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arXiv:2212.03796

Implementation and Learning of Quantum Hidden Markov Models

Vanio Markov, Vladimir Rastunkov, Deshmukh, Amol, Daniel Fry, Charlee Stefanski

Published Dec 7, 2022• 5 citations
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Abstract

In this article, we use the theory of quantum channels and open quantum systems to provide an efficient unitary characterization of a class of stochastic generators known as quantum hidden Markov models (QHMMs). By utilizing the unitary characterization, we demonstrate that any QHMM can be implemented as a quantum circuit with mid-circuit measurement. We prove that QHMMs are more compact and more expressive definitions of stochastic process languages compared to the equivalent classical hidden Markov models (HMMs). Starting with the formulation of QHMMs as quantum channels, we employ Stinespring's construction to represent these models as unitary quantum circuits with mid-circuit measurement. By utilizing the unitary parameterization of QHMMs, we define a formal QHMM learning model. The model formalizes the empirical distributions of target stochastic process languages, defines hypothesis space of quantum circuits, and introduces an empirical stochastic divergence measure - hypothesis fitness - as a success criterion for learning. We demonstrate that the learning model has a smooth search landscape due to the continuity of Stinespring's dilation. The smooth mapping between the hypothesis and fitness spaces enables the development of efficient heuristic and gradient descent learning algorithms. We propose two practical learning algorithms for QHMMs. The first algorithm is a hyperparameter-adaptive evolutionary search. The second algorithm learns the QHMM as a quantum ansatz circuit using a multi-parameter non-linear optimization technique.

DOI
10.48550/arxiv.2212.03796

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In this article, we use the theory of quantum channels and open quantum systems to provide an efficient unitary characterization of a class of stochastic generators known as quantum hidden Markov models (QHMMs). By utilizing the unitary characterization, we demonstrate that any QHMM can be implemented as a quantum circuit with mid-circuit measurement. We prove that QHMMs are more compact and more expressive definitions of stochastic process languages compared to the equivalent classical hidden Markov models (HMMs). Starting with the formulation of QHMMs as quantum channels, we employ Stinespring's construction to represent these models as unitary quantum circuits with mid-circuit measurement. By utilizing the unitary parameterization of QHMMs, we define a formal QHMM learning model. The model formalizes the empirical distributions of target stochastic process languages, defines hypothesis space of quantum circuits, and introduces an empirical stochastic divergence measure - hypothesis fitness - as a success criterion for learning. We demonstrate that the learning model has a smooth search landscape due to the continuity of Stinespring's dilation. The smooth mapping between the hypothesis and fitness spaces enables the development of efficient heuristic and gradient descent learning algorithms. We propose two practical learning algorithms for QHMMs. The first algorithm is a hyperparameter-adaptive evolutionary search. The second algorithm learns the QHMM as a quantum ansatz circuit using a multi-parameter non-linear optimization technique.
Hidden Quantum Markov Models (HQMMs) can be thought of as quantum probabilistic graphical models that can model sequential data. We extend previous work on HQMMs with three contributions: (1) we show how classical hidden Markov models (HMMs) can be simulated on a quantum circuit, (2) we reformulate HQMMs by relaxing the constraints for modeling HMMs on quantum circuits, and (3) we present a learning algorithm to estimate the parameters of an HQMM from data. While our algorithm requires further optimization to handle larger datasets, we are able to evaluate our algorithm using several synthetic datasets. We show that on HQMM generated data, our algorithm learns HQMMs with the same number of hidden states and predictive accuracy as the true HQMMs, while HMMs learned with the Baum-Welch algorithm require more states to match the predictive accuracy.
Extending classical probabilistic reasoning using the quantum mechanical view of probability has been of recent interest, particularly in the development of hidden quantum Markov models (HQMMs) to model stochastic processes. However, there has been little progress in characterizing the expressiveness of such models and learning them from data. We tackle these problems by showing that HQMMs are a special subclass of the general class of observable operator models (OOMs) that do not suffer from the \emph{negative probability problem} by design. We also provide a feasible retraction-based learning algorithm for HQMMs using constrained gradient descent on the Stiefel manifold of model parameters. We demonstrate that this approach is faster and scales to larger models than previous learning algorithms.
In this work, we extend the idea of quantum Markov chains (Gudder in J Math Phys 49(7):072105 [ 3 ]) in order to propose quantum hidden Markov models (QHMMs). For that, we use the notions of transition operation matrices and vector states, which are an extension of classical stochastic matrices and probability distributions. Our main result is the Mealy QHMM formulation and proofs of algorithms needed for application of this model: Forward for general case and Vitterbi for a restricted class of QHMMs. We show the relations of the proposed model to other quantum HMM propositions and present an example of application.
We investigate learning models for implementing arbitrary unitary operations and perform quantum machine learning tasks. First, we discuss efficient constructions for building unitary transformations in U(d), thus providing upper bounds on the minimum time required to implement them. Second, we study the hardness of learning unitary operations in U(d) via gradient descent on the time parameters of an alternating operator sequence, and numerically find that, despite the nonconvexity of the loss landscape, gradient descent always converges to the target unitary when the sequence contains d2 or more parameters. Third, we introduce the first quantum boosting algorithm for converting a weak quantum learner into a strong accurate quantum learner for a Boolean concept class C; thereby achieving a quadratic quantum improvement over classical AdaBoost in terms of VC(C).
Among several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known as the positive realization problem (PRP) [Benvenuti,Farina(2004)] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory [Luenberger(1979)]. We consider the scenario where the latent variables are quantum (e.g., quantum states of a finite-dimensional system), and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument-if any-yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is however devoid from any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [Guta(2011), Guta&Yamamoto(2013)].
A new class of formal latent-variable stochastic processes called hidden quantum models (HQM's) is defined in order to clarify the theoretical foundations of ion channel signal processing. HQM's are based on quantum stochastic processes which formalize time-dependent observation. They allow the calculation of autocovariance functions which are essential for frequency-domain signal processing. HQM's based on a particular type of observation protocol called independent activated measurements are shown to to be distributionally equivalent to hidden Markov models yet without an underlying physical Markov process. Since the formal Markov processes are non-physical, the theory of activated measurement allows merging energy-based Eyring rate theories of ion channel behavior with the more common phenomenological Markov kinetic schemes to form energy-modulated quantum channels. Using the simplest quantum channel model consistent with neuronal membrane voltage-clamp experiments, activation eigenenergies are calculated for the Hodgkin-Huxley K+ and Na+ ion channels. It is also shown that maximizing entropy under constrained activation energy yields noise spectral densities approximating S(f)∼1/fαS(f) \sim 1/f^αS(f)∼1/fα, thus offering a biophysical explanation for the ubiquitous 1/f1/f1/f-type in neurological signals.