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Optimal working point in digitized quantum annealing

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Digitized quantum annealing has an optimal working time proportional to the number of Trotter steps, beyond which the output degrades to a maximally disordered state.

desk verdict Solid numerics identify a practically important optimal time τ ∼ P in digitized QA, but the analytic proof of the large-time plateau has a flawed decorrelation assumption. read the letter →

arxiv 1909.00817 v1 pith:LPALEM7C submitted 2019-09-02 cond-mat.stat-mech cond-mat.dis-nn

Glen Bigan Mbeng , Luca Arceci , Giuseppe E. Santoro This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nn
keywords digitizedquantumannealingTrottererroroptimalworkingpointtransverseIsingchaindefectdensityKibble-Zurekscalingtimediscretization
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

This paper studies digitized quantum annealing, a gate-based version of quantum annealing in which the continuous annealing schedule is replaced by P alternating Trotter rotations. The central claim is that for any fixed number of Trotter steps P, a linear annealing schedule has a sharp optimal total annealing time tau_P^opt that scales linearly with P. Annealing for less time leaves the system relatively far from the target ground state, while annealing for much longer is actively harmful, driving the final state toward a maximally disordered state with defect density 1/2. The paper derives this analytically for the translationally invariant transverse Ising chain and shows numerically that the same optimal-working-point picture survives disorder. The practical message for users of digitized quantum annealing is to choose tau approximately P and not go beyond it.

What carries the argument

The argument is carried by rewriting the digitized evolution on each momentum mode k as a product of 3x3 rotation matrices R_z(4 beta_m) R_{b_k}(4 gamma_m), obtained from the Pauli identity $e^{{-i theta n.tau/2}}$ tau $e^{{i theta n.tau/2}}$ = R_n($\theta$) tau. Each Trotter step is two rotations: one about the z-axis, representing the transverse field, and one about the momentum-dependent axis b_k = (sin k, 0, -cos k), representing the Ising coupling. The large-tau behavior follows from the infinite-time average of this alternating product: for digitized QA the average factorizes over steps, making the relevant projection vanish and giving rho_def^digit = 1/2, whereas for step-QA the symmetric pairs of commensurate frequencies require a nested super-operator contraction and a residue-theorem evaluation.

What would settle it

For a single momentum mode k and small P, compute the exact infinite-time average of the product of alternating rotation matrices in Eq. (A32) and compare it with the factorized product of individual averages used in Eq. (A33); any mismatch invalidates the analytic derivation of the 1/2 plateau. A direct physical check is to run digitized QA on a transverse Ising chain at fixed P (say P = 32) for tau far larger than P, time-average rho_def over many oscillations, and test whether it converges to exactly 1/2.

Watch

Extended reading notes

Core claim

For fixed P, the residual defect density rho_def(tau) of linear-schedule digitized QA has a clear minimum near tau approximately P. For tau smaller than P, the digitized dynamics closely tracks continuous-time annealing and follows the standard finite-time scaling of defect production seen in linear schedules. For tau larger than P, the digital Trotter error dominates and rho_def rises to an irregular plateau whose infinite-time average is exactly 1/2, the value for a maximally disordered state. The paper shows this is a digital error, not a time-discretization error: if the evolution is only time-discretized without Trotter splitting (step-QA), rho_def instead saturates at a P-dependent plateau below 1/2. In the translationally invariant chain the infinite-time average is evaluated analytically: the alternating rotations about the z-axis and the momentum-dependent axis b_k have vanishing average projection, giving rho_def^digit = 1/2 for all P >= 2, while step-QA requires a residue calculation and yields rational plateau values such as 13/72 for P = 4. At the optimal working point, the scaling of the defect density matches the Kibble-Zurek prediction for continuous-time linear QA.

Load-bearing premise

The analytic explanation of the shoot-up assumes that, in the infinite-time average, the many small Trotter rotations effectively forget one another and can be averaged independently, even though the Trotter times s_m = m/P are rationally related and the associated frequencies are commensurate.

Editorial extensions

If this is right

  • For a fixed number of Trotter steps P, the best total annealing time is tau approximately P; increasing tau beyond this optimum is counterproductive and worsens the final state.
  • At the optimal working point, digitized QA with a linear schedule reproduces the same finite-time scaling of defect density as continuous-time linear annealing.
  • Digital Trotter errors always increase the final defect density relative to continuous-time QA, unlike path-integral Monte Carlo simulated annealing, where digital errors can occasionally lower it.
  • The optimal working point persists in disordered transverse Ising chains, so the effect is not an artifact of translational invariance.
  • When increasing P at fixed Trotter time step Delta t = tau/P, the best results occur near Delta t approx 1 (in natural units); smaller steps waste resources, while Delta t approx pi/2 destroys adiabaticity.

Reading between the lines

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

  • A testable extension is to use the sharp minimum in rho_def(tau) as an experimental calibration signal: sweeping tau at fixed P should reveal a clear minimum near tau = P, and the depth of that minimum measures how faithfully the implemented gates realize the intended Trotter rotations.
  • The same decorrelation mechanism suggests that any fixed-depth circuit of alternating non-commuting rotations will have a finite duration beyond which Trotter phase errors dominate; optimized variational schedules such as QAOA may therefore also possess a maximal useful time or depth threshold.
  • Because the infinite-time defect density is claimed to be exactly 1/2 for all P >= 2, the shoot-up is not a finite-size effect; one could probe the size of finite-time fluctuations around the 1/2 plateau and their dependence on P, which the paper does not analyze.
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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

1 major / 4 minor

Summary. The paper studies digitized quantum annealing (dQA) for transverse-field Ising chains. It compares continuous-time QA, step-discretized QA, and fully digitized Trotter QA, and characterizes the residual defect density as a function of total annealing time τ for fixed Trotter-step number P. The central numerical finding is that for each fixed P there is an optimal working point τ_opt ~ P, beyond which the defect density rises toward the maximally disordered value 1/2. This is demonstrated for translationally invariant chains in Figs. 1--3 and for disordered chains in Fig. 4. The paper also derives an exact residue formula for the large-time plateau of step-QA, verified by numerics, and proposes an analytic argument that the large-time dQA average is exactly 1/2 in Appendix A.2.

Significance. If the central claim holds, the result is practically important: users of digitized quantum annealing should set the total annealing time proportional to the number of Trotter steps, and longer annealing times degrade, rather than improve, the output. The step-QA plateau formula is a valuable exact result and is convincingly checked against simulations. The numerical evidence for the optimal working point is clear and parameter-free. However, the analytic derivation of the asymptotic dQA value 1/2 rests on an unproven decorrelation assumption in Appendix A.2, so the theoretical explanation of the large-τ 'shoot-up' is not yet established with the same rigor as the numerical observation.

major comments (1)
  1. [Appendix A.2, Eq. (A33)] The passage from Eq. (A32) to Eq. (A33) is not justified. The text states that the frequencies appearing in the rotation matrices are 'all different', but with Γ=J=1 the z-rotation at step m has frequency 4(1 - s_m + s_{m+1}/2) and the b-rotation at step m has frequency 4 s_m. For P=4, the m=3 z-rotation and the m=3 b-rotation both have frequency 3, while the m=1 z-rotation and the m=4 b-rotation both have frequency 4. More generally, all these frequencies are integer multiples of 2/P, so the integrand is periodic and the frequencies are commensurate. The factorization of the infinite-time average of a product into a product of infinite-time averages is therefore an additional assumption, not a consequence of distinct frequencies. The step-QA analysis in Appendix A.1 explicitly keeps the correlations of degenerate pairs (m and P−m), so dropping such correlations in the dQA case is inconsistent without a separate argument. Consequently Eq. (25) and the claim ρ_digit_def = 1/2 are not proven. The numerical evidence for the optimal working point is independent of this analytic step, but the analytic explanation of the large-τ shoot-up is not.
minor comments (4)
  1. [Introduction] There are typographical errors, e.g., 'adiabatic quantum computating' and the rendering of 'Schrödinger'; these should be corrected.
  2. [Sec. III, Fig. 2] The statement that the best results are obtained for Δt ≈ 1 would be clearer with error bars or a more explicit definition of 'best', since the bottom panel shows a sharp degradation near Δt/π ≈ 0.5.
  3. [Sec. V, Fig. 4] For the disorder-averaged results, adding error bars or a statement about the realization-to-realization spread would strengthen the claim that the optimal working point survives disorder.
  4. [Appendix A.2] The rescaling τ/P → τ used before Eq. (A32) is not spelled out; please state the relation between the physical annealing time and the rescaled variable explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the dQA optimal working point is established by direct numerics, and the Appendix decorrelation step is a rigor gap rather than a circular reduction.

full rationale

The paper's central result, an optimal working point tau_P^opt ~ P for linear-schedule dQA with the residual error growing for larger tau, is established by direct numerical integration of the Schrödinger dynamics for the transverse Ising chain (Figs. 1, 3, and 4), not by fitting a parameter to the target claim. The analytical treatment in Appendix A computes infinite-time averages from the specified Trotter unitaries; even if the decorrelation step between Eqs. (A32) and (A33) is only asserted and is open to the commensurability objection raised in the skeptic summary, that is a mathematical rigor gap rather than circular reasoning: the claimed rho_def^digit = 1/2 is not fed into the derivation as an input. The self-citations, primarily Ref. 23 for the thermodynamic-limit criterion N >= 2P + 2 and Ref. 30 in the PIMC comparison, are technical or contextual and are not used to inject the optimal-point conclusion; the numerics and the separate step-QA residue calculation stand independently. No equation in the paper reduces to an earlier definition by construction, and no fitted quantity is renamed a prediction. Hence there is no significant circularity.

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

The results introduce no fitted free parameters and no new entities. The analytical machinery rests on standard integrability of the transverse-field Ising chain and on a domain assumption about the thermodynamic limit. A third, non-trivial assumption, that digitized-QA rotation matrices decorrelate under infinite-time averaging, is stated but not rigorously proven; it supports the rho=1/2 asymptote.

assumptions (3)
  • standard math Integrability of the transverse-field Ising chain via Jordan-Wigner transformation and Fourier decomposition into independent two-level systems.
    Used throughout Sec. III and Appendix A to reduce the many-body problem to independent k-modes (Eqs. (15)-(18) and surrounding text).
  • domain assumption The thermodynamic limit with a continuum of k-modes is valid; the criterion N >= 2P+2 from ref 23 is adopted to justify finite-N simulations.
    The analytical formulas for defect density use a continuum k-integral (Eq. (21) and Appendix A), and the paper relies on the authors' earlier result for the effective thermodynamic limit.
  • ad hoc to paper The rotation matrices in digitized QA are uncorrelated in the infinite-time average, so the time average of the product equals the product of time averages.
    Assumed between Eq. (A32) and Eq. (A33) in Appendix A.2 and not rigorously proven; since the Trotter times are rational multiples of a common fundamental, the factorization is non-trivial.

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Pith. "Pith review of Optimal working point in digitized quantum annealing." pith.science (2026). https://pith.science/paper/LPALEM7C

@misc{pith2026190900817,
  author       = {Pith},
  title        = {Pith review of: Optimal working point in digitized quantum annealing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPALEM7C}},
  note         = {Machine review of arXiv:1909.00817}
}
abstract

We present a study of the digitized Quantum Annealing protocol proposed by R. Barends et al., Nature 534, 222 (2016). Our analysis, performed on the benchmark case of a transverse Ising chain problem, shows that the algorithm has a well defined optimal working point for the annealing time $\tau^{\mathrm{opt}}_\mathrm{P}$ --- scaling as $\tau^{\mathrm{opt}}_\mathrm{P}\sim \mathrm{P}$, where $\mathrm{P}$ is the number of digital Trotter steps --- beyond which, the residual energy error shoots-up towards the value characteristic of the maximally disordered state. We present an analytical analysis for the translationally invariant transverse Ising chain case, but our numerical evidence suggests that this scenario is more general, surviving, for instance, the presence of disorder.

Figures

Figures reproduced from arXiv: 1909.00817 by the authors.

Figure 1
Figure 1. FIG. 1: Density of defects [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. shows the numerical result we obtained. Ob￾serve that the best results are obtained when ∆t ≈ 1 (in units of ~/J), in a way that is totally consistent with the optimal working point shown in Fig. (1), and with the Kibble-Zurek scaling exponent34,35. For ∆t 1 the Trotter error is negligibly small but we are wasting re￾sources. For ∆t ∼ 1 the Trotter error is not small, but the digitized-QA dynamics is very effective … view at source ↗
Figure 3
Figure 3. FIG. 3: Density of defects [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Density of defects [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

40 extracted references · 37 canonical work pages

  1. [1]

    author A. B. Finnila , author M. A. Gomez , author C. Sebenik , author C. Stenson , and author J. D. Doll , journal Chem. Phys. Lett. volume 219 , pages 343 ( year 1994 )

    work page 1994

  2. [2]

    Kadowaki and author H

    author T. Kadowaki and author H. Nishimori , journal Phys. Rev. E volume 58 , pages 5355 ( year 1998 )

    work page 1998

  3. [3]

    Brooke , author D

    author J. Brooke , author D. Bitko , author T. F. Rosenbaum , and author G. Aeppli , journal Science volume 284 , pages 779 ( year 1999 )

    work page 1999

  4. [4]

    author G. E. Santoro , author R. Marto n \' a k , author E. Tosatti , and author R. Car , journal Science volume 295 , pages 2427 ( year 2002 )

    work page 2002

  5. [5]

    author G. E. Santoro and author E. Tosatti , journal J. Phys. A: Math. Gen. volume 39 , pages R393 ( year 2006 )

    work page 2006

  6. [6]

    Farhi , author J

    author E. Farhi , author J. Goldstone , author S. Gutmann , author J. Lapan , author A. Lundgren , and author D. Preda , journal Science volume 292 , pages 472 ( year 2001 )

    work page 2001

  7. [7]

    Albash and author D

    author T. Albash and author D. A. Lidar , journal Rev. Mod. Phys. volume 90 , pages 015002 ( year 2018 )

    work page 2018

  8. [8]

    Preskill , journal Quantum volume 2 , pages 79 ( year 2018 ), ISSN issn 2521-327X

    author J. Preskill , journal Quantum volume 2 , pages 79 ( year 2018 ), ISSN issn 2521-327X

    work page 2018

Show all 40 references
  1. [9]

    Harris et al

    author R. Harris et al. , journal Phys. Rev. B volume 82 , pages 024511 ( year 2010 )

    2010

  2. [10]

    author M. W. Johnson et al. , journal Nature volume 473 , pages 194 ( year 2011 )

    2011

  3. [11]

    author V. S. Denchev , author S. Boixo , author S. V. Isaksov , author N. Ding , author R. Babbush , author V. Smelyanskiy , and author J. Martinis , journal Phys. Rev. X volume 6 , pages 031015 ( year 2016 )

    2016

  4. [12]

    Lanting et al

    author T. Lanting et al. , journal Phys. Rev. X volume 4 , pages 021041 ( year 2014 )

    2014

  5. [13]

    Boixo , author T

    author S. Boixo , author T. Albash , author F. M. Spedalieri , author N. Chancellor , and author D. A. Lidar , journal Nat. Commun. volume 4 , pages 2067 ( year 2013 )

    2013

  6. [14]

    author N. G. Dickson et al. , journal Nat. Commun. volume 4 , pages 1903 ( year 2013 )

    1903

  7. [15]

    Boixo , author T

    author S. Boixo , author T. F. R nnow , author S. V. Isaksov , author Z. Wang , author D. Wecker , author D. A. Lidar , author J. M. Martinis , and author M. Troyer , journal Nat. Phys. volume 10 , pages 218 ( year 2014 )

    2014

  8. [16]

    Messiah , title Quantum mechanics , vol

    author A. Messiah , title Quantum mechanics , vol. volume 2 ( publisher North-Holland , address Amsterdam , year 1962 )

    1962

  9. [17]

    Barends , author A

    author R. Barends , author A. Shabani , author L. Lamata , author J. Kelly , author A. Mezzacapo , author U. L. Heras , author R. Babbush , author A. G. Fowler , author B. Campbell , author Y. Chen , et al. , journal Nature volume 534 , pages 222 ( year 2016 )

    2016

  10. [18]

    Bravyi , author D

    author S. Bravyi , author D. P. DiVincenzo , author R. I. Oliveira , and author B. M. Terhal , journal Quantum Inf. Comput. volume 8 , pages 361 ( year 2008 )

    2008

  11. [19]

    Troyer and author U.-J

    author M. Troyer and author U.-J. Wiese , journal Phys. Rev. Lett. volume 94 , pages 170201 ( year 2005 )

    2005

  12. [20]

    author S. B. Bravyi and author A. Y. Kitaev , journal arXiv e-prints eid quant-ph/9811052 ( year 1998 ), quant-ph/9811052

    1998 arXiv

  13. [21]

    author A. G. Fowler , author M. Mariantoni , author J. M. Martinis , and author A. N. Cleland , journal Phys. Rev. A volume 86 , pages 032324 ( year 2012 )

    2012

  14. [22]

    Farhi , author J

    author E. Farhi , author J. Goldstone , and author S. Gutmann , journal arXiv e-prints eid arXiv:1411.4028 ( year 2014 ), 1411.4028

    2014 arXiv

  15. [23]

    author G. B. Mbeng , author R. Fazio , and author G. E. Santoro , journal arXiv e-prints eid arXiv:1906.08948 ( year 2019 a ), 1906.08948

    1906 arXiv

  16. [24]

    Ambainis , journal arXiv e-prints eid arXiv:1312.4758 ( year 2013 ), 1312.4758

    author A. Ambainis , journal arXiv e-prints eid arXiv:1312.4758 ( year 2013 ), 1312.4758

    2013 arXiv

  17. [25]

    author T. S. Cubitt , author D. Perez-Garcia , and author M. M. Wolf , journal Nature volume 528 , pages 207 ( year 2015 )

    2015

  18. [26]

    D'Alessandro , title Introduction to quantum control and dynamics ( publisher Chapman and Hall/CRC , year 2007 )

    author D. D'Alessandro , title Introduction to quantum control and dynamics ( publisher Chapman and Hall/CRC , year 2007 )

    2007

  19. [27]

    Brif , author R

    author C. Brif , author R. Chakrabarti , and author H. Rabitz , journal New Journal of Physics volume 12 , pages 075008 ( year 2010 )

    2010

  20. [28]

    Yang , author A

    author Z.-C. Yang , author A. Rahmani , author A. Shabani , author H. Neven , and author C. Chamon , journal Phys. Rev. X volume 7 , pages 021027 ( year 2017 )

    2017

  21. [29]

    Heim , author T

    author B. Heim , author T. F. R nnow , author S. V. Isakov , and author M. Troyer , journal Science volume 348 , pages 215 ( year 2015 )

    2015

  22. [30]

    author G. B. Mbeng , author L. Privitera , author L. Arceci , and author G. E. Santoro , journal Phys. Rev. B volume 99 , pages 064201 ( year 2019 b )

    2019

  23. [31]

    author T. W. B. Kibble , journal J.Phys. A: Math. and Gen. volume 9 , pages 1387 ( year 1976 )

    1976

  24. [32]

    author W. H. Zurek , journal Nature volume 317 , pages 505 ( year 1985 )

    1985

  25. [33]

    Polkovnikov , author K

    author A. Polkovnikov , author K. Sengupta , author A. Silva , and author M. Vengalattore , journal Rev. Mod. Phys. volume 83 , pages 863 ( year 2011 )

    2011

  26. [34]

    author W. H. Zurek , author U. Dorner , and author P. Zoller , journal Phys. Rev. Lett. volume 95 , pages 105701 ( year 2005 )

    2005

  27. [35]

    Dziarmaga , journal Phys

    author J. Dziarmaga , journal Phys. Rev. Lett. volume 95 , pages 245701 ( year 2005 )

    2005

  28. [36]

    Lieb , author T

    author E. Lieb , author T. Schultz , and author D. Mattis , journal Ann. Phys. volume 16 , pages 407 ( year 1961 )

    1961

  29. [37]

    author A. P. Young , journal Phys. Rev. B volume 56 , pages 11691 ( year 1997 )

    1997

  30. [38]

    Caneva , author R

    author T. Caneva , author R. Fazio , and author G. E. Santoro , journal Phys. Rev. B volume 76 , pages 144427 ( year 2007 )

    2007

  31. [39]

    Jordan and author E

    author P. Jordan and author E. Wigner , journal Zeitschrift fur Physik volume 47 , pages 631 ( year 1928 )

    1928

  32. [40]

    Dranov , author J

    author A. Dranov , author J. Kellendonk , and author R. Seiler , journal Journal of Mathematical Physics volume 39 , pages 1340 ( year 1998 )

    1998

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