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CRC Press eBooks

Quantum neural networks for machine learning applied to the tracking and control of the dynamics of stochastic transmission lines

Harish Parthasarathy, Arti Vaish

Published Aug 18, 2023• 1 citation
Publisher

Abstract

In most of the machine learning problems based on classical neural networks, we try to identify a nonlinear system which may or may not have memory by adapting the weights of the neural network so that for the same input process applied to the system as well as to the neural network, the network output matches that of the true system output in the mean square sense or in the sense of some other metric on the space of output processes. If the network is feed forward, then this model can be used to simulate a memory-less nonlinear system. If however the actual system has memory and is described by a stochastic difference equation for its state followed by an output equation that is a function of the state, then we can use a recurrent neural network (RNN) whose state dynamics are described by a nonlinear difference equation involving weights as parameters and the state and weights of the RNN can be estimated from noisy measurements of the actual system output by means of the extended Kalman filter. If however, we are interested in estimating the joint probability density of a vector valued process like the transmission line current and voltage at different points along the line at all times, then a classical neural network will not work. This is because a probability density has several constraints like non-negativity and integrability to unity, and so on, and a classical neural network does not take care of these constraints.

DOI
10.1201/9781003436461-4

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In most of the machine learning problems based on classical neural networks, we try to identify a nonlinear system which may or may not have memory by adapting the weights of the neural network so that for the same input process applied to the system as well as to the neural network, the network output matches that of the true system output in the mean square sense or in the sense of some other metric on the space of output processes. If the network is feed forward, then this model can be used to simulate a memory-less nonlinear system. If however the actual system has memory and is described by a stochastic difference equation for its state followed by an output equation that is a function of the state, then we can use a recurrent neural network (RNN) whose state dynamics are described by a nonlinear difference equation involving weights as parameters and the state and weights of the RNN can be estimated from noisy measurements of the actual system output by means of the extended Kalman filter. If however, we are interested in estimating the joint probability density of a vector valued process like the transmission line current and voltage at different points along the line at all times, then a classical neural network will not work. This is because a probability density has several constraints like non-negativity and integrability to unity, and so on, and a classical neural network does not take care of these constraints.
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.
Recurrent neural networks play an important role in both research and industry. With the advent of quantum machine learning, the quantisation of recurrent neural networks has become recently relevant. We propose fully quantum recurrent neural networks, based on dissipative quantum neural networks, capable of learning general causal quantum automata. A quantum training algorithm is proposed and classical simulations for the case of product outputs with the fidelity as cost function are carried out. We thereby demonstrate the potential of these algorithms to learn complex quantum processes with memory in terms of the exemplary delay channel, the time evolution of quantum states governed by a time-dependent Hamiltonian, and high- and low-frequency noise mitigation. Numerical simulations indicate that our quantum recurrent neural networks exhibit a striking ability to generalise from small training sets.
Artificial neural networks, usually just called neural networks, computing systems indefinitely inspired by the biological neural networks and they are extensive in both research as well as industry. It is critical to design quantum Neural Networks for complete quantum learning tasks. In this project, we suggest a computational neural network model based on principles of quantum mechanics which form a quantum feed-forward neural network proficient in universal quantum computation. This structure takes input from one layer of qubits and drives that input onto another layer of qubits. This layer of qubits evaluates this information and drives on the output to the next layer. Eventually, the path leads to the final layer of qubits. The layers do not have to be of the same breadth, meaning they need not have the same number of qubits as the layer before and/or after it. This assembly is trained on which path to take identical to classical ANN. The intended project can be compiled by the subsequent points provided here: 1. The expert training of the quantum neural network utilizing the fidelity as a cost function, providing both conventional and efficient quantum implementations. 2. Use of methods that enable quick optimization with reduced memory requirements. 3. Benchmarking our proposal for the quantum task of learning an unknown unitary and find extraordinary generality and a remarkable sturdiness to noisy training data.
This paper starts with a basic formulation of a distributed parameter transmission line with memristor per unit length in series and also memristor per unit length in parallel apart from the other standard distributed parameters. The result of having memristors distributed along the line is a nonlinear effect in the form of coupling between charge and current and coupling between charge and voltage. These nonlinear effects result in the generation of an infinite number of higher harmonic components along the line even when the input voltage is of a fixed frequency or a finite superposition of voltages of different frequencies. Taking into account random white noise line loading effects, we formulate the resulting line equations based on KCL and KVL in the form of an infinite dimensional nonlinear stochastic differential equation. We study the problem of quantizing the transmission line dynamics using the well known Hudson-Parthasarathy theory of quantum stochastic differential equations derived from the Hudson-Parthasarathy noisy Schrodinger equation constructed from an appropriate transmission line Hamiltonian and Lindblad operators. We then conclude by discussing the effect of quantum stochastic noise and quantum stochastic supersymmetric noise on the action functional of a field theory and how such a model can be used to obtain corrections to the quantum effective action functional or more precisely, to the TPCP map that evolves an initially mixed state of the field to another mixed state after a finite duration. Some remarks on training and testing of quantum neural networks based on the Belavkin quantum filter are presented. This involves constructing a quantum evolution of a mixed state in such a way that the evolving pdf of a system observable will be obtained from the conditional expectation of the noisy quantum state given non-demolition measurements that are strongly correlated with actually measured signals.
This book examines quantum neural networks through renormalization techniques, supersymmetric field theory, and noisy harmonic oscillator systems. The book's analysis covers adaptive beamforming applications, brain modeling, gravitational control mechanisms, and mixed-state dynamics in superstring theory, and also includes: Comprehensive analysis of quantum neural networks through renormalization techniques and supersymmetric field theory applications in computational modeling Investigation of quantum field dynamics with noise integration, filtering mechanisms, and scattering processes in curved spacetime environments Study of adaptive beamforming methodologies combined with quantum neural networks for brain modeling and evolving field system applications Examination of mixed-state dynamics in superstring theory frameworks with emphasis on quantum noisy fields and supersymmetric effects Analysis of extended Kalman filter integration with quantum neural networks for transmission line control and field estimation optimization The work explores extended Kalman filter methodologies for transmission line control, field estimation, and symmetry-broken dynamics in signal processing systems for advanced computational modeling applications. This title has been co-published with Manakin Press. T&F does not sell or distribute the print editions in Bangladesh, Bhutan, India, Nepal, Pakistan, and Sri lanka.