Quantum-Inspired Classical HMMs

• Quantum-inspired classical HMMs are classical algorithms that mimic quantum probabilistic frameworks to enhance state inference and compress statistical memory.
• They employ techniques like non-negative matrix factorization and operator-based Hilbert space embeddings to efficiently recover hidden states.
• Applications range from qubit readout to financial time series filtering, offering improved fidelity and reduced computational overhead.

Quantum-Inspired Classical Hidden Markov Models (HMM)

Quantum-inspired classical hidden Markov models (HMMs) represent a class of classical algorithms and inference techniques that syntactically and structurally mimic concepts developed in quantum hidden Markov models (QHMMs) and hidden quantum Markov models (HQMMs), leveraging ideas such as non-negative matrix factorization, operator-based learning, projective updates, and statistical memory compression. Such methods retain efficient implementability on classical hardware while aiming to capture modeling or computational advantages hinted at by quantum probabilistic frameworks. The approaches described below combine classical Markov process structures with constrained operator representations, spectral decompositions, and iterative optimization schemes reminiscent of quantum dynamical and inferential procedures.

1. Statistical Learning via Non-negative Matrix Factorization

A structurally distinct method for learning classical HMMs utilizes non-negative matrix factorization (NMF) of high-order empirical Markovian statistics (0809.4086). For a given observation sequence of length $T$, empirical joint statistics are “compressed” into a matrix $F^{(p,s)}$ where each entry approximates the empirical probability $P(V|U)$ of observing a suffix string $V$ after a prefix string $U$ of specified lengths $s$ and $p$, respectively. This matrix admits (under the HMM hypothesis) an approximate factorization:

$F^{(p,s)} \approx C D,$

with $C$ row-stochastic, containing per-prefix probability distributions over $N$ hidden states ($c_{u,k} \approx P(S_k|U,p,\lambda)$), and $D$ row-stochastic, encoding the state-dependent emission distributions for suffixes ($D_{k,\cdot} \approx P(\cdot|S_k,s,\lambda)$). The factors are optimized iteratively by minimizing the I-divergence, an analogue of KL divergence:

$$D_\text{ID}(K \| W) = \sum_{ij} \left[ K_{ij} \log\frac{K_{ij}}{W_{ij}} - K_{ij} + W_{ij} \right]$$

between the empirical matrix and its factorization. Unlike Baum–Welch EM, this approach works on compressed statistics, is naturally order-selecting (via SVD on $F^{(p,s)}$), and is computationally favorable in large-scale or streaming settings.

NMF shares inspiration with quantum decompositions—such as expressing a density matrix as a mixture of pure states—though here the constraint is probabilistic rather than physically quantum. Factor $D$ is analogous to a statistical mixture generator, reminiscent of quantum state tomography (0809.4086).

2. Operator-Based and Hilbert Space Embeddings

Probability Bracket Notation (PBN), formally analogous to Dirac's notation in quantum mechanics, can be exploited to unify HMM, visible Markov model (VMM), and factorial HMM (FHMM) representations as operator chains in Hilbert space (Wang, 2012). For classical models, the Markov Sequence Projector (MSP) $U(Q) = U(q_T) U(q_{T-1})\ldots U(q_1)$ acts as a sequence of projectors encoding a path through hidden states. The evolution equations for the distribution and joint/inference formulas can be written as operator products, highlighting a direct parallel to quantum sequence evolution and measurement.

In the PBN framework, transformations between hidden (state) and observable (output) basis correspond to classical emission matrices, but can be generalized to support quantum-like basis transformations, sequence projections, and even continuous-time or feedback-augmented models. This operator formalism opens the way for classical models that incorporate features inspired by quantum process dynamics—unitary evolution, non-commuting projections, and rich basis inter-conversions—as building blocks of future algorithms (Wang, 2012).

3. Spectral, Memory, and Compression Insights

The quantum-to-classical operational analogy provides a route for memory compression and minimality analysis in sequence models. In classical modeling, the ε-machine is the strongly minimal predictive model, minimizing all Rényi entropy-based memory measures via state majorization (Loomis et al., 2018). Quantum memory models allow for non-orthogonal, overlapping state representations, leading to even further memory compression—however, there is generally no universal quantum analog to the ε-machine’s strong minimality, and different quantum memory measures (e.g., von Neumann–Rényi entropies) may yield different optima.

Recent work has established spectral invariants—the nonzero eigenvalues of the transfer operator $$\mathbb{E}_{AB} = \sum_x \mathcal{A}_x^A \otimes \mathcal{A}_x^B$$ associated with a process. These spectral sets tightly constrain the minimal internal memory (classical or quantum) for any generating model (Zonnios et al., 17 Dec 2024). For a process with spectral set $\Lambda_x$:

• Minimal quantum memory: $$c_Q(\overrightarrow{\mathbf{X}}) \geq \log \lceil |\Lambda_x|^{1/4} \rceil$$
• Minimal classical memory: $$c_C(\overrightarrow{\mathbf{X}}) \geq \log \lceil |\Lambda_x|^{1/2} \rceil$$ showing that quantum models can quadratically compress memory compared to strictly incoherent (classical) models. This bound is not merely formal: for certain processes, quantum models demonstrably operate below classical memory limits, purely due to quantum coherence encoding.

Moreover, quantum-inspired memory compression leads to both reduced computational storage and superior thermodynamic efficiency in “pattern generation” (Elliott, 2021). The link is direct: quantum implementations with non-orthogonal memory states require less work per step in generating a process, whenever the classical implementation suffers modularity-induced thermal dissipation.

4. Quantum-Inspired Filtering and Estimation Algorithms

Quantum-inspired classical HMMs have been applied to real-world tasks including qubit readout (Martinez et al., 2020) and financial time series filtering (Ghysels et al., 28 Jul 2025). In quantum-inspired qubit readout, a classical HMM is constructed reflecting the physics of the measurement process: discrete time-binning of IQ signals, explicit modeling of quantum decay as Markovian transitions, and unsupervised learning (Baum–Welch) to extract state relaxations, priors, and emission statistics. The model achieves higher assignment fidelity than Gaussian or SVM classifiers and removes the dependency on readout time optimization, an essential advantage in quantum device characterization.

In stochastic volatility estimation in finance, the underlying continuous-time SDE for volatility is discretized into a binned state Markov chain (the “quantum-inspired” step), enabling standard HMM machinery—

$$\mathcal{X}_t = \frac{A \mathcal{X}_{t-1} \odot E_t}{\mathbf{1}^\top[A \mathcal{X}_{t-1} \odot E_t]}$$

and explicit likelihood calculation. The non-asymptotic Kullback-Leibler divergence bounds for quantum models are provably tighter relative to classical HMMs with identical approximation quality, an efficiency gain rooted in the quantum encoding’s quadratic compression of state space (Ghysels et al., 28 Jul 2025). Thus, even without quantum hardware, quantum-inspired classical HMMs provide practical accuracy and resource benefits.

5. Applications and Algorithmic Implications

Quantum-inspired classical HMMs and related operator-based methods have broad applicability:

• Time-series modeling: Enhanced expressiveness and compression in sequential data modeling, capturing long-range dependencies and richer output correlations (O`Neill et al., 2012, Adhikary et al., 2019).
• Failure scenario generation in PSA: HQMMs are employed for generative modeling of system failures, outperforming classical HMMs in description accuracy (DA) and leveraging operator-style learning with constraints inspired by quantum information theory (Zaiou et al., 2022).
• Pattern recognition and control: High-fidelity discrimination in quantum device readout, where Markovian dynamics and emission models are physically motivated and robust (Martinez et al., 2020).
• Directed memory compression: Algorithmic techniques for merging and approximating state spaces in classical models take explicit guidance from quantum spectral and coherence theory (Zonnios et al., 17 Dec 2024).

Algorithmic frameworks emerging from this cross-pollination include:

• Operator-based iterative learning (gradient descent or evolutionary methods on Stiefel manifolds, with model parameters represented as constrained operator matrices).
• Kernelized Hilbert space embeddings mapping probability distributions to reproducing kernel Hilbert spaces, linking inference to quantum sum rules and kernel Bayesian regression (Srinivasan et al., 2018).
• Quantum-inspired majorization-based optimization and mixture modeling for memory minimality in classical HMMs (Loomis et al., 2018).
• Spectrally-guided model selection, leveraging explicit transfer operator eigenvalue computation to determine achievable compression and state-space reduction (Zonnios et al., 17 Dec 2024).

6. Numerical Examples and Performance Metrics

Extensive numerical experiments validate these quantum-inspired classical approaches:

• On deterministic and non-deterministic HMMs, NMF-based learning reconstructs transition matrices with high accuracy, matching or exceeding reference methods like CSSR.
• In failure scenario modeling, HQMM-derived generators obtain significantly higher DA scores than classical HMMs. The models can distinguish probable from improbable failure sequences via dual-model strategies and likelihood comparison (Zaiou et al., 2022).
• In qubit readout, quantum-inspired HMMs yield assignment fidelities exceeding 96.5% and approach theoretical limits, with strong robustness to the choice of measurement duration (Martinez et al., 2020).
• Quantum-inspired HMMs in volatility estimation deliver quadratic state-space reduction for maintaining equivalent likelihood KL divergence; quantum bounds are provably tighter for finite sample sizes (Ghysels et al., 28 Jul 2025).

7. Summary and Outlook

Quantum-inspired classical HMMs demonstrate that many of the representational, inferential, and computational gains associated with quantum models—memory compression, expressiveness, robust learning, and thermodynamic efficiency—can be at least partially realized on classical architectures through the adoption of operator-based, spectral, and non-orthogonal frameworks. Spectral invariants, operator constraints, and memory minimality criteria from quantum probability guide the design of state-efficient classical models and learning algorithms in a broad range of sequential data applications. The methodology underscores an overview of quantum and classical methods, where deep structural parallels—not hardware quantum effects alone—drive advances in model design, inference, and computational resource management.