pith. sign in

arxiv: 2510.19039 · v3 · pith:NQJQIBOAnew · submitted 2025-10-21 · 🪐 quant-ph

Hierarchical Fusion Method for Scalable Quantum Eigenstate Preparation

Matthew Patkowski , Onat Ayyildiz , Matja\v{z} Kebri\v{c} , Katharine L. C. Hunt , Dean Lee This is my paper

Pith reviewed 2026-05-18 04:15 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Rodeo Algorithmadiabatic preconditioningeigenstate preparationquantum simulationspin XX modelhierarchical fusionscalable quantum methods
0
0 comments X

The pith

A fusion of adiabatic preconditioning and the Rodeo Algorithm enables scalable eigenstate preparation in large quantum systems.

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

The paper introduces a hierarchical fusion method to prepare eigenstates in large quantum systems. It builds up the system incrementally from exactly solvable smaller subsystems and uses an adiabatic ramp to precondition the state before applying the Rodeo Algorithm. This ensures the Rodeo Algorithm maintains its exponential convergence even when the system size increases. A reader would care because efficient eigenstate preparation is central to quantum simulations of physical systems, and this hybrid overcomes the low-overlap limitation that hinders the basic Rodeo Algorithm on bigger problems.

Core claim

The hybrid approach of adiabatic preconditioning followed by the Rodeo Algorithm, applied hierarchically to subsystems, retains the exponential convergence property of the Rodeo Algorithm across increasing system sizes in the spin-1/2 XX model, offering a decisive computational cost advantage for preparing states with infidelity of 10^{-3} or better compared to using adiabatic ramps or the unmodified Rodeo Algorithm alone.

What carries the argument

The hierarchical fusion method, which incrementally constructs larger systems from smaller solvable ones and applies adiabatic preconditioning to boost overlap for the subsequent Rodeo Algorithm steps.

If this is right

  • The Rodeo Algorithm can be applied to larger quantum systems without losing its exponential convergence property.
  • For target precision of 10^{-3} infidelity or better, the fusion method has lower computational cost than pure adiabatic ramps or the unmodified Rodeo Algorithm.
  • The method works robustly across system sizes in the spin-1/2 XX model.
  • Incremental building from exactly solvable subsystems allows bootstrapping solutions for larger systems.

Where Pith is reading between the lines

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

  • This fusion could be adapted to other models such as the Heisenberg chain by using the same hierarchical preconditioning.
  • It may lower overall circuit depth on quantum hardware by shortening the adiabatic segment while still reaching high accuracy via Rodeo steps.
  • The approach might serve as a better initial-state provider for variational quantum eigensolvers on near-term devices.

Load-bearing premise

The adiabatic ramp on each intermediate subsystem produces a state with high enough overlap to the target eigenstate that the Rodeo Algorithm retains its exponential convergence without needing prohibitive extra resources.

What would settle it

A numerical simulation on a larger XX spin chain in which the infidelity fails to decrease exponentially with additional Rodeo iterations after the adiabatic preconditioning step, or in which the total resource cost exceeds that of a direct adiabatic preparation to the same accuracy.

Figures

Figures reproduced from arXiv: 2510.19039 by Dean Lee, Katharine L. C. Hunt, Matja\v{z} Kebri\v{c}, Matthew Patkowski, Onat Ayyildiz.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Rodeo Algorithm circuit. Ancilla qubits are initialized in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Fusion method results for the XX chain for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: General scheme of the performance of hybrid [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Ground state preparation in the quarter-filled [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Robust and efficient eigenstate preparation is a central challenge in quantum simulation. The Rodeo Algorithm (RA) offers exponential convergence to a target eigenstate but suffers from poor performance when the initial state has low overlap with the desired eigenstate, hindering the applicability of the original algorithm to larger systems. In this work, we introduce a fusion method that preconditions the RA state by an adiabatic ramp to overcome this limitation. By incrementally building up large systems from exactly solvable subsystems and using adiabatic preconditioning to enhance intermediate state overlaps, we ensure that the RA retains its exponential convergence even in large-scale systems. We demonstrate this hybrid approach using numerical simulations of the spin- 1/2 XX model and find that the Rodeo Algorithm exhibits robust exponential convergence across system sizes. We benchmark against using only an adiabatic ramp as well as using the unmodified RA, finding that for state preparation precision at the level of $10^{-3}$ infidelity or better there a decisive computational cost advantage to the fusion method. These results together demonstrate the scalability and effectiveness of the fusion method for practical quantum simulations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper introduces a hierarchical fusion method for scalable quantum eigenstate preparation. It combines adiabatic preconditioning on subsystems built from exactly solvable components with the Rodeo Algorithm (RA) to address the low-overlap limitation of RA. The central claim is that incremental fusion with adiabatic ramps preserves RA's exponential convergence for large systems, demonstrated via numerical simulations on the spin-1/2 XX model showing a decisive computational cost advantage over pure adiabatic evolution or unmodified RA at infidelities of 10^{-3} or better.

Significance. If the performance claims hold with proper scaling analysis, the method could enable practical eigenstate preparation on near-term quantum hardware by leveraging exact solvability of base subsystems. The numerical evidence for the integrable XX model is a strength, as is the focus on reproducible benchmarking against baselines. However, without analytic bounds on overlap scaling or tests beyond modest sizes and integrable cases, the significance for general quantum simulation remains limited.

major comments (2)
  1. [Numerical Simulations] Numerical Simulations section: the reported advantage at 10^{-3} infidelity lacks any specification of system sizes tested, error bars on cost or infidelity metrics, exact benchmarking protocols (e.g., how total resources or overlaps were quantified), or comparison details, making the central scalability claim difficult to verify from the provided information.
  2. [Method and Results] Method and Results sections: the claim that adiabatic preconditioning on fused subsystems produces overlaps high enough for RA to retain exponential convergence without prohibitive resources is load-bearing but unsupported by analytic bounds or scaling arguments. The paper shows this only for the integrable XX model up to modest sizes; no argument is given for how ramp time or overlap behaves with increasing hierarchy depth or for non-integrable Hamiltonians where base subsystems lack exact solvability.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'decisive computational cost advantage' is used without defining the cost metric (e.g., total number of RA shots, gate count, or wall-clock equivalent).
  2. [Introduction] Introduction: additional citations to prior hierarchical or preconditioning techniques in quantum algorithms would help contextualize the novelty of the fusion strategy.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the paper accordingly to improve clarity, verifiability, and scope discussion.

read point-by-point responses

  1. Referee: [Numerical Simulations] Numerical Simulations section: the reported advantage at 10^{-3} infidelity lacks any specification of system sizes tested, error bars on cost or infidelity metrics, exact benchmarking protocols (e.g., how total resources or overlaps were quantified), or comparison details, making the central scalability claim difficult to verify from the provided information.

    Authors: We agree that additional details are needed for verifiability. In the revised manuscript, the Numerical Simulations section now explicitly states the system sizes tested (N=4 to N=16), reports error bars computed from 50 independent runs per data point, describes the benchmarking protocol (total cost quantified as cumulative adiabatic ramp duration plus the number of Rodeo queries needed to reach target infidelity, with overlaps measured via direct state fidelity), and provides side-by-side comparison tables against pure adiabatic evolution and unmodified RA using identical infidelity thresholds and resource metrics. revision: yes

  2. Referee: [Method and Results] Method and Results sections: the claim that adiabatic preconditioning on fused subsystems produces overlaps high enough for RA to retain exponential convergence without prohibitive resources is load-bearing but unsupported by analytic bounds or scaling arguments. The paper shows this only for the integrable XX model up to modest sizes; no argument is given for how ramp time or overlap behaves with increasing hierarchy depth or for non-integrable Hamiltonians where base subsystems lack exact solvability.

    Authors: We acknowledge the absence of analytic bounds. The revised Results section now includes a new paragraph on observed numerical scaling: simulations show that overlap after each adiabatic preconditioning step remains above 0.1 even as hierarchy depth increases to N=16, preserving the exponential convergence rate of RA without ramp times growing faster than linearly in subsystem size. We have added explicit clarification in the Discussion that the method presupposes the availability of exactly solvable base subsystems and therefore applies primarily to integrable models such as the XX chain; for general non-integrable Hamiltonians lacking such subsystems the hierarchical construction does not directly extend, which we now state as a scope limitation rather than a universal claim. revision: partial

standing simulated objections not resolved
  • Deriving analytic bounds on overlap or ramp-time scaling with hierarchy depth for arbitrary non-integrable Hamiltonians

Circularity Check

0 steps flagged

No circularity: method combines existing techniques with new hierarchical fusion, supported by direct numerical demonstration

full rationale

The paper introduces a hierarchical fusion strategy that incrementally builds large systems from solvable subsystems and applies adiabatic preconditioning before each Rodeo Algorithm step. This is presented as a methodological combination rather than a derivation from first principles. The central claim of retained exponential convergence is backed by explicit numerical simulations on the spin-1/2 XX model across system sizes, with benchmarks against pure adiabatic and unmodified RA runs. No equations reduce any claimed advantage to a fitted parameter renamed as prediction, no self-citation chain is load-bearing for the scalability result, and the approach does not rely on ansatz smuggling or renaming of known results. The derivation chain remains self-contained against the reported external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard quantum mechanics and adiabatic theorem assumptions plus the empirical observation that adiabatic ramps improve overlaps in the chosen model; no new entities or fitted parameters are introduced in the abstract description.

axioms (1)
  • domain assumption Adiabatic theorem applies to the intermediate subsystems with controllable ramp speed
    Invoked to justify that the preconditioning step enhances overlap without specifying ramp details or error bounds.

pith-pipeline@v0.9.0 · 5731 in / 1225 out tokens · 41109 ms · 2026-05-18T04:15:15.491779+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 3 internal anchors

  1. [1]

    K. Choi, D. Lee, J. Bonitati, Z. R. Qian, and J. Watkins, Phys. Rev. Lett.127, 040505 (2021)

    work page 2021

  2. [2]

    Dagotto, Rev

    E. Dagotto, Rev. Mod. Phys.66, 763 (1994)

    work page 1994

  3. [3]

    A. W. Sandvik, inAIP Conference Proceedings, Vol. 1297 (American Institute of Physics, 2010) pp. 135–338

    work page 2010

  4. [4]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys.68, 13 (1996)

    work page 1996

  5. [5]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Phys. Rev. Lett.94, 170201 (2005)

    work page 2005

  6. [6]

    Siddiqi, Nature Rev

    I. Siddiqi, Nature Rev. Materials6, 875 (2021)

    work page 2021

  7. [7]

    S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blan- chard, M. Bohn,et al., Phys. Rev. X13, 041052 (2023)

    work page 2023

  8. [8]

    C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, App. Phys. Rev.6, 021314 (2019)

    work page 2019

  9. [9]

    Omran, H

    A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi,et al., Science365, 570 (2019)

    work page 2019

  10. [10]

    Browaeys and T

    A. Browaeys and T. Lahaye, Nature Phys.16, 132 (2020)

    work page 2020

  11. [11]

    Campbell, Nature Rev

    E. Campbell, Nature Rev. Phys.6, 152 (2024)

    work page 2024

  12. [12]

    Google Quantum AI and Collaborators, Nature638, 920 (2024)

    work page 2024

  13. [13]

    Aghaee, A

    Microsoft Azure Quantum, M. Aghaee, A. Al- caraz Ramirez,et al., Nature638, 651 (2025)

    work page 2025

  14. [14]

    A. Y. Kitaev, arXiv preprint quant-ph/9511026 (1995)

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  15. [15]

    A. T. Rezakhani, A. K. Pimachev, and D. A. Lidar, Phys. Rev. A–At., Molec., and Opt. Phys.82, 052305 (2010)

    work page 2010

  16. [16]

    Albash and D

    T. Albash and D. A. Lidar, Rev. Mod. Phys.90, 015002 (2018)

    work page 2018

  17. [17]

    Z. R. Qian, J. Watkins, G. Given, J. Bonitati, K. Choi, and D. Lee, Eur. Phys. J. A60, 151 (2024)

    work page 2024

  18. [18]

    Controlled Gate Networks: Theory and Application to Eigenvalue Estimation

    M. Bee-Lindgren, Z. R. Qian, M. DeCross, N. C. Brown, C. N. Gilbreth, J. Watkins, X. Zhang, and D. Lee, arXiv preprint arXiv:2208.13557 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  19. [19]

    T. D. Cohen and H. Oh, Phys. Rev. A108, 032422 (2023)

    work page 2023

  20. [20]

    J.-S. Chen, E. Nielsen, M. Ebert, V. Inlek, K. Wright, V. Chaplin, A. Maksymov, E. P´ aez, A. Poudel, P. Maunz, et al., Quantum8, 1516 (2024)

    work page 2024

  21. [21]

    Zhang, S

    J. Zhang, S. H. Cant´ u, F. Liu, A. Bylinskii, B. Braver- man, F. Huber, J. Amato-Grill, A. Lukin, N. Gemelke, A. Keesling,et al., Nature Communications16, 712 (2025)

    work page 2025

  22. [22]

    Endres, H

    M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, Science354, 1024 (2016)

    work page 2016

  23. [23]

    E. Lieb, T. Schultz, and D. Mattis, Ann. Phys.16, 407 (1961)

    work page 1961

  24. [24]

    S. R. White, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  25. [25]

    Schollw¨ ock, Ann

    U. Schollw¨ ock, Ann. Phys.326, 96 (2011)

    work page 2011

  26. [26]

    Paeckel, T

    S. Paeckel, T. K¨ ohler, A. Swoboda, S. R. Manmana, U. Schollw¨ ock, and C. Hubig, Ann. Phys.411, 167998 (2019)

    work page 2019

  27. [27]

    Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

    J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes , 005 (2018), code available fromhttps://github.com/ tenpy/tenpy, arXiv:1805.00055

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [28]

    Meden, P

    V. Meden, P. Schmitteckert, and N. Shannon, Phys. Rev. B57, 8878 (1998)

    work page 1998

  29. [29]

    Schollw¨ ock, Matrix product state algorithms: Dmrg, tebd and relatives, inStrongly Correlated Systems: Nu- merical Methods, Springer Series in Solid-State Sciences, Vol

    U. Schollw¨ ock, Matrix product state algorithms: Dmrg, tebd and relatives, inStrongly Correlated Systems: Nu- merical Methods, Springer Series in Solid-State Sciences, Vol. 176, edited by A. Avella and F. Mancini (Springer,

    work page

  30. [30]

    Haegeman, J

    J. Haegeman, J. I. Cirac, T. J. Osborne, I. Piˇ zorn, H. Ver- schelde, and F. Verstraete, Phys. Rev. Lett.107, 070601 (2011)

    work page 2011