Pith. sign in

REVIEW 3 major objections 4 minor 74 references

Newton-Schulz Retraction-Based Inference Enables Hidden Quantum Markov Models to Outperform Classical HMMs

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read A retraction-based HQMM learner can outperform EM-trained classical HMMs on sequence data that are not generated by a quantum process, and it carries the first finite-time convergence guarantee for HQMM inference.

desk verdict NS-RIS is a plausible HQMM learning scheme with a coherent proof skeleton, but Algorithm 2's Newton-Schulz coefficients do not actually orthogonalize—unit singular values map to 0.701 and oscillate—so the central guarantee does not apply to the paper as written. read the letter →

arxiv 2608.06554 v1 pith:GDAWQJQJ submitted 2026-08-06 cs.LG quant-phstat.ME

Ning Ning This is my paper · ORCID
classification cs.LGquant-phstat.ME MSC 65F6090C2662M05
keywords hiddenquantumMarkovmodelsHQMMlearningStiefelmanifoldoptimizationNewton-Schulzorthogonalizationpolardecompositionfinite-timestationarityguaranteesplicejunctionclassificationexpectation-maximizationHMM
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

NS-RIS is a new algorithm for learning hidden quantum Markov models (HQMMs), sequence models that replace the probability vector of a classical HMM with a quantum density matrix and stochastic transitions with quantum operations. The paper claims that NS-RIS, by performing every update on the Stiefel manifold of valid Kraus operators and using Newton–Schulz orthogonalization instead of matrix decompositions, trains HQMMs that beat an EM-trained classical HMM on data that were not generated by a quantum process — the first such benchmark result. It also claims the first finite-time stationarity guarantee for an HQMM inference algorithm, with a bound that separates descent, noise, and retraction-error contributions. If these claims are correct, HQMMs become a practical alternative to HMMs for scientific sequence data rather than only a theoretical generalization.

What carries the argument

The load-bearing object is NSOrth, a Newton–Schulz orthogonalization routine applied to the stacked Kraus matrix $\Gamma \in \mathbb{C}^{swn \times n}$. Given a matrix $Z$, NSOrth normalizes it and iterates $Z \leftarrow aZ + Z(bZ^\dagger Z + c(Z^\dagger Z)^2)$ with fixed coefficients, producing an approximate polar factor without a singular value decomposition. NS-RIS calls it twice per step: on the momentum buffer $M_{k+1}$ to form an approximate operator-norm steepest-descent direction, and on $\Gamma_k - \eta_k \hat{M}_{k+1}$ to retract back to the Stiefel manifold. Proposition 1 identifies the first call as approximating the polar factor solution to the nuclear-norm linearized subproblem and the second as approximating the nearest-Stiefel-matrix projection. Theorem 1 then shows the averaged Riemannian-gradient norm is bounded by a telescoping descent term, a smoothness term, a retraction-residual term, and a momentum-tracking term.

What would settle it

Measure the retraction residual $\rho_k = \|\Gamma_{k+1} - (\Gamma_k - \eta_k \hat{M}_{k+1})\|_\infty$ and the direction error $\|\mathrm{NSOrth}(M_{k+1},T_{\mathrm{NS}}) - \mathrm{Polar}(M_{k+1})\|_{\mathrm{op}}$ across the actual $T_{\mathrm{NS}}$ values and stepsize schedule used in the paper. If $\bar\rho_K$ does not decrease as $O(\eta^2)$ when $\eta$ is halved, or if the direction error is not uniformly below 1, the clean $O(\eta)$ stationarity floor in Theorem 1 — and with it the claim of the first guaranteed HQMM inference algorithm — does not follow.

Watch

Extended reading notes

Core claim

The central claim is that NS-RIS makes HQMM learning both scalable and theoretically grounded, and that this is enough for HQMMs to outperform EM-trained HMMs on non-quantum sequence benchmarks. Learning is posed as optimization over the complex Stiefel manifold of stacked Kraus operators, where the trace-preserving constraint is part of the geometry. The update uses Newton–Schulz orthogonalization twice: once to approximate the polar factor of a momentum-smoothed gradient, giving an operator-norm steepest direction, and once to retract the updated matrix back to the manifold without a singular value decomposition. Theorem 1 gives a finite-time bound on the average nuclear norm of the Riemannian gradient under Lipschitz smoothness, unbiased bounded-variance stochastic gradients, and a uniform finite-accuracy assumption on the Newton–Schulz iterates. Empirically, NS-RIS improves test description accuracy over COSM by 38.5% on average (up to 50.6%) on synthetic HMM data, improves the test metric by 18.9% while running 12.0% faster on a synthetic HQMM benchmark, and lowers splice-junction classification error by 17.9% for latent dimension 6 and 14.9% for latent dimension 8 relative to COSM.

Load-bearing premise

The guarantee relies on Assumption 3, which asserts that the fixed, small number of Newton–Schulz iterations keeps the search direction within a fixed tolerance of the exact polar factor at every step, plus Remark 2's unverified scaling that the retraction error shrinks as the square of the stepsize; neither is proved nor measured for the iteration counts used in the experiments.

Editorial extensions

If this is right

  • Because the stationarity bound separates retraction error from stochastic noise, practitioners can tune the Newton–Schulz iteration count $T_{\mathrm{NS}}$ against a desired accuracy floor without redoing the whole convergence analysis.
  • On the synthetic HMM benchmark, a trained HQMM can beat the EM-trained HMM whose generating process is classical, so HQMMs should be considered viable default sequence models rather than tools restricted to quantum-generated data.
  • On the Splice benchmark, latent dimensions 6 and 8 consistently beat dimension 4, so the four-letter nucleotide alphabet does not cap the useful latent size for sequence classification.
  • The decomposition-free Newton–Schulz updates keep runtime comparable to COSM while improving accuracy, which supports applying NS-RIS to larger Kraus ranks and longer sequences without a prohibitive computational penalty.

Reading between the lines

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

  • Going beyond the paper: the polar-factor search direction implies that NS-RIS's benefit may come specifically from replacing the Frobenius-norm steepest direction with the operator-norm steepest direction; an ablation that substitutes the raw normalized momentum for the first NSOrth call would isolate that mechanism.
  • Going beyond the paper: the Splice results suggest a testable prediction — datasets whose splice signals are longer-range and more context-dependent should show larger NS-RIS gains, while datasets dominated by local motifs should show smaller gains, since the paper attributes the gains to latent states encoding motifs, positional context, and long-range dependencies.
  • Going beyond the paper: the same two-step Newton–Schulz retraction pattern could transfer to other Stiefel-constrained latent-variable models, such as orthogonal dictionary learning or tensor decompositions, provided those objectives are smooth in the same operator–nuclear sense as the HQMM negative log-likelihood.
Share X Bluesky LinkedIn Reddit HN

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 NS-RIS, a Newton–Schulz retraction-based algorithm for learning trace-preserving hidden quantum Markov models (HQMMs) on a complex Stiefel manifold. The algorithm replaces matrix decompositions with two calls to a finite Newton–Schulz orthogonalization subroutine: one to polarize the momentum direction and one to retract the updated Kraus matrix to the Stiefel manifold. The authors state a finite-time stationarity bound (Theorem 1) under smoothness, stochastic-gradient, and finite Newton–Schulz accuracy assumptions, and they report numerical results on synthetic HMM, synthetic HQMM, and real Splice benchmarks claiming consistent improvement over EM-trained HMMs and the COSM baseline.

Significance. If the algorithm as written were correct, the paper would be a meaningful step for HQMM learning: it proposes a decomposition-free Stiefel procedure, gives an explicit stationarity bound whose proof separates descent, stochastic noise, and feasibility residuals, and it would provide the first benchmark evidence that HQMMs can beat EM-trained HMMs on non-quantum-generated data. The proof structure is transparent and does not appear to smuggle fitted constants into the convergence rate. However, the significance is contingent on a load-bearing algorithmic subroutine; as I detail below, the Newton–Schulz iteration in Algorithm 2 does not approximate the polar factor, so the stated guarantee does not apply to the submitted implementation and the reported experiments would need to be regenerated after a correction.

major comments (3)
  1. [§3.1.2, Algorithm 2] The Newton–Schulz coefficients (a,b,c)=(3.4445,-4.7750,2.0315) do not implement a convergent polar-factor or orthogonalization iteration. For a matrix with a unit singular value, the scalar update is s←a s + b s^3 + c s^5, and at s=1 this gives a+b+c=0.701, not 1. Subsequent iterates oscillate (for example 1→0.701→1.114→0.722) rather than converging to 1. Consequently NSOrth(M_{k+1},T_NS) does not approximate Polar(M_{k+1}), and the second call NSOrth(Γ_{k+1/2},T_NS) does not retract Γ_{k+1/2} to St(p,n). Lemma 4's hypothesis Γ_{k+1}∈St(p,n) is therefore not met by Algorithm 1 as written, the residual ρ_k in Theorem 1 is O(1), and both the theoretical guarantee and the numerical results are unsupported by the submitted implementation.
  2. [§3.2, Assumption 3 and Remark 2] Assumption 3 is load-bearing but unverified. It postulates a uniform bound ∥\hat M_{k+1}−P_{k+1}∥_op≤ε_NS for all k, and Remark 2 further asserts that the retraction residuals satisfy \bar ρ_K=O(η^2) and \bar q_K=O(η^4) in order to obtain the clean O(η) stationarity floor. The paper gives no T_NS-dependent error bound, no proof of these scalings, and no numerical check for the chosen T_NS. The classical convergence results cited in Section 3.2 concern exact polar iterations, not the finite NSOrth with the stated coefficients; without a proof or empirical verification of Assumption 3, the claim that NS-RIS is the first HQMM inference algorithm with a mathematical performance guarantee does not follow.
  3. [Theorem 1 versus Algorithm 1] There is a mismatch between the theorem and the algorithm as stated. Theorem 1 assumes a constant stepsize η_k=η, while Algorithm 1 explicitly decays the stepsize after every epoch, η←αη. No argument is provided showing that the theorem applies to the epoch-decaying schedule actually used in the experiments. If the authors intend the theorem only for the α=1 case, the text should say so; otherwise the finite-time stationarity guarantee does not cover the implemented procedure.
minor comments (4)
  1. [§4.2] The 'scenario id=4' used for the 50.6% relative improvement is not defined; please specify the corresponding (n,w) configuration so the claim is reproducible.
  2. [§4.1] The sentence describing COSM says it 'performs gradient-based updates directly on the Stiefel manifold using a retraction that we propose'; this appears to be a wording error and should read 'that they propose'.
  3. [§4.2 and §5.2] The language 'significantly outperform' is not backed by a statistical significance test; the reported error bars are standard deviations across test sequences rather than variability across training runs. A paired test or confidence interval on the relative improvements would strengthen the headline empirical claim.
  4. [References] Several references are dated 2026, including arXiv items, relative to the 2026 arXiv date of this manuscript; please verify that all citations are complete and correctly dated.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the NS-RIS stationarity bound is derived from stated assumptions, and the benchmark comparisons are external; the only self-citation is contextual and non-load-bearing.

full rationale

NS-RIS's central claims are not circular. Theorem 1 is a conditional finite-time stationarity bound: under Assumptions 1-3 (Lipschitz smoothness, unbiased bounded-variance stochastic gradients, and a uniform finite Newton-Schulz accuracy tolerance), the proof in Appendix A derives the displayed inequality by algebraic manipulation of the one-step descent lemma and the momentum-tracking lemma. No parameter in the bound is fitted to benchmark data, and the proof does not invoke the theorem's conclusion. The empirical claims are comparisons against external baselines (EM, GS, and COSM) using established synthetic and Splice benchmarks; the reported improvements are measured quantities, not identities forced by construction. The only self-citation is Ning (2025), used in the introduction alongside Srinivasan et al. (2018a) to support the contextual premise that HQMMs are expressive; this is not load-bearing for the NS-RIS algorithm, its convergence guarantee, or its benchmark results. The reviewer concern that the Newton-Schulz coefficients could fail to converge to the polar factor, and the unverified residual scaling rho_bar = O(eta^2) in Remark 2, are correctness and validity risks about whether Assumption 3 holds for the implemented algorithm; they affect whether the theorem's hypotheses are satisfied, not whether the derivation is equivalent to its inputs. Therefore no circular step is identified.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central proof uses standard optimization assumptions and prior results on polar decomposition. No new physical entities are introduced. The main uncharged premises are the smoothness and finite-accuracy assumptions.

free parameters (3)
  • Newton-Schulz coefficients a, b, c = a=3.4445, b=-4.7750, c=2.0315
    Hand-specified constants in Algorithm 2; no derivation or citation is given for these values, and they differ from the classical Newton-Schulz coefficients.
  • Hyperparameters eta0, beta, alpha, TNS = Not reported in text (adopted from COSM)
    Empirical results depend on these hyperparameters; the paper says it reused COSM's configuration but does not list the values.
  • Finite NS accuracy tolerance epsilon_NS = Unspecified in [0,1)
    Assumption 3 postulates that the approximate polar direction is within epsilon_NS of exact; no value or verification is provided, making it a free premise of Theorem 1.
assumptions (4)
  • domain assumption F is L-smooth in the operator-nuclear norm (Assumption 1)
    Required for Theorem 1; the paper does not verify that the HQMM negative log-likelihood satisfies this Lipschitz condition on the Stiefel manifold.
  • domain assumption Unbiased stochastic gradients with bounded second moment (Assumption 2)
    Standard SGD premise; plausible for a differentiable loss on a compact manifold but not checked in experiments.
  • ad hoc to paper Finite Newton-Schulz iterations achieve uniform polar-direction accuracy epsilon_NS (Assumption 3)
    Load-bearing for Theorem 1; no proof, no TNS-dependent bound, and Remark 2's O(eta^2) residual scaling is asserted.
  • standard math Newton-Schulz iteration converges to the polar factor (cited prior work)
    Relies on Bjorck-Bowie, Higham, and Nakatsukasa; not re-proved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Newton-Schulz Retraction-Based Inference Enables Hidden Quantum Markov Models to Outperform Classical HMMs." pith.science (2026). https://pith.science/paper/GDAWQJQJ

@misc{pith2026260806554,
  author       = {Pith},
  title        = {Pith review of: Newton-Schulz Retraction-Based Inference Enables Hidden Quantum Markov Models to Outperform Classical HMMs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDAWQJQJ}},
  note         = {Machine review of arXiv:2608.06554}
}
read the original abstract

Hidden Markov models (HMMs) are widely used probabilistic models for discrete sequential data but can be limited when hidden dynamics are complex. Hidden quantum Markov models (HQMMs) generalize HMMs by replacing probability vectors with density matrices and stochastic transitions with quantum operations, enabling richer latent representations. However, existing HQMM learning methods have not consistently outperformed Expectation--Maximization (EM)-trained HMMs on data not generated by quantum processes, limiting their practical applicability. We introduce NS-RIS, Newton--Schulz Retraction-based Inference on the Stiefel manifold, a scalable algorithm for learning trace-preserving HQMMs. NS-RIS uses Newton--Schulz orthogonalization to compute a polar-factor search direction while preserving Stiefel-manifold feasibility, avoiding costly matrix decompositions. We further establish a finite-time stationarity guarantee under standard assumptions on smoothness, stochastic gradients, and finite Newton--Schulz accuracy. Empirically, NS-RIS provides the first benchmark evidence that an HQMM can significantly outperform an EM-trained HMM on data not generated by a quantum model. On synthetic HMM-generated benchmarks, NS-RIS outperforms both EM and the state-of-the-art HQMM method COSM, improving the evaluation metric by an average of 38.5% and by up to 50.6%. On a synthetic HQMM benchmark, it improves the test metric over COSM by 18.9% while reducing runtime by 12.0%. On the real-world Splice classification benchmark, NS-RIS also surpasses both EM and COSM in higher-dimensional latent regimes, reducing mean classification error by 17.9% for latent dimension 6 and 14.9% for latent dimension 8 relative to COSM. These results move HQMMs beyond a theoretical generalization of HMMs and establish them as practical and expressive models for scientific sequence data.

Figures

Figures reproduced from arXiv: 2608.06554 by the authors.

Figure 1
Figure 1. Test description accuracy for HQMM learning methods under varying hidden dimensions [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Running time comparison for HQMM learning methods under varying hidden dimensions [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Comparison of NS-RIS, GS, COSM, and EM on the synthetic HQMM benchmark with [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Mean error comparison of COSM and NS-RIS on the Splice dataset for latent state sizes [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Nested pie comparison of COSM and NS-RIS errors across three metrics. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Running-time comparison of NS-RIS and COSM on the Splice dataset across latent [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

74 extracted references · 65 canonical work pages

  1. [1]

    Adhikary, S., Srinivasan, S., Gordon, G., and Boots, B. (2020). Expressiveness and learning of hidden quantum Markov models. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics , volume 108 of Proceedings of Machine Learning Research , pages 4151--4161. PMLR

    work page 2020

  2. [2]

    Agarwal, V., Reddy, N. J. K., and Anand, A. (2019). Unsupervised representation learning of DNA sequences. arXiv preprint arXiv:1906.03087

    arXiv 2019

  3. [3]

    Beck, A. (2017). First-order methods in optimization . SIAM

    2017

  4. [5]

    Burge, C. B. and Karlin, S. (1997). Modeling dependencies in pre-m RNA splicing signals. Proceedings of the National Academy of Sciences , 94(2):387--392

    work page 1997

  5. [6]

    A., Huang, W., Barlow, T

    Clark, L. A., Huang, W., Barlow, T. M., and Beige, A. (2015). Hidden quantum Markov models and open quantum systems with instantaneous feedback. In ISCS 2014: Interdisciplinary Symposium on Complex Systems , pages 143--151. Springer

    work page 2015

  6. [7]

    Deb, B., Pandey, S., and Pakray, P. (2026). Quantum hidden Markov model for sequential analysis. Evolutionary Intelligence , 19:90

    work page 2026

  7. [8]

    and Karra Taniskidou, E

    Dheeru, D. and Karra Taniskidou, E. (2017). UCI machine learning repository. http://archive.ics.uci.edu/ml

    work page 2017

  8. [9]

    Eddy, S. R. (1998). Profile hidden Markov models. Bioinformatics , 14(9):755--763

    work page 1998

Show all 74 references
  1. [10]

    Ghamsari, M. S. (2025). Quantum computing applications in biology. Discover Computing , 28(1):1--33

    2025

  2. [11]

    Higham, N. J. (1986). Computing the polar decomposition---with applications. SIAM Journal on Scientific and Statistical Computing , 7(4):1160--1174

    1986

  3. [12]

    Higham, N. J. (2008). Functions of Matrices: Theory and Computation . SIAM

    2008

  4. [13]

    Jaeger, H. (2000). Observable operator models for discrete stochastic time series. Neural Computation , 12(6):1371--1398

    2000

  5. [14]

    Jaggi, M. (2013). Revisiting Frank-Wolfe : Projection-free sparse convex optimization. In International conference on machine learning , pages 427--435. PMLR

    2013

  6. [15]

    Kim, S. W. P., von Keyserlingk, C., and Lamacraft, A. (2026). Measurement-induced phase transitions in quantum inference problems and quantum hidden Markov models. Physical Review Research , 8(2):023155

    2026

  7. [16]

    K \"o soglu-Kind, B. et al. (2023). A biological sequence comparison algorithm using quantum computers. Scientific Reports , 13(1):14077

    2023

  8. [17]

    S., Sjolander, K., and Haussler, D

    Krogh, A., Brown, M., Mian, I. S., Sjolander, K., and Haussler, D. (1994). Hidden Markov models in computational biology: Applications to protein modeling. Journal of Molecular Biology , 235(5):1501--1531

    1994

  9. [18]

    Leifer, M. S. and Poulin, D. (2008). Quantum graphical models and belief propagation. Annals of Physics , 323(8):1899--1946

    2008

  10. [19]

    Leifer, M. S. and Spekkens, R. W. (2013). Towards a formulation of quantum theory as a causally neutral theory of B ayesian inference. Physical Review A , 88(5):052130

    2013

  11. [20]

    Monras, A., Beige, A., and Wiesner, K. (2010). Hidden quantum Markov models and non-adaptive read-out of many-body states. arXiv preprint arXiv:1002.2337

    2010 arXiv

  12. [21]

    Nakatsukasa, Y., Bai, Z., and Gygi, F. (2010). Optimizing H alley's iteration for computing the matrix polar decomposition. SIAM Journal on Matrix Analysis and Applications , 31(5):2700--2720

    2010

  13. [22]

    Nalecz-Charkiewicz, K., Charkiewicz, K., and Nowak, R. M. (2024). Quantum computing in bioinformatics: a systematic review mapping. Briefings in Bioinformatics , 25(5):bbae391

    2024

  14. [23]

    Nesterov, Y. (2013). Introductory lectures on convex optimization: A basic course , volume 87. Springer Science & Business Media

    2013

  15. [24]

    Ning, N. (2025). Robust iterative learning hidden quantum Markov models. https://arxiv.org/abs/2510.23237

    2025

  16. [25]

    Outeiral, C. et al. (2021). The prospects of quantum computing in computational molecular biology. WIREs Computational Molecular Science , 11(1):e1481

    2021

  17. [26]

    Poddar, A., Uzun, V., Tunbridge, E., Haerty, W., and Nevado-Holgado, A. (2023). Identifying DNA sequence motifs using deep learning. arXiv preprint arXiv:2311.12884

    2023 arXiv

  18. [27]

    Schuld, M., Sinayskiy, I., and Petruccione, F. (2015). An introduction to quantum machine learning. Contemporary Physics , 56(2):172--185

    2015

  19. [28]

    and Andolsi, A

    Souissi, A. and Andolsi, A. (2026). A hidden quantum Markov model framework for entanglement and topological order in the AKLT chain. The European Physical Journal Plus , 141(4):404

    2026

  20. [29]

    Srinivasan, S., Downey, C., and Boots, B. (2018a). Learning and I nference in H ilbert space with Q uantum G raphical M odels. In Advances in Neural Information Processing Systems 31

    2018

  21. [30]

    Srinivasan, S., Gordon, G., and Boots, B. (2018b). Learning hidden quantum Markov models. In International Conference on Artificial Intelligence and Statistics , pages 1979--1987

    2018

  22. [31]

    Towell, G. G. and Shavlik, J. W. (1991). Refinement of approximate domain theories by knowledge-based neural networks. Proceedings of the Eighth International Workshop on Machine Learning , pages 861--865

    1991

  23. [32]

    Varsamis, G. D. et al. (2023). Quantum gate algorithm for reference-guided DNA sequence alignment. Briefings in Functional Genomics

    2023

  24. [33]

    Warmuth, M. K. and Kuzmin, D. (2006). A bayesian probability calculus for density matrices. In UAI

    2006

  25. [34]

    Yeang, C.-H. (2010). A probabilistic graphical model of quantum systems. In 2010 Ninth International Conference on Machine Learning and Applications , pages 155--162. IEEE

    2010

  26. [35]

    Zhu, Q.-S., Chen, G., Yu, L.-H., Xing, X., and Li, X.-Y. (2025). Channel-constrained Markovian quantum diffusion model from open system perspective. arXiv preprint arXiv:2511.12221

    2025

  27. [36]

    Åke Björck and Bowie, C. (1971). An iterative algorithm for computing the best estimate of an orthogonal matrix. SIAM Journal on Numerical Analysis , 8(2):358--364

    1971

  28. [37]

    Robust Iterative Learning Hidden Quantum

    Ning, Ning , journal=. Robust Iterative Learning Hidden Quantum

  29. [38]

    A Hidden Quantum

    Souissi, Abdessatar and Andolsi, Amenallah , journal=. A Hidden Quantum. 2026 , doi=

    2026

  30. [39]

    Quantum Hidden

    Deb, Bitanuka and Pandey, Shyambabu and Pakray, Partha , journal=. Quantum Hidden. 2026 , doi=

    2026

  31. [40]

    and von Keyserlingk, Curt and Lamacraft, Austen , journal=

    Kim, Sun Woo P. and von Keyserlingk, Curt and Lamacraft, Austen , journal=. Measurement-Induced Phase Transitions in Quantum Inference Problems and Quantum Hidden. 2026 , doi=

    2026

  32. [41]

    Channel-Constrained

    Zhu, Qin-Sheng and Chen, Geng and Yu, Lian-Hui and Xing, Xiaodong and Li, Xiao-Yu , journal=. Channel-Constrained

  33. [42]

    Discover Computing , volume=

    Quantum Computing Applications in Biology , author=. Discover Computing , volume=. 2025 , doi=

    2025

  34. [43]

    SIAM Journal on Numerical Analysis , volume =

    Åke Björck and Charles Bowie , title =. SIAM Journal on Numerical Analysis , volume =. 1971 , publisher =

    1971

  35. [44]

    Higham , title =

    Nicholas J. Higham , title =. SIAM Journal on Scientific and Statistical Computing , volume =. 1986 , publisher =

    1986

  36. [45]

    Optimizing

    Yuji Nakatsukasa and Zhengji Bai and Fran. Optimizing. SIAM Journal on Matrix Analysis and Applications , volume =. 2010 , publisher =

    2010

  37. [46]

    Higham , title =

    Nicholas J. Higham , title =

  38. [47]

    WIREs Computational Molecular Science , year=

    The prospects of quantum computing in computational molecular biology , author=. WIREs Computational Molecular Science , year=

  39. [48]

    Scientific Reports , volume=

    A biological sequence comparison algorithm using quantum computers , author=. Scientific Reports , volume=

  40. [49]

    and others , journal=

    Varsamis, Georgios D. and others , journal=. Quantum gate algorithm for reference-guided

  41. [50]

    Briefings in Bioinformatics , volume=

    Quantum computing in bioinformatics: a systematic review mapping , author=. Briefings in Bioinformatics , volume=. 2024 , publisher=

    2024

  42. [51]

    Proceedings of the Eighth International Workshop on Machine Learning , pages=

    Refinement of Approximate Domain Theories by Knowledge-Based Neural Networks , author=. Proceedings of the Eighth International Workshop on Machine Learning , pages=. 1991 , publisher=

    1991

  43. [52]

    Expressiveness and Learning of Hidden Quantum

    Adhikary, Sandesh and Srinivasan, Siddarth and Gordon, Geoff and Boots, Byron , booktitle=. Expressiveness and Learning of Hidden Quantum. 2020 , series=

    2020

  44. [53]

    Learning Hidden Quantum

    Srinivasan, Siddarth and Gordon, Geoff and Boots, Byron , booktitle=. Learning Hidden Quantum

  45. [54]

    Learning and

    Srinivasan, Siddarth and Downey, Carlton and Boots, Byron , booktitle=. Learning and. 2018 , url=

    2018

  46. [55]

    Hidden Quantum

    Monras, Alex and Beige, Almut and Wiesner, Karoline , journal=. Hidden Quantum

  47. [56]

    and Huang, Wei and Barlow, Thomas M

    Clark, Lewis A. and Huang, Wei and Barlow, Thomas M. and Beige, Almut , booktitle=. Hidden Quantum. 2015 , organization=

    2015

  48. [57]

    Annals of Physics , volume=

    Quantum Graphical Models and Belief Propagation , author=. Annals of Physics , volume=. 2008 , publisher=

    2008

  49. [58]

    2010 Ninth International Conference on Machine Learning and Applications , pages=

    A probabilistic graphical model of quantum systems , author=. 2010 Ninth International Conference on Machine Learning and Applications , pages=. 2010 , organization=

    2010

  50. [59]

    and Spekkens, Robert W

    Leifer, Matthew S. and Spekkens, Robert W. , journal=. Towards a formulation of quantum theory as a causally neutral theory of. 2013 , publisher=

    2013

  51. [60]

    , author=

    A Bayesian Probability Calculus for Density Matrices. , author=. UAI , year=

  52. [61]

    Contemporary Physics , volume=

    An Introduction to Quantum Machine Learning , author=. Contemporary Physics , volume=. 2015 , publisher=

    2015

  53. [62]

    arXiv preprint arXiv:1611.09347 , year=

    Quantum Machine Learning , author=. arXiv preprint arXiv:1611.09347 , year=

    arXiv

  54. [63]

    Neural Computation , volume=

    Observable Operator Models for Discrete Stochastic Time Series , author=. Neural Computation , volume=

  55. [64]

    2017 , howpublished =

    Dheeru, Dua and Karra Taniskidou, Efi , title =. 2017 , howpublished =

    2017

  56. [65]

    Saira and Sjolander, Kimmen and Haussler, David , journal=

    Krogh, Anders and Brown, Michael and Mian, I. Saira and Sjolander, Kimmen and Haussler, David , journal=. Hidden. 1994 , publisher=

    1994

  57. [66]

    , journal=

    Eddy, Sean R. , journal=. Profile Hidden. 1998 , publisher=

    1998

  58. [67]

    and Karlin, Samuel , journal=

    Burge, Christopher B. and Karlin, Samuel , journal=. Modeling dependencies in pre-m. 1997 , publisher=

    1997

  59. [68]

    Jayanth Kumar and Anand, Ashish , journal=

    Agarwal, Vishal and Reddy, N. Jayanth Kumar and Anand, Ashish , journal=. Unsupervised Representation Learning of

  60. [69]

    Identifying

    Poddar, Asmita and Uzun, Vladimir and Tunbridge, Elizabeth and Haerty, Wilfried and Nevado-Holgado, Alejo , journal=. Identifying

  61. [70]

    The Hierarchical Hidden

    Fine, Shai and Singer, Yoram and Tishby, Naftali , journal=. The Hierarchical Hidden. 1998 , publisher=

    1998

  62. [71]

    and Neal, Radford M

    Shestopaloff, Alexander Y. and Neal, Radford M. , journal=. Sampling Latent States for High-Dimensional Non-Linear State Space Models with the Embedded

  63. [72]

    Introductory lectures on convex optimization:

    Nesterov, Yurii , volume=. Introductory lectures on convex optimization:. 2013 , publisher=

    2013

  64. [73]

    2017 , publisher=

    First-order methods in optimization , author=. 2017 , publisher=

    2017

  65. [74]

    Revisiting

    Jaggi, Martin , booktitle=. Revisiting. 2013 , organization=

    2013

  66. [75]

    Weakly Convex Optimization over

    Li, Xiao and Chen, Shixiang and Deng, Zengde and Qu, Qing and Zhu, Zhihui and So, Anthony Man-Cho , journal=. Weakly Convex Optimization over. 2021 , doi=

    2021

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.