidelity per k-step protocol.

Load-bearing premise

The scheme assumes that a photon reflected from the cavity acts as a perfect oracle: it flips the sign of exactly the Dicke component whose shifted resonance matches the photon frequency and leaves every other component untouched, which real cavities only approximate because resonances have finite width and reflectivity less than one.

Editorial extensions

If this is right

  • Every Dicke state with $3 \le N \le 500$ qubits is preparable in at most four Grover steps in the ideal limit, and the W state in one step, independent of N.
  • GHZ states are preparable in $O(N^{1/4})$ steps starting from a product state, with the same constant resource per step as Dicke states.
  • The physical resources per Grover step are constant — two photon reflections for Dicke states, three for GHZ and Cat states — plus global rotations, with no individual addressing, ancillas, or measurements.
  • With realistic cooperativity $C=100$, predicted fidelities are 70–80% (unheralded) and 80–90% (heralded) for small-m Dicke states; infidelity scales as $O(C^{-1/2})$ unheralded and $O(C^{-2/3})$ heralded, plus a mode-matching penalty.
  • Heralding on detection of the reflected photon also gives a success probability $1 - O(C^{-1/3})$, so the deterministic protocol can be made near-unity efficiency at large cooperativity.

Reading between the lines

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

  • A natural extension, not pursued in the paper, is to treat the overlap condition as a general design rule: any target state whose overlap with a globally rotated product state can be tuned to $\sin(\pi/[2(2k+1)])$ is preparable in k steps by the same two-reflection iteration, so the method likely generalizes beyond Dicke, GHZ, and Cat families.
  • Because the fidelity of $\chi_m$ decreases with m while $\chi_0$ is much more accurate, the paper's trick of preparing $|N-m\rangle$ and flipping all qubits means symmetric states inherit the error of the smaller excitation number; this asymmetry could be exploited experimentally to target high-m states with better fidelity than direct preparation.
  • The scheme's Hamiltonian is formally equivalent to the Tavis–Cummings model, so the constant-depth Grover iteration should transfer to circuit QED, trapped ions, and Rydberg ensembles; the paper names these platforms but leaves their specific error behavior as future work.
  • A testable prediction of the error model is that for fixed cooperativity the optimal detuning parameter moves from $d\sim (C/m)^{1/4}$ (unheralded) to $d\sim (C/m)^{1/3}$ (heralded); an experiment measuring phase-inversion fidelity as a function of detuning could confirm the trade-off between resolution and spontaneous emission.
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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

3 major / 3 minor

Summary. The paper proposes a deterministic state-preparation protocol in which Grover iterations, built from global rotations and conditional phase inversions, prepare Dicke states in O(m^{1/4}) unitary steps, GHZ states in O(N^{1/4}) additional steps, and approximate Cat states in a few steps. The algorithm part gives an exact overlap condition, Eq. (11), and a transcendental equation, Eq. (21), for the required global rotation angle, with proofs for m=1 and m=N/2 and numerical verification up to N=500. The physical part realizes the phase inversion by reflecting a single photon off a one-sided cavity in the dispersive regime, so that one Grover step costs two or three photon reflections and global rotations. Appendices D-F present an error analysis including spatial mode mismatch, finite photon bandwidth, spontaneous emission, and cavity losses, and derive infidelity scalings in the cooperativity C, namely O(C^{-1/2}) unheralded and O(C^{-2/3}) heralded.

Significance. The ideal unitary construction is a genuine contribution: the overlap-condition approach in Eq. (11) is simple, exact, and leads to a single-parameter protocol with no individual addressing or ancillas. The proofs for the W state and the m=N/2 Dicke state, together with the detailed Kraus/input-output error analysis and the analytic scalings in Appendix F, are concrete and falsifiable. If the physical resource claim is properly qualified, the scheme would be an attractive alternative to prior carving and amplitude-amplification proposals. The main weakness is that the advertised O(C^{-1/2}) infidelity and O(N^{1/4}) photon count hide an N-dependence in the required cooperativity and mode matching, so the central practical resource claim needs revision.

major comments (3)
  1. [§VI.C.1, Eq. (F35), Table I] The single-oracle infidelity is 1−F(χ_m) ∼ a/d² + am d²/C. With the optimized d ∼ (C/m)^{1/4}, both terms are ∼√(m/C), so the per-oracle error contains an explicit factor √m. For the marquee case m=N/2, each Grover step therefore has infidelity ∼√(N/C) (χ_0 contributes the smaller 1/C), and after k ≈ 0.88N^{1/4} − 1/2 steps the accumulated infidelity grows as N^{3/4}/√C. The abstract's and Table I's 'O(C^{-1/2})' scaling is thus valid only at fixed m; maintaining a fixed total error ε requires C ∼ N^{3/2}/ε². Equation (39) imposes a similar N-dependent mode-matching requirement, 1−ζ ∼ ε/k ∼ ε/N^{1/4}. The physical resource claim 'O(N^{1/4}) photon reflections' is therefore incomplete unless these growing resource requirements are stated alongside.
  2. [§V and Discussion] The Cat-state protocol is explicitly approximate: χ_cat is replaced by χ_φ χ_{−φ}, valid only when the two coherent spin states are nearly orthogonal, and in the Discussion the authors state that a proof of the scaling of the Cat-state preparation steps is left for future work. Since the abstract lists Cat states among the main results, the paper should either prove the stated scaling or clearly label the Cat-state claim as a conjecture and quantify the approximation error.
  3. [§III.C and Discussion] The statement that Grover's algorithm 'can always prepare a Dicke state |m⟩ perfectly in O(m^{1/4}) steps' goes beyond the support provided. Equation (22) is an exact existence condition, but closed-form bounds are proven only for m=1 and m=N/2, the scaling for N≫1 with m≪N/2 is asymptotic, and the general claim is verified numerically only for N≤500. A general proof, for example via a uniform bound on the binomial overlap in Eq. (22), is needed before claiming 'always' for all N and m.
minor comments (3)
  1. [§VI.A, Eq. (41)] The text says r_n(δω_n) = −1, but Eq. (41) defines δω_n through the real part of a complex expression, so with dissipation one has only r_n(δω_n) ≈ −1; please rephrase to avoid implying an exact phase inversion in the physical model.
  2. [§VI.C.1, Eq. (F33)] The notation in Eq. (F33), in particular the term r_n r_l^* + a_n a_l^*, would be clearer if it were stated explicitly that only off-diagonal terms n≠l contribute to the infidelity.
  3. [§VI.C.4, Fig. 13] The text states that a four-orders-of-magnitude change in C reduces the error by two or three orders of magnitude, citing Fig. 13 for N=15 and m=1..6; this confirms the scaling only for small fixed m and should not be presented as evidence for the m=N/2 case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ideal protocol is self-contained and the physical error analysis is derived from an independent input-output model.

full rationale

The paper's ideal algorithm is not circular. The central construction is to choose a global product-state rotation angle ϕ such that the overlap with the target Dicke state is exactly sin(π/[2(2k+1)]) (Eq. 21). This is an explicit algebraic constraint solved by the authors, not a quantity fitted to the target fidelity. The Grover iteration G = R(ϕ)^⊗N χ0 R(−ϕ)^⊗N χm is then shown by the standard Grover rotation formula (Eq. 5) to produce the target after k steps; this is a direct derivation, not a restatement of the goal. The existence condition (Eq. 22) and the proofs for m=1 and m=N/2 are independent mathematical statements. The physical implementation in Sec. VI uses the independent dispersive Hamiltonian H = ℏΩ m̂ n̂_c (cited to [33,48]) and the input-output scattering amplitudes of Eqs. (40a)-(40d) (cited to [54]); the phase-inversion operation χm is not defined in terms of the final Dicke-state fidelity. The error analysis in Appendix F computes fidelities from these scattering amplitudes and Kraus superoperators, optimizing the detuning parameter d; the claimed scalings 1−F(χ_m)∼C^{-1/2} and C^{-2/3} are derived from Eqs. (F35), (F45), and related expressions, not from assuming the target fidelity. While the skeptical concern that the per-step infidelity of χ_m grows with m (e.g., √(m/C) for fixed optimized d) is a valid correctness/resource-scaling caveat, it is not circularity: the paper's own equations expose this m-dependence, and the resource claim is an extrapolation, not a tautology. The only explicit limitation is the Cat-state scaling, which the authors state is unproved ('We leave it for future work to prove the scaling of the steps for Cat states with the number of qubits'), but an admitted open problem is not a circular step. Self-citations (e.g., companion letter [45] and Ref. [56]) are not load-bearing assumptions of the derivation. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Overall the derivation chain is self-contained and externally benchmarked against standard Grover/Long results and previous cavity-carving schemes.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The protocol depends on standard Grover and Long machinery plus a set of cavity-QED modeling choices. One control angle phi is solved exactly and the detuning parameter d is optimized; there are no data-fitted constants or invented physical entities. The error-scaling results assume a noiseless rotation layer and a single-photon dispersive interaction.

free parameters (3)
  • global rotation angle phi = solutions of Eq. (21); two roots in [0, pi]
    Chosen so the product state R(phi)^N |0> has overlap sin(pi/(2(2k+1))) with the target Dicke state, enabling exact preparation in k steps. It is a searched control parameter, not a measured data fit.
  • optimized detuning parameter d = g^2/(Delta kappa) = d ~ (C/m)^(1/4) unheralded, d ~ (C/m)^(1/3) heralded
    In the physical error model, the cavity detuning is optimized, first by analytic estimates and then numerically, to balance off-resonant leakage and spontaneous emission; this choice enters the predicted infidelity scalings.
  • modified Grover phase alpha = alpha = 2 arcsin( sin(pi/(4k+6)) / sin(theta/2) )
    Optional phase used to achieve unit-fidelity Dicke and GHZ preparation in integer steps, taken from Long's method (Eq. 10). It is a free control phase if used.
assumptions (6)
  • domain assumption Dispersive cavity-QED Hamiltonian H = hbar Omega m n_c with Omega = g^2/Delta and Omega much less than Delta.
    Invoked in Sec. VI A to justify the linear frequency shift omega_m = omega_0 + m Omega and the phase-inversion implementation; taken from Refs. [33,48].
  • domain assumption Input-output scattering amplitudes r_n, t_n, a_n, m_n in Eqs. (40a)-(40d) describe reflection, transmission, spontaneous emission, and mirror scattering.
    Standard input-output theory used for the error model in Sec. VI C and Appendix F; these amplitudes are assumed without derivation in this paper.
  • domain assumption Global rotations R(phi) are assumed noiseless in the error-scaling analysis.
    The infidelity scalings 1/sqrt(C) and 1/C^(2/3) ignore rotation errors, which would add a constant error floor in a real implementation.
  • ad hoc to paper The Cat-state phase inversion can be approximated by chi_phi chi_(-phi), valid when the two coherent spin states are nearly orthogonal.
    Used in Sec. V, Eq. (34). For finite N this is an approximation; exactness is not guaranteed and the scaling with N is left unproved.
  • domain assumption All atoms couple identically to the cavity mode with a single coupling constant g.
    Assumed by the collective Dicke-state description and by the scattering amplitudes; position-dependent coupling would break the perfect frequency selectivity of chi_m.
  • domain assumption Each reflection event uses a single photon or a heralded weak coherent pulse.
    The analysis in Sec. VI and Appendix E is for one photon per reflection; multi-photon components would add errors not captured by the Kraus model.

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Pith. "Pith review of Deterministic carving of quantum states with Grover's algorithm." pith.science (2026). https://pith.science/paper/WK3K2JZT

@misc{pith2026250118884,
  author       = {Pith},
  title        = {Pith review of: Deterministic carving of quantum states with Grover's algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WK3K2JZT}},
  note         = {Machine review of arXiv:2501.18884}
}
abstract

We show that iteration of a few ( $\sim N^{1/4}$) unitary steps of Grover's algorithm suffices to perfectly prepare a Dicke state of $N$ atoms in a cavity. We also show that a few subsequent Grover steps can be employed to generate GHZ and Cat states. The Grover iteration is physically realized by global qubit rotations and by the phase shift of single photons reflected on the cavity. Our protocols are deterministic and require no individual addressing of the atoms. A detailed error analysis accounting for spatial mode matching of the photon to the cavity, spontaneous emission, mirror scattering, and the finite bandwidth of the photon mode is used to predict the fidelity of the prepared states as a function of system parameters and atom-cavity cooperativity. The fidelity can be increased by heralding on detection of the reflected photon.

Figures

Figures reproduced from arXiv: 2501.18884 by the authors.

Figure 1
Figure 1. FIG. 1. The Grover steps [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Grover steps [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Density plot of the required global rotation angle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The energy level scheme for the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Fidelity after applying Grover’s algorithm versus the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The fidelity in implementing the phase inversion operator [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The infidelity of the phase inversion [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The fidelity in implementing the phase inversion operator [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The fidelity in preparing the Dicke state [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The fidelity in preparing the Dicke state [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The efficiency (success probability) in preparing the Dicke state [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Scaling of the infidelity with the atom-cavity coop [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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    Analysis of the Cat states preparation Starting from the Cat states |cat±, ϕ⟩ = 1p 2 ± 2 ⟨ϕ| −ϕ⟩ (|ϕ⟩⊗N ± |−ϕ⟩⊗N ) (B13) where the CSS states |ϕ⟩⊗N = R(ϕ)⊗N |m = 0⟩ and |−ϕ⟩⊗N = R(−ϕ)⊗N |m = 0⟩ are given by |ϕ⟩⊗N = cos(ϕ/2) |0⟩ + sin(ϕ/2) |1⟩ ⊗N (B14) = NX m=0 s N m cosN −m(ϕ/...

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    ∼ 4 n2C 2 + 4 n2d2 (F47) Again, just like in the unheralded case above, we find no tradeoff between the two error terms, so the ideald value is d = ∞(∆ = 0), and we get1 − F (χ0) ∼ 1/C 2. To summarize, the heralding case gives the scaling: 1 − F (χm) ∼ 1 C 2/3 + w2, m ̸= 0, d ...

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    Grover iteration a. Not heralding on detection of the reflected photon The effect of one Grover step,G = R(ϕ)⊗N χ0R(−ϕ)⊗N χn, under this description would involve hitting the cavity with two photon wavepackets and not heralding on detection of the reflected photons. Therefore,...

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    (F13)] and the Grover iteration [Eq

    Numerical simulation We outline the procedure used in this work to numerically simulate the effects of the physical phase inversion operator [Eq. (F13)] and the Grover iteration [Eq. (F56)]. For N qubits, the corresponding operators/matrices are(N + 1) × (N + 1) in the Dicke b...

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    K(N +1)2 )

    Construct the diagonal entries ofKavg as a vector(K1 K2 . . .K(N +1)2 )

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    To model the action of the rotation superoperatorR(−ϕ) after the action ofKavg, reshape ρ = (K1ρ1 K2ρ2 . . .K(N +1)2 ρ(N +1)2 ) (F61) into the original density matrix format,ρ, then compute RN (−ϕ)ρRN (ϕ). (F62) which costs O(N 3) operations

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    Therefore, the total computational cost of this approach isO(N 3) operations

    To computeR(ϕ)Kavg(0), repeat steps 1 and 2. Therefore, the total computational cost of this approach isO(N 3) operations. We note that further optimizations, based on the symmetry properties of the operators and the density matrix, could be possible

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