Pith. sign in

REVIEW 3 major objections 5 minor 56 references

The Hopf ansatz claims that one binary-tree circuit gives arbitrary pure-state optimization a full navigable geometry: coordinates, a diagonal metric, and preparable tangent directions.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:44 UTC pith:PCECKBSX

load-bearing objection A clean and genuinely useful variational chart with exact tangent-state geometry, but the main text never benchmarks the finite-shot gradient estimator on which the practical 'compass' claim rests. the 3 major comments →

arxiv 2607.14231 v1 pith:PCECKBSX submitted 2026-07-15 quant-ph

A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization

Ruge Lin , Guangxi Li This is my paper
classification quant-ph MSC 81P68 PACS 03.67.Lx
keywords Hopf ansatzstate preparationpure-state optimizationvariational quantum eigensolverquantum Fisher informationgradient estimationbinary treeFubini-Study metric
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.

This paper introduces the Hopf ansatz, a binary-tree circuit that parameterizes every normalized real or complex quantum state vector. The same tree provides state preparation, an explicit inverse map from amplitudes to angles, a diagonal pullback metric, and normalized coordinate tangents that can be prepared as quantum states. For Hamiltonian objectives and related transition-moment objectives, each gradient component becomes a known metric factor times a measurable transition moment, and the compiled gradient-access settings are organized into O(log N) circuit families, each costing O(nN) CNOTs. If correct, this turns universal state preparation into a practical optimization framework with closed-form geometry, with applications to VQE and variational metrology.

Core claim

The paper claims that the Hopf binary-tree construction is a complete variational chart for the real or complex pure-state sphere, not just a state-preparation routine. It proves that the pullback metric is diagonal and analytic: magnitude directions are weighted by subtree probability mass, and phase directions by leaf occupations. It further proves that every nonzero coordinate tangent can be prepared exactly by the same circuit skeleton using shifted or clamped angle assignments, and that gradient components factor into a metric factor times a transition moment between the current state and this tangent state. Signed branch states then express these transition moments through expectation-

What carries the argument

The central object is the Hopf binary tree: a complete rooted binary tree whose internal nodes carry split angles (sine-cosine factors) and whose terminal leaves carry independent phases in the complex case. Generalized Hopf coordinates are recursively defined by splitting probability mass between orthogonal subtrees, and the circuit realization uses multi-controlled Ry or RC gates. The diagonal metric theorem makes the geometry analytic and componentwise, while the tangent-state theorem gives explicit gate assignments for normalized coordinate derivatives. The signed branch-state construction converts transition moments into measurable expectation values, and layerwise index-controlled batc

Load-bearing premise

The practical usefulness of the gradient-access construction assumes that the finite-shot signed-branch estimator (Section III, Eqs. 24-30) delivers its claimed precision; the benchmarks only validate a deterministic exact-gradient optimizer layer, not the shot-noise-limited estimator.

What would settle it

Simulate the signed-branch gradient estimator on a small real-Hopf VQE instance (e.g., n=4) at a fixed finite shot count, comparing the estimated gradient components to the exact analytical gradient; if the estimator exhibits bias or variance substantially worse than the predicted statistical cost from the branch probabilities, the central efficiency claim for hardware optimization would be falsified.

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

If this is right

  • Arbitrary pure-state optimization can be equipped with a closed-form coordinate map, local metric, and exact tangent-state preparation, removing the need for dense metric estimation or inversion.
  • The number of compiled circuit families needed to access a full gradient grows only logarithmically with Hilbert-space dimension, with each setting costing O(nN) CNOTs.
  • The gradient-access construction extends beyond VQE to objectives whose derivatives are transition moments of Hermitian chain-rule observables, including local pure-state QFI and fixed-readout Ramsey CFI objectives.
  • Metric-aware optimizers using the diagonal Hopf metric and state-sphere geodesics reach numerical-precision median gaps in deterministic real-state benchmarks, with clearer gains over coordinate Adam baselines in VQE mean final gaps.
  • The measurement budget needed for a chosen gradient precision remains a separate statistical cost, decoupled from the compiled-setting count.

Where Pith is reading between the lines

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

  • Editorial inference: The same layerwise batching could be adapted to higher-order derivative information, such as Hessian-vector products, since the tangent states are already indexed and preparable.
  • Editorial inference: The explicit inverse map and diagonal geometry suggest that classical optimization over the state sphere can be tightly coupled with circuit parameter updates, potentially enabling hybrid classical-quantum optimizers with better conditioning than coordinate descent.
  • Editorial inference: The finite-shot estimator remains the natural stress point; a testable extension is to benchmark the signed-branch gradient estimator on noisy simulators or hardware for small n, measuring bias and variance against exact gradients.
  • Editorial inference: The phase-block batching in the complex ansatz highlights a symmetry not present in the real case; exploiting the single-leaf structure of phase tangents could lead to specialized readout strategies for phase-sensitive metrology objectives.

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

3 major / 5 minor

Summary. The paper introduces the Hopf ansatz, a binary-tree circuit that parameterizes arbitrary real and complex pure states. The same tree supports state preparation, an explicit inverse map from amplitudes to angles, a diagonal pullback metric, and exact preparation of normalized coordinate tangents. For Hamiltonian (and chain-rule) objectives, each gradient component is expressed as a known metric factor times a transition moment, estimated via signed branch states; the compiled gradient-access configurations are organized into O(log N) circuit families. Numerical experiments with deterministic exact gradients on real-state VQE and metrology-inspired tasks show that the metric-aware Hopf optimizers reach near-machine-precision median gaps and outperform coordinate-Adam baselines in several respects.

Significance. If the results hold, this is a valuable conceptual contribution: a universal state-preparation circuit that serves simultaneously as a coordinate chart, a metric provider, and a source of exact tangent states. The proofs of Theorems 1–3 are self-contained and appear correct; the inverse map is constructive with O(N) classical cost; the diagonal metric is explicit and removes the need to estimate or invert a dense quantum geometric tensor. The compiled-setting scaling O(log N) is elegant, and the deterministic numerical results are a useful sanity check. However, the practical 'compass' claim rests on the finite-shot signed-branch gradient estimator, whose variance and optimizer behavior under shot noise are not analyzed or benchmarked in the main text. That gap is load-bearing for the hardware-efficiency implications.

major comments (3)
  1. [Section III, Eqs. (29)–(30); Section IV, first paragraph] The finite-shot signed-branch estimator is the only route from the exact gradient identity (13) to a shot-noise-limited quantum device, yet it is never exercised in the numerical section. Section IV explicitly states that each step evaluates the objective and first derivatives exactly from the current state. The repository is said to contain 'finite-shot signed-branch gradient tests,' but no results or variance diagnostics are reported in the manuscript. Without evidence on bias, variance, and optimizer robustness under finite sampling, the claim that the Hopf ansatz provides a practically usable 'compass' for optimization is unsupported.
  2. [Section III, Eqs. (29)–(30) and 'Complete estimator and scaling analysis'] Each gradient component is 2√g times a difference of separately measured expectation values. When Re⟨e_i|H|ψ⟩ is small compared with E_ψ and E_tan_i — which is generic near a minimum or when g_i,i is small — shot noise in the baseline and tangent energies dominates, and the per-component sample count for a target gradient accuracy ε scales as roughly 1/ε². The layer-batched scheme collects K label-conditioned samples per parameter, so a full gradient costs O(KN) repetitions even though the number of compiled circuit families is O(log N). The text acknowledges this in one sentence, but the 'logarithmically with Hilbert-space dimension' framing is likely to be misread as a measurement-efficiency claim. This needs either a quantitative variance analysis or a deliberate de-emphasis of the practical efficiency claim.
  3. [Section IV and repository statement] The numerical benchmarks cover only the real Hopf ansatz with deterministic exact gradients; the complex ansatz, the leaf-phase gradient block, and the finite-shot branch estimator are not tested in the main text. The mathematical construction for these cases is plausible and My reading of the proofs is favorable, but the paper's 'arbitrary pure-state optimization' claim is broader than what is demonstrated. The auxiliary repository checks are not a substitute for reported results.
minor comments (5)
  1. [Figure 1 and accompanying text] Several metric-aware traces are reported with median final gap 0 or 1.02×10⁻²⁵ on logarithmic axes with a positive floor. Please state the exact clipping floor and how zeros and negative roundoff values are represented, so the box plots are reproducible.
  2. [Theorem 3, Eq. (14)] There is a typographical artifact in the display: 'HopfComplex(θ^(i))|0⟩^{⊗n} , .' contains a stray comma and period. Please clean up the equation.
  3. [Section IV, inverse-map clipping] The clipping floors (10⁻⁶ for non-final angles, 10⁻⁹ near final-layer sign singularities) are described only briefly. State whether these floors affect any of the reported final gaps, especially the near-machine-precision medians.
  4. [Reference [49]] The repository reference is cited as 'Hopf ansatz GitHub repository (2026)' without a URL or persistent identifier. Please provide a DOI or stable link.
  5. [Appendix D, Eq. (D3)] The notation B_χ[O] is used before it is explicitly defined as ⟨χ|O|χ⟩. Please move or repeat the definition at first use.

Circularity Check

0 steps flagged

No significant circularity: inverse map, metric, tangent synthesis, and gradient identities are all directly proved; the only self-citation (Ref. [24]) is minor and non-load-bearing.

full rationale

The Hopf derivation chain is self-contained. Definition 2 defines the forward amplitude recursion; Lemma 1 (Appendix A) constructs the inverse bottom-up from subtree norms and signed leaf pairs, proving it reproduces x by induction, so the inverse map is the algebraic inverse of the forward map, not a fitted or assumed result. Theorem 1 computes the pullback metric from the product amplitude formula (Lemma 3) and the orthogonality of subtree supports; no metric entry is imported from data. Theorem 3 gives explicit gate assignments (ancestors clamped to 0 or pi/2, target angle shifted by pi/2, descendants kept) and proves the resulting Hopf-circuit state equals |e_i>; this is a construction, not a self-referential prediction. The gradient statement Theorem 2 is the ordinary product rule rewritten through the normalized tangent; Eqs. (27)-(30) are exact rearrangements of the signed-branch expansion, so the estimator is an identity for exact expectations. The compiled-setting count (39)-(41) follows by counting layers, and the paper explicitly states that the measurement budget is a separate statistical cost. Section IV uses deterministic exact gradients and known optima, and the repository's finite-shot signed-branch tests are not quantified in the main text; this is an unvalidated practical claim, but it is not a circular derivation. The only self-citation of any weight is Ref. [24], used for the state-sphere update layer and the ambient-sphere metric convention; those formulas are restated in Appendix C and the central Hopf claims do not depend on it.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No fitted parameters enter the central derivations. The listed axioms are standard math and domain assumptions; the only hand-chosen numbers are numerical clipping floors. No new physical entities are postulated.

free parameters (1)
  • Boundary clipping floors for inverse map and gradient lift = 1e-6 (non-final angles), 1e-9 (final-layer signs), 1e-12 (state-gradient floor)
    Hand-chosen numerical stabilizers in Section IV; not fitted to data and not load-bearing for the analytic claims, but they influence reported final gaps on boundary traces.
axioms (5)
  • standard math Hopf/polyspherical coordinates cover S^{N-1} (real) and S^{2N-1} (complex) with the stated angle ranges.
    Definition 2 (A)/(B) invokes Refs. [21,22]; the universality claim depends on this coverage.
  • standard math The no-ancilla multi-controlled rotation CNOT counts of Lemma 2 and Theorem 5 are valid.
    Used to claim each compiled setting costs O(nN) and hence that the O(log N) setting organization is meaningful; the counts come from Refs. [51–53].
  • domain assumption Objectives considered are Hermitian-expectation costs whose derivatives reduce to transition moments Re<e_i|O|psi>.
    Theorem 2 and Appendix D; this excludes noise-model-dependent and global-phase-estimation regimes.
  • domain assumption Coherent controlled preparation of the Hopf ansatz with fixed relative phase is available, and ancilla X-basis measurement yields signed branch states without postselection.
    Section III Eqs. (24)–(35); the branch-state estimator and batching rely on this.
  • domain assumption Deterministic exact-gradient real-state state-vector simulations are representative of the optimizer layer's behavior.
    Section IV uses exact costs and gradients to test Riemannian updates; extrapolation to finite shots is not directly benchmarked in the main text.

pith-pipeline@v1.3.0-alltime-deepseek · 103 in / 16799 out tokens · 190170 ms · 2026-08-02T02:44:55.931165+00:00 · methodology

0 comments
read the original abstract

Optimizing arbitrary quantum state vectors is like navigating the unit sphere in Hilbert space: beyond target reachability, optimization asks for coordinates, local distance information, and measurable directions. We introduce the Hopf ansatz, a binary-tree circuit for arbitrary normalized real and complex state vectors. Internal angles steer probability between subtrees, leaf phases carry the complex degrees of freedom, and the same tree gives state preparation and an explicit inverse map from amplitudes to physical angles. Together these structures act as a compass for the search: the inverse map gives coordinates, the diagonal induced metric gives local distance information, and nonzero coordinate tangents become preparable normalized states. For Hamiltonian objectives, and for objectives with the same local transition-moment form, each gradient component is a known scale factor times a transition moment between the current state and a tangent state. A branch-state construction expresses these moments through expectation-value measurements, while the tree organizes the compiled gradient settings by magnitude layer and leaf-indexed phase family. Thus the number of distinct compiled gradient-access circuit families grows only logarithmically with Hilbert-space dimension, while the measurement budget required for a chosen precision remains a separate statistical cost. In deterministic real-state benchmarks with exact costs, exact gradients, and known global optima, metric-aware Hopf optimizers reach numerical-precision median gaps, with the clearest baseline gains in smaller VQE mean final gaps and stronger concentration of metrology-inspired traces at numerical precision. The Hopf ansatz turns universal state preparation into a navigable framework for arbitrary pure-state optimization.

Figures

Figures reproduced from arXiv: 2607.14231 by Guangxi Li, Ruge Lin.

Figure 1
Figure 1. Figure 1: Aggregate final precision and threshold hitting over all stress tests and system sizes n = 6, . . . , 10. The top row shows final optimality gaps after 200 accepted updates; the gray dashed line marks the 10−8 reference threshold. The bottom row shows the first accepted update at which each crossing trace reaches that threshold; the parenthetical labels report the number crossing within 200 updates out of … view at source ↗
Figure 2
Figure 2. Figure 2: Per-task convergence at n = 10, with ten deterministic pseudo-random initial-state seeds per task. The top row shows the parent-Hamiltonian, scrambled-Hamming-spectrum, and small-gap-spectrum VQE tasks. The bottom row shows the single-target Fisher, QFI extremal-superposition, and balanced two-target Fisher tasks. Each curve is the mean optimality gap across the ten seeds, and each translucent band is the … view at source ↗
Figure 3
Figure 3. Figure 3: Binary tree for preparing a 4-qubit state in Hopf coordinates. A bit-0 branch multiplies the amplitude by cos θj , and a bit-1 branch multiplies it by sin θj . The displayed θj labels are the Hopf magnitude parameters associated with internal nodes, while the blue labels θ16, . . . , θ31 are the complex leaf phases. Circled numbers indicate the gate-application order generated by Algorithms 1 and 2. The bi… view at source ↗
Figure 4
Figure 4. Figure 4: Four-qubit Hopf ansatz circuit derived from the tree in [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Indexed derivative preparation for the d = 2 magnitude layer of the four-qubit real Hopf ansatz. The index register I1I2 selects one of the tangent-state assignments θ (j) with j ∈ {4, 5, 6, 7} according to Eq. (B13). Conditioned on the observed index, the system register is in the corresponding normalized coordinate tangent [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Indexed signed-branch preparation for the same d = 2 magnitude layer. The branch a = 0 prepares the baseline state HopfReal(θ)|0⟩ ⊗4 , while the branch a = 1 prepares the index-selected tangent state HopfReal(θ (j) )|0⟩ ⊗4 . Measuring a in the X basis, equivalently by adding a Hadamard before Z-basis readout, yields the signed branch states used in the estimator. Appendix C: Geometry-native optimizers for … view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 11 canonical work pages

  1. [1]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, Nature communications 10.1038/ncomms5213 (2014)

    doi:10.1038/ncomms5213 2014

  2. [3]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio,et al., Nature Reviews Physics 10.1038/s42254- 021-00348-9 (2021)

    doi:10.1038/s42254- 2021

  3. [4]

    Koczor, S

    B. Koczor, S. Endo, T. Jones, Y. Matsuzaki, and S. C. Benjamin, New Journal of Physics 10.1088/1367- 2630/ab965e (2020)

    doi:10.1088/1367- 2020

  4. [6]

    J. J. Meyer, J. Borregaard, and J. Eisert, npj Quantum Information 10.1038/s41534-021-00425-y (2021)

    doi:10.1038/s41534-021-00425-y 2021

  5. [7]

    Kaubruegger, A

    R. Kaubruegger, A. Shankar, D. V. Vasilyev, and P. Zoller, PRX Quantum 10.1103/PRXQuan- tum.4.020333 (2023)

    doi:10.1103/prxquan- 2023

  6. [8]

    D. V. Vasilyev, A. Shankar, R. Kaubruegger, and P. Zoller, (2024), arXiv:2404.14194 [quant-ph]

    Pith/arXiv arXiv 2024

  7. [9]

    T. K. Le, H. Q. Nguyen, and L. B. Ho, Scientific Reports 10.1038/s41598-023-44786-0 (2023). 13

    work page doi:10.1038/s41598-023-44786-0 2023

  8. [10]

    MacLellan, P

    B. MacLellan, P. Roztocki, S. Czischek, and R. G. Melko, npj Quantum Information 10.1038/s41534-024-00914-w (2024)

    work page doi:10.1038/s41534-024-00914-w 2024

  9. [11]

    Knill, (1995), arXiv:quant-ph/9508006

    E. Knill, (1995), arXiv:quant-ph/9508006

    Pith/arXiv arXiv 1995

  10. [12]

    V. V. Shende, S. S. Bullock, and I. L. Markov, IEEE Transactions on Computer-Aided Design of Inte- grated Circuits and Systems 10.1109/TCAD.2005.855930 (2005)

    arXiv 2005

  11. [13]

    M¨ ott¨ onen, J

    M. M¨ ott¨ onen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, (2004), arXiv:quant-ph/0407010

    Pith/arXiv arXiv 2004

  12. [14]

    Plesch and ˇC

    M. Plesch and ˇC. Brukner, Physical Review A 10.1103/PhysRevA.83.032302 (2011)

    doi:10.1103/physreva.83.032302 2011

  13. [15]

    Zhang, T

    X.-M. Zhang, T. Li, and X. Yuan, Physical Review Let- ters 10.1103/PhysRevLett.129.230504 (2022)

    doi:10.1103/physrevlett.129.230504 2022

  14. [16]

    X. Sun, G. Tian, S. Yang, P. Yuan, and S. Zhang, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 10.1109/TCAD.2023.3244885 (2023)

    arXiv 2023

  15. [17]

    I. F. Araujo, D. K. Park, T. B. Ludermir, W. R. Oliveira, F. Petruccione, and A. J. Da Silva, Quantum Information Processing 10.1007/s11128-023-03869-7 (2023)

    work page doi:10.1007/s11128-023-03869-7 2023

  16. [18]

    Schuld, R

    M. Schuld, R. Sweke, and J. J. Meyer, Physical Review X11, 041011 (2021)

    2021

  17. [19]

    G. Li, R. Ye, X. Zhao, and X. Wang, Advances in Neural Information Processing Systems35, 19456 (2022)

    2022

  18. [20]

    R. M. Farias, T. O. Maciel, G. Camilo, R. Lin, S. Ramos-Calderer, and L. Aolita, Physical Review Ap- plied 10.1103/PhysRevApplied.23.044014 (2025)

    doi:10.1103/physrevapplied.23.044014 2025

  19. [21]

    H. S. Cohl, SIGMA. Symmetry, Integrabil- ity and Geometry: Methods and Applications 10.3842/SIGMA.2013.042 (2013)

    work page doi:10.3842/sigma.2013.042 2013

  20. [22]

    Sch¨ obel, SIGMA

    K. Sch¨ obel, SIGMA. Symmetry, Integrabil- ity and Geometry: Methods and Applications 10.3842/SIGMA.2016.041 (2016)

    doi:10.3842/sigma.2016.041 2016

  21. [23]

    Stokes, J

    J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum 10.22331/q-2020-05-25-269 (2020)

    doi:10.22331/q-2020-05-25-269 2020

  22. [24]

    A. J. Ferreira-Martins, R. M. S. Farias, G. Camilo, T. O. Maciel, A. Tosta, R. Lin, A. Alhajri, T. Haug, and L. Aolita, (2025), arXiv:2506.17395 [quant-ph]

    arXiv 2025

  23. [25]

    Schuld, V

    M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Kil- loran, Physical Review A 10.1103/PhysRevA.99.032331 (2019)

    doi:10.1103/physreva.99.032331 2019

  24. [26]

    Banchi and G

    L. Banchi and G. E. Crooks, Quantum 10.22331/q-2021- 01-25-386 (2021)

    doi:10.22331/q-2021- 2021

  25. [27]

    Wierichs, J

    D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, Quan- tum 10.22331/q-2022-03-30-677 (2022)

    doi:10.22331/q-2022-03-30-677 2022

  26. [28]

    Abbas, R

    A. Abbas, R. King, H.-Y. Huang, W. J. Hug- gins, R. Movassagh, D. Gilboa, and J. McClean, Advances in Neural Information Processing Systems 10.5555/3666122.3668062 (2023)

    arXiv 2023

  27. [29]

    Bowles, D

    J. Bowles, D. Wierichs, and C.-Y. Park, Quantum 10.22331/q-2025-10-02-1873 (2025)

    work page doi:10.22331/q-2025-10-02-1873 2025

  28. [30]

    Coyle, S

    B. Coyle, S. Raj, N. Mathur, E. A. Cherrat, N. Jain, S. Kazdaghli, and I. Kerenidis, npj Quantum Information 10.1038/s41534-025-01099-6 (2025)

    work page doi:10.1038/s41534-025-01099-6 2025

  29. [31]

    Chinzei, S

    K. Chinzei, S. Yamano, Q. H. Tran, Y. Endo, and H. Oshima, npj Quantum Information 10.1038/s41534- 025-01036-7 (2025)

    doi:10.1038/s41534- 2025

  30. [32]

    S. L. Braunstein and C. M. Caves, Physical Review Let- ters 10.1103/PhysRevLett.72.3439 (1994)

    doi:10.1103/physrevlett.72.3439 1994

  31. [33]

    M. G. A. Paris, International Journal of Quantum Infor- mation 10.1142/S0219749909004839 (2009)

    doi:10.1142/s0219749909004839 2009

  32. [34]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho- tonics 10.1038/nphoton.2011.35 (2011)

    doi:10.1038/nphoton.2011.35 2011

  33. [35]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics 10.1103/RevModPhys.89.035002 (2017)

    doi:10.1103/revmodphys.89.035002 2017

  34. [36]

    Pezz` e, A

    L. Pezz` e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Reviews of Modern Physics 10.1103/RevModPhys.90.035005 (2018)

    doi:10.1103/revmodphys.90.035005 2018

  35. [37]

    N. F. Ramsey, Physical Review 10.1103/PhysRev.78.695 (1950)

    doi:10.1103/physrev.78.695 1950

  36. [38]

    Barzilai and J

    J. Barzilai and J. M. Borwein, IMA journal of numerical analysis 10.1093/imanum/8.1.141 (1988)

    doi:10.1093/imanum/8.1.141 1988

  37. [39]

    D. C. Liu and J. Nocedal, Mathematical programming 10.1007/BF01589116 (1989)

    doi:10.1007/bf01589116 1989

  38. [40]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre,Optimiza- tion algorithms on matrix manifolds(Princeton Univer- sity Press, 2008)

    2008

  39. [41]

    D. P. Kingma and J. Ba, (2014), arXiv:1412.6980 [cs.LG]

    Pith/arXiv arXiv 2014

  40. [42]

    Bae and L.-C

    J. Bae and L.-C. Kwek, Journal of Physics A: Mathe- matical and Theoretical 10.1088/1751-8113/48/8/083001 (2015)

    doi:10.1088/1751-8113/48/8/083001 2015

  41. [43]

    Subramanian and S

    M. Subramanian and S. Vinjanampathy, Physical Review A 10.1103/PhysRevA.110.032424 (2024)

    work page doi:10.1103/physreva.110.032424 2024

  42. [44]

    Kaubruegger, P

    R. Kaubruegger, P. Silvi, C. Kokail, R. van Bijnen, A. M. Rey, J. Ye, A. M. Kaufman, and P. Zoller, Physical Re- view Letters 10.1103/PhysRevLett.123.260505 (2019)

    doi:10.1103/physrevlett.123.260505 2019

  43. [45]

    X. Yang, X. Chen, J. Li, X. Peng, and R. Laflamme, Scientific Reports 10.1038/s41598-020-80070-1 (2021)

    work page doi:10.1038/s41598-020-80070-1 2021

  44. [46]

    Rabitz, R

    H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, Science 10.1126/science.288.5467.824 (2000)

    work page doi:10.1126/science.288.5467.824 2000

  45. [47]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K¨ ockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr¨ uggen,et al., The European Physical Journal D 10.1140/epjd/e2015-60464-1 (2015)

    doi:10.1140/epjd/e2015-60464-1 2015

  46. [48]

    Khatri, R

    S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sorn- borger, and P. J. Coles, Quantum 10.22331/q-2019-05-13- 140 (2019)

    work page doi:10.22331/q-2019-05-13- 2019

  47. [49]

    Lin and G

    R. Lin and G. Li, Hopf ansatz GitHub repository (2026)

    2026

  48. [50]

    Efthymiou, S

    S. Efthymiou, S. Ramos-Calderer, C. Bravo-Prieto, A. P´ erez-Salinas, D. Garc´ ıa-Mart´ ın, A. Garcia-Saez, J. I. Latorre, and S. Carrazza, Quantum Science & Technol- ogy 10.1088/2058-9565/ac39f5 (2022)

    doi:10.1088/2058-9565/ac39f5 2058

  49. [51]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVin- cenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Physical review A 10.1103/Phys- RevA.52.3457 (1995)

    doi:10.1103/phys- 1995

  50. [52]

    V. V. Shende and I. L. Markov, (2008), arXiv:0803.2316 [quant-ph]

    Pith/arXiv arXiv 2008

  51. [53]

    R. Vale, T. M. D. Azevedo, I. C. Ara´ ujo, I. F. Araujo, and A. J. Da Silva, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 10.1109/T- CAD.2023.3327102 (2023)

    arXiv 2023

  52. [54]

    Wilkens, Quantum Circuit Library (2023)

    J. Wilkens, Quantum Circuit Library (2023)

    2023

  53. [55]

    Dai and Y

    Y.-H. Dai and Y. Yuan, SIAM Journal on optimization 10.1137/S1052623497318992 (1999)

    work page doi:10.1137/s1052623497318992 1999

  54. [56]

    M. R. Hestenes, E. Stiefel,et al., Journal of research of the National Bureau of Standards 10.6028/JRES.049.044 (1952). 14 Appendix A: Algorithms and subroutines Inverse Hopf map Lemma 1(Inverse Hopf map and classical cost).Letn∈NandN=2 n. (A) Real case. Letx= (x 0,...,x N−1 )∈R N satisfy ∑N−1 ℓ=0 x2 ℓ =1. For each internal nodej∈{1,...,N−1}, define its l...

    doi:10.6028/jres.049.044 1952

  55. [57]

    For an internal nodej, the real metric entryg R j,j is the squared amplitude weight accumulated along the path from the root to the parent ofj

    (B5) The same tree also gives the metric entries. For an internal nodej, the real metric entryg R j,j is the squared amplitude weight accumulated along the path from the root to the parent ofj. Let cj :=cos 2θj,s j :=sin 2θj. Then the diagonal real Hopf metric forn=4 is listed in Table II. In the complex ansatz, the magnitude block has the same entries, w...

  56. [58]

    Inside the subtree, the same trigonometric shift at nodeiproduces the same linear combination (E5), now with the correct inherited leaf phases

    Since Step 1 makesx k(θ(i)) =0 fork/∈L(i), those outside phases are irrelevant. Inside the subtree, the same trigonometric shift at nodeiproduces the same linear combination (E5), now with the correct inherited leaf phases. Hence, HopfComplex ( θ(i)) |0⟩⊗n =|e i(θ)⟩. Finally, if nodeihas depthd(root at depth 0), its subtree contains exactlyN/2 d leaves. T...