Pith. sign in

REVIEW 4 major objections 3 minor 13 references

On Quantum and Quantum-Inspired Maximum Likelihood Estimation and Filtering of Stochastic Volatility Models

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A quantum hidden Markov model can match a classical one using only the square root of the classical number of hidden states.

desk verdict The paper's genuinely new application—stochastic volatility estimation via quantum hidden Markov models—comes with a misspecified likelihood and a central theorem whose proof relies on a tautology, so the main claims are not yet supported. read the letter →

arxiv 2507.21337 v1 pith:TKPGLLYS submitted 2025-07-28 quant-ph

Eric Ghysels , Jack Morgan , Hamed Mohammadbagherpoor This is my paper · ORCID
classification quant-ph MSC 62M0581P6862F12
keywords quantumhiddenMarkovmodelsstochasticvolatilitymaximumlikelihoodestimationKullback-Leiblerdivergencedimensionalityreductionnon-asymptoticboundsfilteringCIRdiffusion
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 tries to establish that stochastic volatility, a latent process that makes maximum likelihood estimation hard, can be estimated and filtered through hidden Markov models built for quantum hardware, and that the quantum version needs far fewer hidden states to reach the same accuracy. The central result is a dimension reduction: a classical HMM with $n^2$ hidden volatility states can be matched, in Kullback-Leibler divergence to the true data-generating process, by a quantum HMM with only $n$ hidden quantum states. The paper also claims that non-asymptotic error bounds for the quantum model are tighter than for the classical model, and supports this with simulations on S&P 500 returns. If true, this gives econometricians a quasi-closed-form likelihood and volatility filter for stochastic volatility, with a quadratic state reduction that becomes a runtime advantage whenever the number of states dominates the sample length.

What carries the argument

The machinery is the generalized Hankel matrix of a stochastic process and the rank bound for unitary quantum hidden Markov models. A generalized Hankel matrix records probabilities of finite observation strings, and its rank controls the minimal number of hidden states a hidden Markov model needs. For a classical HMM of order $n_c$, Anderson's realization theory identifies the model with a bi-infinite Hankel matrix of rank $n_c$, and a minimal realization of that rank exists. A unitary quantum HMM with $n_q$ hidden states has Hankel rank at most $(n_q)^2$, which delivers the quadratic reduction when $(n_q)^2 = n_c$. KL-equidivergence is then obtained by comparing canonical decompositions of the two Hankel matrices and invoking a Pythagorean identity for KL divergence. The non-asymptotic comparison runs through a cited penalized-MLE bound for misspecified HMMs, with the quantum model evaluated at the reduced state count.

What would settle it

Run a long simulation ($T=10{,}000$) from a known CIR stochastic volatility process, fit a classical HMM with 16 hidden states and a KL-equidivergent quantum HMM with 4 hidden states, and estimate each model's KL divergence from the true density. Theorem 4.2 predicts the quantum divergence is no larger; repeated samples in which the quantum model is farther from the truth would refute the central claim. A direct check of whether the concentration constants in Assumption A.14 can actually be derived for the quantum channel would settle the tighter-bound claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that quantum hidden Markov models dominate classical ones in state count without sacrificing approximation quality. For every classical HMM of order $n_c = n^2$, Theorem 4.1 asserts there exists at least one KL-equidivergent unitary quantum HMM with $n_q = n$ hidden states: the two models have the same minimal Kullback-Leibler divergence from the true stochastic volatility data-generating process. Theorem 4.2 strengthens this asymptotically, stating that for $n_q = \sqrt{n_c}$ the quantum model's KL divergence is no larger than the classical model's, so the expected log-likelihood ratio favors the quantum model. Theorem 6.1 then claims that, under transferability of the constants in the non-asymptotic bounds, the KL-equidivergent quantum HMM has a tighter probability bound on parameter estimation and filtered volatility error than the classical penalized MLE. The empirical section reports that a 4-state quantum HMM beats or matches 16-state classical HMMs in almost all of 1000 simulated trials, even though the classical parametric model is matched to the data-generating process.

Load-bearing premise

Assumption A.14 asserts, without proof, that the concentration constants for the classical HMM non-asymptotic bounds also apply to the KL-equidivergent quantum HMM; if that transfer fails, Theorem 6.1's tighter quantum bound does not follow.

Editorial extensions

If this is right

  • A practitioner who needs $n^2$ classical volatility states can instead use $n$ quantum states and keep the same KL accuracy, cutting the state dimension quadratically.
  • The likelihood and volatility filter become quasi-closed-form expressions in Kraus operators and partial traces, avoiding numerical integration and MCMC accept-reject loops.
  • In large samples the quantum model is expected to fit the true DGP at least as well as the classical model with the square of its state count, and the simulations at $T=500$ show this mostly already holds.
  • The quantum filter inherits a tighter non-asymptotic high-probability bound on the error of estimated parameters and filtered integrated volatility, provided the concentration constants transfer.
  • On quantum hardware the training complexity can become polylogarithmic in the hidden state count, though it grows exponentially in the number of time steps.

Reading between the lines

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

  • The quadratic state reduction is proved through Hankel-rank arguments that use no detail specific to stochastic volatility, so the same KL-equidivergence claim should transfer to any latent-state time series model admitting a finite-rank HMM approximation; testing this on regime-switching or discrete-time stochastic volatility models would be a natural extension.
  • Because the numerical comparison shows the quantum model winning most often at $T=500$, the asymptotic result may hold well before the large-sample limit; a finite-sample version of Theorem 4.2 giving a rate for the expected log-likelihood ratio would be a testable strengthening the paper leaves implicit.
  • The runtime analysis shows an exponential speedup in hidden state count offset by an exponential slowdown in sample length, so the practical window for the quantum advantage is short samples with many states; shrinking the $T$-dependence, for example by quantum amplitude estimation of the likelihood, would widen that window.
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

4 major / 3 minor

Summary. The paper proposes classical and quantum hidden Markov model approximations for stochastic volatility diffusions. Its central theoretical claims are (i) a classical HMM with n_c = n^2 hidden states can be replaced, without loss of KL divergence, by a quantum HMM with n_q = n hidden states (Theorem 4.1), yielding a quadratic state reduction; (ii) this implies an asymptotic KL comparison favoring the quantum model (Theorem 4.2); and (iii) non-asymptotic bounds from Lehéricy (2021) are tighter for the KL-equidivergent quantum HMM (Theorem 6.1). The paper also reports likelihood-based estimation, filtering, and simulation comparisons on S&P 500 data. The manuscript contains a number of self-identified limitations, including the lack of a fully developed quantum likelihood-ratio theory and the provisional nature of Assumption A.14, but these are presented as caveats rather than as obstacles to the main claims.

Significance. If the central claims were valid, the quadratic reduction in hidden-state dimension and the tighter non-asymptotic bounds would be of substantial interest to both quantum computing and financial econometrics. The paper also has genuine strengths: it provides a concrete ansatz-circuit QHMM construction, a reproducible Python implementation, and a clear focus on classical statistical inference rather than Bayesian MCMC. However, the central theoretical results are not established. The likelihood functions are mis-specified, the proof of the main dimensionality-reduction theorem relies on an invalid invariance argument, and the non-asymptotic comparison is carried by an unproved assumption that transfers classical concentration constants to the quantum model. The empirical simulations cannot compensate for these defects because they use the same incorrect likelihoods.

major comments (4)
  1. [Section 2.4, Eqs. (2.7) and (2.8)] The log-likelihood is written with the logarithm inside the double summation: ℓ_t = Σ_i Σ_j log[P_y(...) g(...) X_{t-1}^i]. This is not the log of the marginal likelihood of an HMM, which would be log Σ_i Σ_j P_y(...) g(...) X_{t-1}^i. The displayed expression is a weighted average of log terms, not a log-likelihood. Since the paper's KL-divergence measures, MLE, and likelihood-ratio simulations in Sections 4 and 5 all use this object, the error is load-bearing and invalidates the estimation and comparison results that follow.
  2. [Appendix A.3.3, Theorem 4.1 and Theorem A.2] The proof of Theorem 4.1 does not establish KL-equidivergence. Theorem A.2 gives only an upper bound, rank(H_Q) ≤ (n_q)^2, for any QHMM; it does not assert that for every classical HMM of rank n_c = n^2 there exists a QHMM with n_q = n hidden states generating the same finite-dimensional distributions. The proof then defines canonical decompositions and claims D_KL((C_c)^{-1}H_c(D_c)^{-1} || (C_q)^{-1}H_q(D_q)^{-1}) = 0 because both transformed matrices have the same identity-block form. This is a tautology about rank, not a statement about the stochastic processes. KL divergence is not invariant under arbitrary invertible row/column transformations, the transformed matrices are not nonnegative probability matrices, and equality of rank does not imply equality of the Hankel entries or of D*_KL. Consequently, Theorems 4.1, 4.2, and the non-asymptotic comparison in Section 6 are unsupported.
  3. [Appendix A.1.2, Assumption A.14 and Theorem 6.1] Assumption A.14 asserts that the constants A, C_Λ, and T0 from Lehéricy's non-asymptotic bounds for the classical HMM also apply to the KL-equidivergent quantum HMM. This assumption is load-bearing for Theorem 6.1 and Corollary 6.1, which claim tighter quantum bounds, but it is stated without proof. The accompanying remark that the assumption is 'reasonable' in light of Theorems A.2 and 4.1 is not a derivation, and Theorem 4.1 is itself unproved. The tighter quantum bound is therefore obtained by postulating exactly the concentration behavior that needs to be demonstrated.
  4. [Appendix A.3.2, Lemma A.3 and proof of Theorem 4.2] Lemma A.3 claims that if two models M1 and M2 are KL-equidivergent, then E*[log L(M1)/L(M2)] = 0, citing the Pythagorean identity. Equality of D*_KL(M1) and D*_KL(M2) does not, by itself, imply that the cross-entropy or cross-KL between M1 and M2 vanishes; the Pythagorean identity holds only for specific projection geometries, not for arbitrary pairs with equal divergence to a third distribution. The proof of Theorem 4.2 relies on this step, so the asymptotic comparison D*_KL(M_nq) ≤ D*_KL(˜M_nc) is not established even conditional on Theorem 4.1.
minor comments (3)
  1. [Throughout] There are numerous typographical errors and inconsistent notations, e.g., 'Obsersable states' in Definition 2.2, 'entires' in Definition 2.4, 'distribtuions' in Section 5.2, and the title 'V olatility' on the first page. These should be corrected carefully.
  2. [Section 5.4, Figure 4] The caption and text refer to 'Log Likelihood Ratio' but the horizontal axis is labeled log10(L_i/L_j); please clarify whether the plotted quantity is the natural-log likelihood ratio or its base-10 logarithm, and state this consistently in the text.
  3. [Appendix A.1.2, Assumption A.14] Assumption A.14 is introduced in a regularity-conditions section without any discussion of how the quantum HMM satisfies the underlying mixing or forgetting conditions that the classical constants encode; even as a provisional assumption, this deserves explicit justification or a reference.

Circularity Check

4 steps flagged · score 6.0 of 10

Quadratic reduction and tighter quantum bounds reduce to rank-counting by construction and to Assumption A.14; Theorem 4.1's proof equates transformed Hankel matrices rather than processes.

  1. self definitional [Appendix A.3.3, eqs. (A.6)-(A.8)]
    "Hc(d) = Cc(d) diag(I_nc,0) Dc(d) ... Hq(d) = Cq(d) diag(I_(nq)^2,0) Dq(d) (nq)^2 = nc ... DKL((Cc(d))−1Hc(d)(Dc(d))−1 ∥ (Cq(d))−1Hq(d)(Dq(d))−1) = 0 for any d."

    Since (n_q)^2 = n_c, both transformed Hankel matrices equal the same identity-block matrix, so the displayed KL is zero by construction. KL divergence is not invariant under arbitrary invertible row/column transformations, and the transformed matrices are not the probability matrices of the two processes. Thus Definition 4.1 is satisfied by rank counting, not by equivalence of the stochastic processes; no QHMM with the same finite-dimensional distributions is constructed.

  2. ansatz smuggled in via citation [Appendix A.1.3, after Theorem A.2]
    "One implication is that if we have a classical HMM of rank ncL, as defined in Definition 2.5, we can find a QHMM with nqL = sqrt(ncL) of hidden states with the same stochastic properties. One might call this a quadratic dimensionality reduction."

    The cited Theorem A.2 only states rank(H_Q) ≤ (n_q)^2; it does not state that every rank-n_c classical process is realizable by a QHMM with sqrt(n_c) states, nor that the realized process has the same finite-dimensional distributions. The 'same stochastic properties' conclusion is smuggled in by interpreting a rank bound as a realization theorem, and this unsupported equivalence then becomes the input to Theorem 4.1.

2 more flagged steps
  1. self definitional [Appendix A.3.4, proof of Theorem 4.2]
    "For models in the set Mq√nc(M ), ... we have: D∗KL(˜MncL) = D∗KL(MnqL) and therefore the expectation of the log likelihood ratio under the true DGP equals zero. However, Mq√nc(M ) ⊂ M√nc, and therefore D∗KL(MnqL) ≤ D∗KL(˜MncL) in the larger set."

    This inequality is a set-inclusion tautology once KL-equidivergence is assumed: if a quantum model is defined to have the same D* as the classical model, the minimum over the quantum class is automatically no larger. The claimed quantum advantage therefore inherits all its content from Theorem 4.1 and from the definition of KL-equidivergence, rather than from any property of quantum dynamics.

  2. fitted input called prediction [Appendix A.1.2 (Assumption A.14) and Appendix A.3.5, eq. (A.15)]
    "ASSUMPTION A.14. The constants A and CΛ ... which apply to the classical HMM non-asymptotic bound also apply to the KL-equidivergent quantum HMM. ... In light of Theorems A.2 and 4.1 the above assumption is reasonable to make ... By analogy, under Assumption A.14, for the quantum HMM we have: D∗KL(Mq√nc,T(ˆMncL)) ≤ NABQ(τ,T,√nc,Mq√nc,T(ˆMncL))."

    The quantum non-asymptotic bound is not derived from a quantum concentration result; it is written 'by analogy' with the classical bound and then combined with Assumption A.14, which simply asserts that the classical constants carry over. The only structural difference is replacing nc by sqrt(nc) in the penalty, so the 'tighter bound' is an algebraic restatement of the quadratic reduction already assumed, not an independent prediction.

full rationale

The central dimensionality-reduction claim rests on a proof that makes KL-equidivergence hold by construction: after canonical transformations both Hankel matrices equal diag(I_{n_c},0) because (n_q)^2=n_c, so the KL in (A.8) is zero regardless of process equivalence; KL is not invariant under these transformations. Theorem 4.2 then only repackages this definition via set inclusion. The non-asymptotic comparison is likewise built from Assumption A.14, which simply transfers the classical constants to the quantum model, and from a quantum bound written 'by analogy' with sqrt(n_c) in place of n_c; the tighter bound is therefore an algebraic consequence of the assumed quadratic reduction, not an independent concentration result. No load-bearing self-citation chain was found: the paper cites Markov et al. (2022) and Lehéricy (2021) externally, and the empirical section is self-contained, but it cannot rescue the definitional nature of Theorems 4.1, 4.2, and 6.1. Score 6 rather than 8 because the simulation study and the classical HMM likelihood machinery are independent contributions; the circularity is localized to the claimed quantum speedup and tighter-bound comparisons.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a known rank bound for QHMMs, but the paper adds a strong ad hoc assumption (A.14) to transfer non-asymptotic constants. The empirical parameters are fitted, and the comparison is between models with different effective dimensionality, which drives the claimed advantage.

free parameters (4)
  • CIR parameters (alpha, beta, sigma) = alpha = 2.2, beta = 0.077, sigma = 1.1
    Estimated from daily S&P 500 returns (1990-2024) and used as the data generating process for simulations.
  • QHMM circuit rotation angles = Not reported
    Parameters of the EfficientSU2 Ansatz are fit by Nelder-Mead optimization to maximize the quantum likelihood.
  • Non-parametric classical HMM transition matrix entries = 240 parameters for nc_L = 16
    All entries of the transition matrix are estimated for the non-parametric classical HMM.
  • Number of hidden states and bins = nc_L = 16, nq_L = 4, nO = 2, k = 4
    Chosen for the simulations; these are modeling choices that affect the comparison.
assumptions (5)
  • domain assumption Assumption A.10: the order of an HMM equals its Hankel rank (minimality)
    Used in the proof of Theorem 4.1 to equate hidden state count with rank.
  • ad hoc to paper Assumption A.14: constants from Lehéricy's bounds apply to the KL-equidivergent QHMM
    This is the key assumption that allows the tighter quantum non-asymptotic bound in Theorem 6.1.
  • standard math Theorem A.2 (Markov et al. 2022): rank of QHMM Hankel matrix is at most (nq_L)^2
    Borrowed from prior literature and used as the basis for the quadratic reduction claim.
  • domain assumption Assumptions A.2 through A.5: regularity conditions on the SV diffusion
    Standard conditions from Genon-Catalot et al. (2000) to ensure the HMM representation exists.
  • domain assumption Assumption 4.1: laws of large numbers for the KL divergences
    Needed for the asymptotic statements in Theorems 4.1 and 4.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Quantum and Quantum-Inspired Maximum Likelihood Estimation and Filtering of Stochastic Volatility Models." pith.science (2026). https://pith.science/paper/TKPGLLYS

@misc{pith2026250721337,
  author       = {Pith},
  title        = {Pith review of: On Quantum and Quantum-Inspired Maximum Likelihood Estimation and Filtering of Stochastic Volatility Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKPGLLYS}},
  note         = {Machine review of arXiv:2507.21337}
}
read the original abstract

Stochastic volatility models are the backbone of financial engineering. We study both continuous time diffusions as well as discrete time models. We propose two novel approaches to estimating stochastic volatility diffusions, one using Quantum-Inspired Classical Hidden Markov Models (HMM) and the other using Quantum Hidden Markov Models. In both cases we have approximate likelihood functions and filtering algorithms that are easy to compute. We show that the non-asymptotic bounds for the quantum HMM are tighter compared to those with classical model estimates.

Figures

Figures reproduced from arXiv: 2507.21337 by the authors.

Figure 1
Figure 1. For our practical implementation, we chose to use a linear entanglement layer as our initial [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Example circuit of QMM using an EfficientSU2 Ansatz circuit for one time step. We use [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Probability distributions of the final three states from two models. The blue model [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Log Likelihood Ratio (LLR) of the estimated models in each of the 1000 trials. The [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Hidden processes and hidden Markov processes: classical and quantum

    ACCARDI , L., E. G. S OUEIDY , Y. G. L U, AND A. S OUISSI (2023): “Hidden processes and hidden Markov processes: classical and quantum,” arXiv preprint arXiv:2302.07058. ADHIKARY , S., S. S RINIVASAN , G. G ORDON , AND B. B OOTS (2020): “Expressiveness and learn- ing of hidden quantum markov models,” in International Conference on Artificial Intelligence ...

    work page Pith review arXiv 2023

  2. [3]

    Let Mnq L be a QHMM with hidden state dimension nq L and let H Q be its associated Hankel matrix, then rank(H Q) ≤ (nq L)2. One implication is that if we have a classical HMM of rank nc L, as defined in Definition 2.5, we can find a QHMM with nq L = pnc L of hidden states with the same stochastic properties. One might call this a quadratic dimensionality ...

    work page 2000

  3. [4]

    such that for all t ∈ T ⊂ N,, for all k, k′ ∈ N, and for all ∆yt t−(k∨k′), | log ℓ∗(∆yt|∆Y t−1 t−k ) − log ℓ∗(∆yt|∆Y t−1 t−k′)| ≤C∗ρk∧k′−1 ∗ Assumption A.8 ensures that the process forgets its initial distribution exponentially fast, which is not guaranteed by ρ-mixing. A.1.2 Regularity Conditions - Classical HMM We present the regularity conditions for c...

    work page 2021

  4. [6]

    A shorthand notation for models will be Mnc L simply emphasizing the number of hidden states

    Recall that the classical HMM according Definition 2.4 is defined by a 5-tuple: (SO, HL, A , E , X0) , where X0 is a stochastic vector defining the initial latent process states and E the emission matrix. A shorthand notation for models will be Mnc L simply emphasizing the number of hidden states. First, two assumptions that apply to both classical and qu...

    work page 2021

  5. [7]

    We use the argu- ments in the proof of Theorem 3.1 of Genon-Catalot, Jeantheau, and Lar´edo (2000) to obtain the result

    (A.1) Appendix - 4 A.3 Proofs A.3.1 Proof of Lemma A.1 We need to show first that under Assumptions A.2 - A.5, the hidden discrete time Markov processUt = ( ¯Vt, Vt∆) defined in equation (2.2) has a time homogeneous transition density dependent only on Vt∆. We use the argu- ments in the proof of Theorem 3.1 of Genon-Catalot, Jeantheau, and Lar´edo (2000) ...

    work page 2000

  6. [8]

    log ℓ∗(∆Y T t=1) ℓ(YT t=1, Mnc L) # (A.2) and: DKL(ℓ∗ T ∥ ℓ∆y T ( ˜Mnc L)) = E∗

    only depends on V. Finally, the strict stationarity and ergodicity of ∆yt follows from Proposition 3.1 in GJL. □ A.3.2 Proofs of Lemmas A.3 and A.2 We start with the proof of Lemma A.2. First we want to establish the existence of ℓ∗. Using Theorems 2.3 and 3.1 of Genon-Catalot, Jeantheau, and Lar ´edo (2000), we know that under Assumptions A.2 through A.5...

    work page 2000

  7. [9]

    number of hidden states) equal to nc L = n2 for n integer

    □ A.3.3 Proof of Theorem 4.1 We start from a classical HMM with order (i.e. number of hidden states) equal to nc L = n2 for n integer. Using Assumptions A.9 through A.10, we know from Anderson (1999) and Huang, Ge, Kakade, and Dahleh (2014) we know that there is a minimal classical HMM with rank, defined in Definition 2.5, equal tonc L and it is unique ac...

    work page 1999

  8. [10]

    □ A.3.4 Proof of Theorem 4.2 We apply again the Pythagorean identity, see Csisz´ar (1975)

    This means that a (computable and invertible) trans- formation of the Hankel matrix of the classical HMM is KL-equidivergent to a (computable and invertible) transformation of the corresponding reduced dimensionality quantum HMM. □ A.3.4 Proof of Theorem 4.2 We apply again the Pythagorean identity, see Csisz´ar (1975). In the general case we have: D∗ KL( ...

    work page 1975

Show all 13 references
  1. [11]

    Unlike Leh´ericy (2021), in our application the emissions E do not involve any parameters and are only driven by the coarseness of the bins selected in BO

    □ A.3.5 Proof of Theorem 6.1 We start with Theorem 6 from Leh´ericy (2021), assuming Assumptions A.1 through A.13 hold. Unlike Leh´ericy (2021), in our application the emissions E do not involve any parameters and are only driven by the coarseness of the bins selected in BO. O...

    2021

  2. [12]

    Note that |v⟩ ⟨v| = Pn−1 i=0 |ai|2Pi, where Pi ≡ |ui⟩ ⟨ui| , which is the projection operator associated with|ui⟩ . The so called Born rule determines that the probability of the system collapsing state |ui⟩ after measurement equals the square of the magnitude of the amplitude...

    2022

  3. [13]

    θ nc l −1 1 − Pnc l −1 i=0 θi θnc l

    parameters θ to construct a spot volatility transition matrix A which takes the form A =   θ0 . . . θ nc l −1 1 − Pnc l −1 i=0 θi θnc l . . . θ2(nc l −1) 1 − Pi=2(nc l −1) i=nc l θi ... ... ... ... θ(nc l −1) 2 . . . θnc l (nc l −1) 1 − Pi=nc l (nc l −1) i=(nc l −1...

    2025

  4. [291]

    The Variational Quantum Eigensolver: A review of methods and best practices,

    TILLY, J., H. C HEN , S. C AO, D. P ICOZZI , K. S ETIA , Y. LI, E. G RANT , L. W OSSNIG , I. RUNGGER , G. H. B OOTH , AND J. T ENNYSON (2022): “The Variational Quantum Eigensolver: A review of methods and best practices,” Physics Reports, 986, 1–128, The Variational Quantum Ei...

    2022 arXiv

  5. [5056]

    Accelerating Variational Quantum Algorithms Using Circuit Concurrency,

    RESCH , S., A. G UTIERREZ , J. S. H UH, S. B HARADWAJ , Y. E CKERT , G. L OH, M. O SKIN , AND S. TANNU (2021): “Accelerating Variational Quantum Algorithms Using Circuit Concurrency,” . RIVAS, A., AND S. F. H UELGA (2012): Open Quantum Systems. Springer Berlin, Heidelberg. 35 ...

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.