Pith. sign in

REVIEW 3 major objections 4 minor 29 references

Adaptive Observation Cost Control for Variational Quantum Eigensolvers

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read This paper claims that adaptive shot-count control—keeping the optimized subspace inside a Gaussian-process confident region—can substantially reduce the total quantum measurement budget of a variational quantum eigensolver without…

desk verdict A clean uniformity theorem and a sensible adaptive shot-count heuristic for SMO-based VQE, but the 'minimum shots' claim is not actually proven and the paper overstates its guarantee. read the letter →

arxiv 2502.01704 v1 pith:4G3CHPMU submitted 2025-02-03 quant-ph cs.LG

Christopher J. Anders , Kim A. Nicoli , Bingting Wu , Naima Elosegui , Samuele Pedrielli , Lena Funcke , Karl Jansen , Stefan Kühn
show 1 more author
Shinichi Nakajima
This is my paper · ORCID
classification quant-phcs.LG
keywords variationalquantumeigensolversequentialminimaloptimizationGaussianprocessconfidentregionadaptiveshotcontrolnoisemeasurementcostVQEkernel
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 claims that the quantum measurement cost of running a variational quantum eigensolver (VQE) can be substantially reduced by adapting, in every sequential-minimal-optimization step, the number of measurement shots to the current need for accuracy. Its method, SubsCoRe, keeps the whole one-dimensional subspace being optimized inside the 'confident region' of a Gaussian-process surrogate, where the surrogate's uncertainty stays below a threshold $κ_t^{2}$. The threshold is tuned to the optimization progress: coarse early, fine near convergence. If the claim holds, users of VQE-type algorithms can spend far fewer total shots—the dominant cost on quantum hardware—without losing energy accuracy or state fidelity.

What carries the argument

The confident region (CoRe), Equation (11), is the set of inputs where GP posterior variance is below a threshold $κ^{2}$. Theorem 3.1 shows that with the VQE kernel and 1+2V_d equidistant measurements along one axis, the posterior variance is a constant (Eq. 14) independent of the test point along that axis. That constant, together with Corollary 3.2 (posterior variance < observation noise $σ^{2}$), turns the hard shot-allocation problem (Eq. 13) into a simple choice of per-point noise variances σ˘, solved by the SubsCoRe-Center subroutine (Algorithm 2). The threshold κ_t is updated by a linear regression on the recent best values (Eq. 16), tying the required accuracy to the measured progress.

What would settle it

Run SubsCoRe on a simulator while tracking the empirical GP posterior variance at points in the subspace: if, with the shots chosen from a deliberately misestimated $η^{2}$ (say, half the true value), the posterior variance exceeds $κ_t^{2}$ on any tested point, the claimed guarantee fails. The same check with the true $η^{2}$ should keep the subspace inside the CoRe.

Watch

Extended reading notes

Core claim

The central discovery is that, for VQE objective functions, observing 1+2V_d equidistant points along the axis being optimized makes the GP posterior variance exactly uniform along the subspace and independent of the test position α'. That uniformity is the key to the cost allocation: to guarantee the entire subspace lies inside the CoRe at accuracy $κ_t^{2}$, it suffices to set the per-point observation noise to a value tied to $κ_t^{2}$ and distribute shots among the 1+2V_d points, because uniform uncertainty is min-max optimal. The paper's SubsCoRe-Center variant exploits that the previous best point is already inside the CoRe from earlier iterations, thus spending fewer shots there and the minimum total number of shots overall while satisfying Equation (12), the 'subspace in CoRe' constraint. The consequence is an SMO loop whose quantum cost adapts to progress, and which the paper reports to outperform fixed-shot NFT, EMICoRe, and the adaptive-shot SGLBO on the Ising benchmark.

Load-bearing premise

The whole shot-allocation rule assumes the single-shot measurement variance $η^{2}$ is known exactly; if it is misestimated, the computed shot counts will not actually place the subspace inside the confident region and the accuracy guarantee is void.

Editorial extensions

If this is right

  • If SubsCoRe is right, fixed-shot SMO methods waste a large fraction of their quantum budget: shots spent early, where accuracy is not yet needed, can be moved to convergence without degrading the final energy.
  • The same CoRe-criterion applies to any VQE ansatz built from parameterized rotations, since the trigonometric-polynomial form (Eq. 9) holds for any such circuit.
  • SubsCoRe-Center combined with EMICoRe's point-selection would, by the paper's own account, be possible and likely further reduce total shots (the authors list it as future work).
  • The per-step shot count becomes a direct function of the required accuracy κ_t^2 times the known single-shot variance, which gives practitioners a principled way to set measurement budgets rather than tuning fixed shot counts.

Reading between the lines

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

  • If the single-shot variance η^2 is state-dependent and only approximately known, the guarantee in Eq. (12) degrades gracefully: the actual uncertainty scales with η^2/σ^2, so misestimation by a factor of four changes the required shots by the same factor. A robust variant would estimate η^2 online from repeated measurements of the same point.
  • The min-max argument suggests the equidistant point choice α=2π/(1+2V_d) is the right default for any cost-controlled SMO, not just for the Ising Hamiltonian, and could be tested on molecular Hamiltonians where V_d > 1.
  • A testable extension: monitor the GP's leave-one-out predictive error against the claimed CoRe guarantee; if the observed error regularly exceeds κ, the assumed η^2 is too small, and the algorithm should raise its shot allocation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes SubsCoRe, an adaptive shot-control method for sequential minimal optimization (SMO) in variational quantum eigensolvers. In each SMO step, SubsCoRe measures the fixed set of 1+2V_d equidistant points along the updated axis and chooses the per-point number of quantum measurement shots so that the GP posterior variance along the entire 1D subspace stays below an adaptively updated threshold κ_t^2. The theoretical core is Theorem 3.1, which gives a closed-form expression for the posterior variance after observing 1+2V_d equidistant points with equal variance and no prior training points, showing that this variance is uniform along the subspace. The paper then proposes SubsCoRe-Bound and SubsCoRe-Center variants that use this theory to set observation noise levels, and it reports experiments on a 5-qubit Ising Hamiltonian comparing SubsCoRe-Center with NFT, EMICoRe, and SGLBO over 100 seeded trials, with statistically significant improvements in energy and fidelity at a fixed cumulative shot budget.

Significance. If the claims are correct, the paper makes a useful practical contribution: an adaptive shot-allocation rule for SMO that reduces total quantum measurement cost while maintaining a controlled GP uncertainty level. The strengths are concrete: the analytic derivation in Appendix C is detailed, the experimental comparison uses 100 independent trials and an external significance test, the code is publicly linked, and the comparison includes three state-of-the-art baselines. However, the headline claim of using the minimum total number of shots with an optimal distribution is not established by the supplied proof, and the actual Algorithm 2 is a heuristic feasibility search. The significance is therefore conditional on revising these claims or supplying the missing optimality argument.

major comments (3)
  1. [Section 3, Theorem 3.1 and Eq. (14)] The theorem proves that, for a fixed set of 1+2V_d equidistant points with equal observation variance and no previous training data, the GP posterior variance is uniform along the subspace. Uniformity alone does not imply min-max optimality: no comparison is made to other placements, unequal shot splits, or configurations with previous data. The footnote in Section 3 asserts that the equidistant design is 'min-max optimal', but Appendix C contains no extremal argument. Either provide a proof of this optimality or remove/soften the optimality claim.
  2. [Section 3, Eq. (13) and Algorithm 2] The actual method does not solve Eq. (13): Algorithm 2 uses previous training data X_t (lines 6 and 21) and performs a grid search over observation variances, so Theorem 3.1 and Corollary 3.2, which assume no prior points and equal variances, do not apply to the posterior that determines feasibility. The resulting shot count is therefore a feasible solution to the relaxed problem (15), not a proven minimum. The paper's central assertion that SubsCoRe 'uses the minimum total number of measurement shots and distributes them optimally' is unsupported by the supplied theory and should be revised, or supported by an additional optimality argument or a numerical optimality-gap study.
  3. [Section 3, Eq. (12) and abstract] The phrase 'guaranteed accuracy' overstates what the CoRe condition provides: the constraint S_d(x) ⊆ Z(κ_t^2) controls the posterior variance, not the error of the resulting SMO minimum. The energy error also depends on the GP mean bias, kernel hyperparameters, the threshold schedule, and the fact that the threshold κ_t is updated heuristically via Eq. (16). In addition, Algorithm 2 assumes the single-shot variance η^2 is known exactly; if η^2 is misestimated, the computed shot counts do not realize the intended observation noise, and the posterior variance condition is not actually enforced. These caveats should be stated explicitly where the accuracy guarantee is claimed.
minor comments (4)
  1. [Appendix D.2, Algorithm 1, line 10] The quantum-circuit call uses 'shots= ¯N0' rather than the newly computed per-point shot counts from Algorithm 2; this appears to be a typo, since lines 8-12 of Algorithm 2 return the vector ¯N^{t+1} with potentially different values for the center and the shifted points.
  2. [Section 4.2, Figure 3] The statistical test is reported only as 'p-value < 0.05 according to the Wilcoxon signed-rank test'; please specify the exact test setup (comparison at the final budget or at each cumulative budget, one-sided or two-sided, and what is being compared, final ΔEnergy/ΔFidelity or trajectories).
  3. [Section 3, text after Eq. (15)] SubsCoRe-Center is described as solving the problem by a 2D search over ˘σ_0^2 and ˘σ^2, but Algorithm 2 performs two sequential 1D grid searches: first for the shifted points, then for the center. Please clarify whether this is meant to be an approximate 2D search or a specific decoupled heuristic.
  4. The abstract says SubsCoRe 'outperforms the state-of-the-art methods' without qualification; the experiments cover a single Ising benchmark, one circuit family, and one qubit/layer setting. Please qualify the empirical claim accordingly, e.g., 'on the tested Ising benchmark'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SubsCoRe's shot allocation enforces its CoRe constraint by construction and the central posterior-variance theorem is proven in-appendix; empirical claims rest on external benchmarks.

full rationale

The paper's key theoretical result, Theorem 3.1, is not an input to the method: it is derived in Appendix C from the VQE Fourier-basis property, the VQE kernel definition, and matrix-inversion algebra, with no use of the target claim. The adaptive shot-control loop (Algorithm 2) works by searching for the smallest observation variance whose posterior violates none of the CoRe constraints, then converting variance to shots via sigma^2 = eta^2/Nshots; this is a feasibility construction, not a fitted parameter renamed as prediction, and the paper's headline comparisons are measured against NFT, SGLBO, and EMICoRe under a common cumulative-shot budget. The self-citation to Nicoli et al. (2023a) supplies the VQE kernel and CoRe notion, but the present paper proves the relevant uniform-variance statement itself and does not rely on an unverified uniqueness theorem. The over-strong phrasing 'min-max optimal' in the Section 3 footnote and the heuristic solution of relaxed problem (15) are completeness/correctness concerns (global minimality is not established), but they are not reductions of the claim to its own inputs. Therefore no circular step is exhibited.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method rests on standard GP/VQE mathematics plus two domain assumptions: known noise variance and no hardware noise. The min-max optimality of equidistant points is an unproven assertion, and several algorithm hyperparameters (C0, C1, TAve, kappa_0, gamma, sigma0) are hand-chosen or fitted, which affects the strength of the 'guaranteed accuracy' claim.

free parameters (6)
  • eta^2 (single-shot measurement variance)
    Algorithm 2 sets shot counts as N_i = eta^2 / sigma_i^2; if eta^2 is misestimated, the chosen shots do not achieve the target noise level. In the simulations it is computed exactly from the state and Hamiltonian; on hardware it would need estimation.
  • C0 (coremin_scale) = 1024 shots per point
    Lower bound on kappa_t via max(C0, ...); equivalently caps shots at 1024 per data point. Hand-chosen; controls late-stage shot budget.
  • C1 (corethresh_scale) = 1.0
    Linear scaling between CoRe threshold and estimated slope in Eq. (16).
  • TAve (corethresh_width) = 40
    Number of recent best values used to estimate the slope for the kappa update.
  • kappa_0 (initial CoRe threshold) = 512 initial Nshots
    Initial threshold; sets the initial observation noise via sigma^2 = eta^2 / 512.
  • kernel hyperparameters gamma and sigma0 = gamma grid-searched via LOO-CV in [sqrt(2), 20]; sigma0 from approximate ground-state energy
    They define the GP prior and therefore the posterior variance used in the CoRe condition.
assumptions (5)
  • standard math The VQE objective along each 1D subspace is exactly a Vd-th order trigonometric polynomial (Eq. 9).
    Proven in Nakanishi et al. 2020, cited; underlies the SMO/NFT update and the exactness of the GP-VQE kernel.
  • domain assumption Observation noise is i.i.d. Gaussian with known variance sigma*^2(Nshots) = sigma*^2 / Nshots.
    Used in Eqs. (1) and in mapping desired variance to shot counts (Alg. 2); approximate via CLT only in the large-shot limit.
  • domain assumption Hardware noise is negligible.
    Explicitly assumed in the footnote on p.2 and Appendix A. On real devices, hardware noise breaks the model.
  • domain assumption The GP with the VQE kernel and fitted hyperparameters is a faithful probabilistic model of the objective.
    Needed for the CoRe posterior variance to be a meaningful accuracy guarantee; gamma and sigma0 are fit rather than known.
  • ad hoc to paper Equidistant observations are min-max optimal for the worst-case posterior variance over the subspace.
    Stated in Section 3 without proof; used to justify fixing the observation points. Theorem 3.1 establishes uniformity, not optimality.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Adaptive Observation Cost Control for Variational Quantum Eigensolvers." pith.science (2026). https://pith.science/paper/4G3CHPMU

@misc{pith2026250201704,
  author       = {Pith},
  title        = {Pith review of: Adaptive Observation Cost Control for Variational Quantum Eigensolvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4G3CHPMU}},
  note         = {Machine review of arXiv:2502.01704}
}
read the original abstract

The objective to be minimized in the variational quantum eigensolver (VQE) has a restricted form, which allows a specialized sequential minimal optimization (SMO) that requires only a few observations in each iteration. However, the SMO iteration is still costly due to the observation noise -- one observation at a point typically requires averaging over hundreds to thousands of repeated quantum measurement shots for achieving a reasonable noise level. In this paper, we propose an adaptive cost control method, named subspace in confident region (SubsCoRe), for SMO. SubsCoRe uses the Gaussian process (GP) surrogate, and requires it to have low uncertainty over the subspace being updated, so that optimization in each iteration is performed with guaranteed accuracy. The adaptive cost control is performed by first setting the required accuracy according to the progress of the optimization, and then choosing the minimum number of measurement shots and their distribution such that the required accuracy is satisfied. We demonstrate that SubsCoRe significantly improves the efficiency of SMO, and outperforms the state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2502.01704 by the authors.

Figure 1
Figure 1. Illustration of SubsCoRe. (a) After iteration t − 1, the function value at the best point xb t−1 , which is in the CoRe (yellow rectangle) due to the previous observations, is predicted by the GP with sufficiently low uncertainty (red curves). Two equidistant shifts (blue points) at xb t−1 ± 2π 3 edt are chosen, yet not mea￾sured, along the line parallel to the dt axis at the current step. (b) SubsCoRe finds the min… view at source ↗
Figure 2
Figure 2. Uncertainty of GP with the VQE kernel trained on equidistant (darker green, round markers) and non-equidistant (olive, triangle markers) observation points. The left and right plots show the Vd = 1 and Vd = 3 cases, respectively. The green and olive solid lines refer to the GP posterior uncertainty obtained after observing 1 + 2Vd points with an equidistant and non-equidistant spacing. For instance, in the Vd = 1 ca… view at source ↗
Figure 3
Figure 3. Energy (left) and fidelity (right), as the difference from the ground-truth (see Eqs. (18) and (19)), achieved by our SubsCoRe￾Center and the baselines, NFT, SGLBO, and EMICoRe for the Ising model with (Q, L) = (5, 3). Both plots are on a logarithmic scale, and the horizontal axis indicates the cumulative number of shots (per operator group) as the total quantum computation cost. The fidelity by SGLBO is not shown s… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Prior (top) and posterior (middle) GP with SubsCoRe￾Center (left) and SubsCoRe-Bound (right) in an SMO step. The red solid lines are the true function f ∗ (x), and the blue solid lines and shadows are the GP mean and uncertainty, respectively. The verti￾cal lines repre…
Figure 5
Figure 5. Figure 5: Energy (left) and fidelity (right), as the difference from the ground-truth (see Eqs. (18) and (19)), achieved by SubsCoRe-Center and SubsCoRe-Bound. The inferior efficiency of SubsCoRe-Bound is due to the fact that it relies on a looser bound. taking the average over …
Figure 6
Figure 6. Figure 6: Number of measurement shots (Readout) taken by SubsCoRe-Center for each SMO step. The x-axis indicates the SMO step while the y-axis represents the total number of measurement shots taken at every step (i.e., distributed over the three observed points per step). 22 [P…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 19 canonical work pages

  1. [1]

    Abraham, H. et al. Qiskit: An Open-source Framework for Quantum Computing . Zenodo, 2019. doi:10.5281/zenodo.2562111

    doi:10.5281/zenodo.2562111 2019

  2. [2]

    Bauer, B., Bravyi, S., Motta, M., and Chan, G. K.-L. Quantum algorithms for quantum chemistry and quantum materials science. Chemical Reviews, 120 0 (22): 0 12685--12717, 2020

    work page 2020

  3. [3]

    J., Geim, A

    Bluvstein, D., Evered, S. J., Geim, A. A., Li, S. H., Zhou, H., Manovitz, T., Ebadi, S., Cain, M., Kalinowski, M., Hangleiter, D., et al. Logical quantum processor based on reconfigurable atom arrays. Nature, pp.\ 1--3, 2023

    work page 2023

  4. [4]

    Potential of quantum computing for drug discovery

    Cao, Y., Romero, J., and Aspuru-Guzik, A. Potential of quantum computing for drug discovery. IBM Journal of Research and Development, 62 0 (6): 0 6--1, 2018

    work page 2018

  5. [5]

    M., Figgatt, C., Landsman, K

    Debnath, S., Linke, N. M., Figgatt, C., Landsman, K. A., Wright, K., and Monroe, C. Demonstration of a small programmable quantum computer with atomic qubits. Nature, 536 0 (7614): 0 63--66, 2016. doi:10.1038/nature18648. URL https://doi.org/10.1038/nature18648

    doi:10.1038/nature18648 2016

  6. [6]

    Di Meglio, A. et al. Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group , 7 2023

    work page 2023

  7. [7]

    and Gelfand, M

    Fedorov, A. and Gelfand, M. Towards practical applications in quantum computational biology. Nature Computational Science, 1 0 (2): 0 114--119, 2021

    work page 2021

  8. [8]

    Review on Quantum Computing for Lattice Field Theory

    Funcke, L., Hartung, T., Jansen, K., and K\"uhn, S. Review on Quantum Computing for Lattice Field Theory . PoS, LATTICE2022: 0 228, 2023. doi:10.22323/1.430.0228

    doi:10.22323/1.430.0228 2023

Show all 29 references
  1. [9]

    R., Economou, S

    Grimsley, H. R., Economou, S. E., Barnes, E., and Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature communications, 10 0 (1): 0 3007, 2019

    2019

  2. [10]

    S., Delgado, A., Pooser, R., Seck, C., Bennink, R., Leyton-Ortega, V., Wang, C

    Humble, T. S., Delgado, A., Pooser, R., Seck, C., Bennink, R., Leyton-Ortega, V., Wang, C. C. J., Dumitrescu, E., Morris, T., Hamilton, K., Lyakh, D., Date, P., Wang, Y., Peters, N. A., Evans, K. J., Demarteau, M., McCaskey, A., Nguyen, T., Clark, S., Reville, M., Di Meglio, A...

    2022

  3. [11]

    and Jansen, K

    Iannelli, G. and Jansen, K. Noisy bayesian optimization for variational quantum eigensolvers. ArXiv e-prints, 2021

    2021

  4. [12]

    M., and Gambetta, J

    Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., and Gambetta, J. M. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature, 549 0 (7671): 0 242--246, 2017

    2017

  5. [13]

    Kielpinski, D., Monroe, C., and Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature, 417 0 (6890): 0 709--711, 2002. doi:10.1038/nature00784. URL https://doi.org/10.1038/nature00784

    2002 doi

  6. [14]

    Risk-averse heteroscedastic bayesian optimization

    Makarova, A., Usmanova, I., Bogunovic, I., and Krause, A. Risk-averse heteroscedastic bayesian optimization. In Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Advances in Neural Information Processing Systems, volume 34, pp.\ 17235--17245. Cur...

    2021

  7. [15]

    R., Romero, J., Babbush, R., and Aspuru-Guzik, A

    McClean, J. R., Romero, J., Babbush, R., and Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 0 (2): 0 023023, 2016

    2016

  8. [16]

    Mickaël Binois, Jiangeng Huang, R. B. G. and Ludkovski, M. Replication or exploration? sequential design for stochastic simulation experiments. Technometrics, 61 0 (1): 0 7--23, 2019. doi:10.1080/00401706.2018.1469433. URL https://doi.org/10.1080/00401706.2018.1469433

    2019

  9. [17]

    Quantum circuit learning

    Mitarai, K., Negoro, M., Kitagawa, M., and Fujii, K. Quantum circuit learning. Physical Review A, 98 0 (032309), 2018

    2018

  10. [18]

    M., Fujii, K., and Todo, S

    Nakanishi, K. M., Fujii, K., and Todo, S. Sequential minimal optimization for quantum-classical hybrid algorithms. Phys. Rev. Res., 2: 0 043158, 2020. doi:10.1103/PhysRevResearch.2.043158. URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.043158

    2020 doi

  11. [19]

    uhn, S., M \

    Nicoli, K., Anders, C. J., Funcke, L., Hartung, T., Jansen, K., K\"uhn, S., M \"u ller, K.-R., Stornati, P., Kessel, P., and Nakajima, S. Physics-informed bayesian optimization of variational quantum circuits. In Advances in Neural Information Processing Systems (NeurIPS2023), 2023 a

    2023

  12. [20]

    A., Anders, C

    Nicoli, K. A., Anders, C. J., et al. Emicore: Expected maximum improvement over confident regions. https://github.com/angler-vqe/emicore, 2023 b

    2023

  13. [21]

    J., Kandala, A., Chen, C.-F., Barkoutsos, P

    Ollitrault, P. J., Kandala, A., Chen, C.-F., Barkoutsos, P. K., Mezzacapo, A., Pistoia, M., Sheldon, S., Woerner, S., Gambetta, J. M., and Tavernelli, I. Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor. Phys. Rev. Res., 2: 0 ...

    2020 doi

  14. [22]

    A variational eigenvalue solver on a photonic quantum processor

    Peruzzo, A., McClean, J., Shadbolt, P., et al. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 0 (1): 0 4213, 2014. doi:10.1038/ncomms5213. URL https://doi.org/10.1038/ncomms5213

    2014 doi

  15. [23]

    Sequential minimal optimization : A fast algorithm for training support vector machines

    Platt, J. Sequential minimal optimization : A fast algorithm for training support vector machines. Microsoft Research Technical Report, 1998

    1998

  16. [24]

    Rasmussen, C. E. and Williams, C. K. I. Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA, USA, 2006

    2006

  17. [25]

    Principles of Mathematical Analysis

    Rudin, W. Principles of Mathematical Analysis. McGraw-Hill, 1964

    1964

  18. [26]

    Evaluating analytic gradients on quantum hardware

    Schuld, M., Bergholm, V., Gogolin, C., et al. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99 0 (3): 0 032331, 2019

    2019

  19. [27]

    and Yamasaki, H

    Tamiya, S. and Yamasaki, H. Stochastic gradient line bayesian optimization for efficient noise-robust optimization of parameterized quantum circuits. npj Quantum Information, 8 0 (1): 0 90, 2022. doi:10.1038/s41534-022-00592-6. URL https://doi.org/10.1038/s41534-022-00592-6

    2022 doi

  20. [28]

    H., et al

    Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., et al. The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986: 0 1--128, 2022

    2022

  21. [29]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.