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REVIEW 3 major objections 6 minor 1 cited by

Quantum Dimension Reduction of Hidden Markov Models

T0 review · 3 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Any finite ergodic hidden Markov model can be compressed into a smaller quantum model by first relabeling its non-deterministic transitions as deterministic ones.

desk verdict The dilation idea is genuinely new and mostly correct, but the universal normality claim in Sec. III B is false as stated, and the paper's own Appendix A contradicts it. read the letter →

arxiv 2601.16126 v2 pith:ADE37X2O submitted 2026-01-22 quant-ph cond-mat.stat-mech

Rishi Sundar , Thomas J. Elliott This is my paper
classification quant-phcond-mat.stat-mech
keywords hiddenMarkovmodelsquantumdimensionreductionmatrixproductstatestensornetworksq-samplesdeterministicdilationco-emissiondivergenceratestochasticprocesses
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 attempts to close a gap between real-world hidden Markov models and quantum compression methods. Existing tensor-network compression works cleanly only for deterministic hidden Markov models, but models learned from data are typically non-deterministic. The paper introduces a dilation that adds auxiliary output symbols to label each branch of a non-deterministic transition, making the model deterministic without changing any statistics on the original symbols. That deterministic model's quantum q-sample is a normal infinite matrix product state, so standard variational truncation applies, and the truncated tensors yield a compressed quantum model of the original process. If the construction is sound, every finite ergodic HMM becomes a candidate for quantum dimension reduction.

What carries the argument

The label-augmenting dilation: for each allowed transition (s,x) with multiple successors, assign a unique auxiliary symbol y=f(s,s',x) to each successor, yielding a deterministic transition tensor T^{(x,y)}. This is load-bearing because it turns any ergodic HMM into a deterministic presentation with the same hidden-state space, guaranteeing that the q-sample iMPS built from square roots of the dilated transition probabilities is normal—a primitive transfer operator with a unique canonical form. Normality is what makes tangent-space variational truncation at fixed bond dimension stable. The other key object is the co-emission divergence rate (CDR), computed from transfer operators on the pro

What would settle it

Construct a small ergodic non-deterministic HMM whose dilated iMPS has an exactly flat Schmidt spectrum, truncate at bond dimension two, and compute the co-emission divergence rate; if the measured divergence does not decrease roughly in proportion to the discarded tail weight (or does not decrease at all), then the central scaling lemma is not operative for the dilated construction.

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Extended reading notes

Core claim

The central claim is that the obstruction to compressing a general finite stationary ergodic HMM with tensor networks is not intrinsic complexity but presentation: by tagging each branch of every non-deterministic transition with a fresh auxiliary output symbol, the process becomes deterministic on an enlarged alphabet while marginalizing to the original statistics. The q-sample of this dilated deterministic process is a normal iMPS with a primitive transfer operator, so existing variational iMPS truncation routines stably reduce the bond dimension. Coarse-graining the truncated tensors over the auxiliary symbols yields a quantum instrument on the original alphabet—a dimension-reduced QHMM—w

Load-bearing premise

The theoretical error control rests on a companion-paper lemma—co-authored by one of the current authors—stating that after variational truncation the quantum fidelity divergence rate is bounded by a constant times the discarded Schmidt tail; if that scaling fails in the dilated setting, or the constant is impractically large, the guaranteed compression quality collapses.

Editorial extensions

If this is right

  • Every finite stationary ergodic HMM—not just deterministic ones—can be fed into existing iMPS compression pipelines.
  • The compressed bond dimension becomes a tunable resource parameter trading memory against sequence-level distortion.
  • A compressed quantum model of a learned non-deterministic HMM can be reconstructed directly from the truncated tensors by grouping Kraus operators by original symbol.
  • Analytic bounds relate the classical fidelity divergence rate to the discarded Schmidt tail and to the rank of the slice matrix, so the choice of labeling enters the error certificate.
  • The labeling function is a genuine free parameter that can change Schmidt-spectrum decay and hence compression quality.

Reading between the lines

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

  • If the labeling function can be optimized rather than chosen heuristically, the method becomes a search over dilations for the most compressible representation; the paper demonstrates sensitivity but leaves this optimization open.
  • The same dilation idea may apply to other stochastic models beyond HMMs—any finite-state process with ambiguous transitions could be made unifilar and fed through the same pipeline.
  • The slice-rank bound suggests a cheap pre-screen for compressibility of a learned HMM before running the variational optimisation.
  • The gap between the certified fidelity divergence rate and the reported CDR indicates that a tighter theoretical bridge between these measures would strengthen the method's guarantees.
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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

3 major / 6 minor

Summary. The paper proposes a pipeline for compressing arbitrary finite, stationary, ergodic HMMs into quantum models of reduced memory. The core construction adds an auxiliary output label to each transition (the ``dilation''), producing a deterministic HMM on an enlarged alphabet that marginally reproduces the original statistics. From this deterministic process the authors construct a q-sample iMPS, variationally truncate it to a smaller bond dimension, and then reinterpret the truncated tensors as a quantum instrument for the original output alphabet. The method is tested on a tunable non-deterministic source and on a speech-derived HMM, reporting the co-emission divergence rate (CDR) as a function of retained bond dimension. The paper also derives analytic bounds on a classical fidelity divergence rate in terms of the discarded Schmidt tail and the bond entropy.

Significance. If the central claim held, the paper would remove a well-known obstruction to tensor-network compression of stochastic processes: non-deterministic HMMs do not directly give normal iMPS, and existing variational truncation tools require normality. The dilation idea is natural, and the statistical-preservation part (Lemmas 1-3) is clean and correct. The numerical demonstrations are suggestive, and the paper is careful to distinguish the reported CDR from the analytically bounded fidelity rate. However, the central theoretical assertion that every finite ergodic deterministic HMM, including the dilated one, has a normal q-sample iMPS is false as stated, and the attached counterexample is a direct obstruction. Because normality is the stated prerequisite for the variational truncation step, the main advertised universality claim is not established. The analytic bound in Eq. (30) also rests on a self-cited prior result with only a sketch, and it inherits the normality assumption. These issues require a substantial revision, but the underlying idea may be salvageable by adding an explicit minimality/predictivity step or by narrowing the class of HMMs to which the theorem applies.

major comments (3)
  1. [Sec. III B, Eq. (15)] The claim that a finite, stationary, ergodic deterministic HMM always has a primitive transfer operator and hence a normal iMPS is false. Counterexample: S={0,1}, X={0,1}, a in (0,1), b=1-a, T^0 = a I, T^1 = b X with X the swap matrix. This HMM is deterministic by Eq. (2), ergodic, and stationary; the dilation is trivial (|Y|=1). The constructed tensors are A^0=sqrt(a) I, A^1=sqrt(b) X, and the transfer operator E in Eq. (15) has two fixed points, I and X, so the leading eigenvalue 1 is degenerate. Blocking sites does not remove the degeneracy because E^L has the same two fixed points. Thus the iMPS is not normal. This contradicts the universal claim in Sec. III B and the statement that ``blocking a finite number of sites yields an injective MPS.'' The manuscript itself flags the issue at the end of Appendix A (``for general nonpredictive unifilar presentations one should not assume this
  2. [Appendix B, Lemma 6, Eq. (30)] The key error bound R_F <= c epsilon_d~ is imported from Ref. [32], co-authored by the current author Elliott, with only a sketch supplied. This is a load-bearing step: without Lemma 6, there is no analytic guarantee that variational truncation of the dilated iMPS controls the fidelity divergence rate. Moreover, Lemma 6 is stated for a normal iMPS with primitive transfer operator; as shown above, the construction in Sec. III B does not always produce such an iMPS. The paper should either provide a self-contained proof of Lemma 6, or explicitly restrict the theorem to cases where normality is proven (e.g., minimal/predictive deterministic presentations), or characterize when the dilated presentation is minimal.
  3. [Sec. III F vs Sec. IV] The analytical bounds are for the classical fidelity divergence rate R_F, while all numerical figures (Figs. 3, 5, 6) report the co-emission divergence rate R_C. The paper explicitly warns that R_F bounds do not directly translate to R_C, which is honest, but the claim that the numerics are ``in line with the analytic picture'' (Sec. IV A) is not supported by any quantitative connection. This mismatch does not by itself invalidate the numerical results, but it means the main practical metric is currently without a theoretical certificate. The authors should either compute R_F in the examples, or prove a relation between R_F and R_C under the conditions used.
minor comments (6)
  1. [Appendix A, final paragraph] This paragraph effectively retracts the stronger identification used in Sec. III B, but the main text never reconciles the two. The contradiction should be resolved by revising Sec. III B to state the precise conditions under which primitivity/normality holds.
  2. [Sec. III B, Eq. (14)] The notation A^(x,y) uses the same symbol for the site tensor as for the square root of the transition tensor; after gauge fixing and truncation this identification may be lost. Please clarify that Eq. (14) defines the initial exact tensors only.
  3. [Fig. 4] The y-axis labels appear garbled: repeated ``6.66667 x 10^-2'' values are likely a plotting artifact. Please replace with a clear axis label such as ``Eigenvalue lambda_i''.
  4. [Fig. 5] The legend ``quantum classical'' is not formatted with a separator; it should read ``quantum'' and ``classical'' as two entries. Also, values below 1e-9 are capped without indicating that the cap is a display choice; this can mislead the visual comparison.
  5. [Sec. IV C] The sentence ``the probability ascending strategies values exactly coincide with the sequential strategy'' is grammatically unclear and should be rephrased; it is also not evident from Fig. 6 that the curves coincide.
  6. [References] Ref. [15] is listed as a technical report without full bibliographic data; please provide the published or arXiv version if available.

Circularity Check

1 steps flagged · score 4.0 of 10

Core dilation-compression construction is self-contained; however the analytic error bound of Sec. III F is deferred to an overlapping-author reference, giving a load-bearing self-citation without full proof.

  1. self citation load bearing [Sec. III F, Lemma 6 (Appendix B), Eqs. (30) and (B6)]
    "Ref. [32] then implies that, for sufficiently small ε̃_d, there exists a constant c>0 (depending on the local physical dimension and the spectral gap of the transfer operator) such that R_F(P, P̃)≤c ε̃_d. ... The full argument is given in Ref. [32]."

    The paper's central analytic guarantee—that variational truncation error is controlled by the discarded Schmidt tail—is not proved in this work. It is imported from Ref. [32], which is co-authored by the present author T. J. Elliott. The paper supplies only a sketch and explicitly defers the 'full argument' to that overlapping-author source. This makes the theoretical error bound load-bearing on a self-citation rather than on a self-contained derivation. The numerical CDR calculations are computed directly and are not themselves reduced to this bound, so the circularity is partial.

full rationale

The main construction—dilation to a deterministic process, square-root q-sample tensors, variational truncation, and reconstruction of a quantum instrument—is presented with explicit equations and does not reduce to its inputs by definition. The preservation of observable statistics under dilation is proved in Lemmas 1–3, and Lemma 4 shows the square-root tensors reproduce word statistics for unifilar generators. The numerical demonstrations on the TNS and speech-derived HMM compute CDR directly from the constructed generators, so those results are not circular. The principal circularity concern is Lemma 6: the QFDR-versus-Schmidt-tail bound is stated as a consequence of Ref. [32], a paper sharing an author with the current work, and only a sketch is given here. This is load-bearing for the analytic error-control claims (Eqs. (30), (33), (36)) but not for the empirical compression curves. I also note, per the reviewing rule, that Appendix A itself concedes a limitation: for 'general nonpredictive unifilar presentations one should not assume this stronger identification' with the q-sample. That caveat undermines the universality of the normality assertion in Sec. III B, but it is a correctness concern rather than a circularity. Overall, the central claim retains independent content, so the score is moderate rather than high.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The paper introduces one genuine design degree of freedom (the labelling function) and relies on several prior theoretical results, including a load-bearing truncation theorem from a self-cited paper. No free numerical parameters are fitted to make the central construction work.

free parameters (1)
  • Labelling function f = varied: sequential, probability-descending, probability-ascending, random permutation
    The choice of auxiliary label assignment is a hand-chosen degree of freedom. Fig. 6 shows it materially changes the compression trade-off, so it is not fixed by the theory.
assumptions (4)
  • domain assumption Finite-state, stationary, ergodic HMM assumption
    The paper restricts to stationary ergodic HMMs throughout (Sec. II A); if the hidden chain is not ergodic, the transfer operator may not be primitive and the normal iMPS construction can fail.
  • domain assumption q-sample iMPS representation theorem from Ref. [31]
    The paper uses the result that unifilar HMMs admit a q-sample iMPS with primitive transfer operator; cited to Yang et al. and not proved here.
  • domain assumption Truncation error scaling, Lemma 6 from Ref. [32]
    The analytic bounds rely on the result that variational truncation error is linear in the discarded Schmidt tail; this is only sketched in Appendix B and the full proof is deferred to a self-cited previous paper.
  • domain assumption CDR validity for generalized HMMs from Ref. [59]
    The co-emission divergence rate formula (Eq. (26)) is taken to be valid for any stationary ergodic processes represented by finite linear generators; cited to prior work.
invented entities (1)
  • Auxiliary output alphabet Y and labelling function f
    purpose: Makes any HMM deterministic by encoding the successor state into the emitted symbol, enabling normal iMPS construction and stable truncation.
    A mathematical construction, not a physical entity; it has no independent falsifiable handle outside the paper.

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Cite this review

Pith. "Pith review of Quantum Dimension Reduction of Hidden Markov Models." pith.science (2026). https://pith.science/paper/ADE37X2O

@misc{pith2026260116126,
  author       = {Pith},
  title        = {Pith review of: Quantum Dimension Reduction of Hidden Markov Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADE37X2O}},
  note         = {Machine review of arXiv:2601.16126}
}
read the original abstract

Hidden Markov models (HMMs) are ubiquitous in time-series modelling, with applications ranging from chemical reaction modelling to speech recognition. These HMMs are often large, with high-dimensional memories. A recently-proposed application of quantum technologies is to execute quantum analogues of HMMs. Such quantum HMMs (QHMMs) are strictly more expressive than their classical counterparts, enabling the construction of more parsimonious models of stochastic processes. However, state-of-the-art techniques for QHMM compression, based on tensor networks, are only applicable for a restricted subset of HMMs, where the transitions are deterministic. In this work we introduce a pipeline by which \emph{any} finite, ergodic HMM can be compressed in this manner, providing a route for effective quantum dimension reduction of general HMMs. We demonstrate the method on both a simple toy model, and on a speech-derived HMM trained from data, obtaining favourable memory--accuracy trade-offs in the examples studied, relative to a simple classical state-merging baseline.

Figures

Figures reproduced from arXiv: 2601.16126 by the authors.

Figure 1
Figure 1. FIG. 1. Tensor network representations of stochastic mod [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. HMM representation of the generalised [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Schmidt spectra of the dilated iMPS of the TNS for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: FIG. 3. TNS model compression results: co-emission diver [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Speech-derived HMM compression results: compar [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Impact of labelling function on CDR for TNS [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Causal Architecture in Hidden Quantum Markov Models

    math-ph 2026-02 unverdicted novelty 6.0 of 10

    Reversing the order of hidden updates and emissions in quantum Markov models produces distinct quantum processes distinguishable at late times, except when both arise from entangled liftings of classical hidden Markov models.

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    F rom quantum fidelity rate to a classical Bhattacharyya rate Let|P xy⟩and ˜Pxy E denote the infinite-chainq–sample states of the exact and truncated dilated processes over X × Y. For concreteness we define the quantum fidelity divergence rate (QFDR) as RF (|Pxy⟩, ˜Pxy E ) =−l...

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    T runcation and quantum fidelity rate Let{λ k}ds k=1 be the Schmidt coefficients of the dilated iMPS across a fixed bond, ordered so thatλ 1 ≥λ 2 ≥ · · · ≥λds and P k λk = 1. For a target bond dimension ˜dwe define the discarded tail weight ε ˜d := X k> ˜d λk.(B5) The effect o...

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    Then ε ˜d ≤ H(λ) log2 ˜d .(B9) Proof.Fork > ˜dwe haveλ k ≤λ ˜d

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    (B12) with Eq

    Combined slice-rank bound Combining Eq. (B12) with Eq. (B16) gives the slice- rank bound RF (P, ˜P)≤c log2 rankK log2 ˜d ,(B17) which appears as Eq. (36) in the main text. The rank ofKdepends explicitly on the chosen labelling function fthrough the set of nonzero slicesA (x,y)...

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