The aim of this thesis is to study adaptive controllers in the context of a Pro-portional Integral Derivative (PID) controller. The PID controller is tuned via loop shaping techniques to ensure desired robustness and performance characteristics with respect to a target loop shape. There are two problems that this work addresses: Consider a system that is controlled via an adaptive PID controller. If in absence of or under lack of excitation, the system or controller parameters drift to an arbitrary system (that may or may not be stable). Then, once the system gets sufficient ex- citation, there are two questions to be addressed: First, how quickly is the system able to recover to the target system, and in the process of recovery, how large are the transient overshoots and what factors affect the recovery of the drifted system? Second, continuous online adaptation of the controller may not always be necessary (and economical). So, is there a means to monitor the performance of the current controller and determine via robustness conditions whether to continue with the same controller or reject it and adapt to a new controller? Hence, this work is concerned with robust performance monitoring and recovery of an adaptive PID control system that had drifted to another system in absence of sufficient excitation or excessive noise.

Quantum entanglement is a phenomenon in which a group of particles that are either generated, interacting with each other, or close in proximity to each other, have a property that the quantum states of each particle cannot be described independently of the states of the other particles. This phenomenon was initially investigated by Albert Einstein, Boris Poldosky, and Nathan Rosen in their landmark paper known as the EPR paradox, in which Einstein described this behavior as "spooky action at a distance''. This thesis presents a mathematical and theoretical approach in defining quantum entanglement by detecting photons in their entangled state through a set of photon-number-resolving (PNR) detectors and threshold detectors. This theoretical approach is made rigorous by including the notion of a dark count, a phenomenon in which a detector incidentally detects a photon when it should not have been detected. With this dark count model, we define the probabilities of finding a coincidence of such entangled photons through a combination and configuration of PNR detectors and threshold detectors. Then, we find the coincidence probabilities of detecting a single coincidence of photons within the detector system, the total coincidence probabilities of detecting this coincidence with respect to ground truth, and the effective density matrix that characterizes how well each combination and configuration of detectors can detect photon coincidences. By making mathematical and probabilistic assumptions on the distribution of photon types and counts with respect to ground truth, we are able to compute these quantities and analyze their expressions based on a mathematical and conceptual context.

Quantum entanglement, a phenomenon first introduced in the realm of quantum mechanics by the famous Einstein-Podolsky-Rosen (EPR) paradox, has intrigued physicists and philosophers alike for nearly a century. Its implications for the nature of reality, particularly its apparent violation of local realism, have sparked intense debate and spurred numerous experimental investigations. This thesis presents a comprehensive examination of quantum entanglement with a focus on probing its non-local aspects. Central to this thesis is the development of a detailed project document outlining a proposed experimental approach to investigate the non-local nature of quantum entanglement. Drawing upon recent advancements in quantum technology, including the manipulation and control of entangled particles, the proposed experiment aims to rigorously test the predictions of quantum mechanics against the framework of local realism. The experimental setup involves the generation of entangled particle pairs, such as photons or ions, followed by the precise manipulation of their quantum states. By implementing a series of carefully designed measurements on spatially separated entangled particles, the experiment seeks to discern correlations that defy explanation within a local realistic framework.

In this thesis, applications of sparsity, specifically sparse-tensors are motivated in physics.An algorithm is introduced to natively compute sparse-tensor's partial-traces, along with direct implementations in popular python libraries for immediate use. These applications include the infamous exponentially-scaling (with system size) Quantum-Many-Body problems (both Heisenberg/spin-chain-like and Chemical Hamiltonian models). This sparsity aspect is stressed as an important and essential feature in solving many real-world physical problems approximately-and-numerically. These include the original motivation of solving radiation-damage questions for ultrafast light and electron sources.

Dynamical decoupling (DD) is a promising approach to mitigate the detrimental effects that interactions with the environment have on a quantum system. In DD, the finite-dimensional system is rotated about specified axes using strong and fast controls that eliminate system-environment interactions and protect the system fromdecoherence. In this thesis, the framework of DD is theoretically studied, and later it discusses how this framework can be implemented on an infinite-dimensional system that amplifies system components rather than suppressing them through quadrature squeezing operations. It begins by studying the impact of system-environment interactions on a quantum system, and then it analyzes how DD suppresses these interactions. The conditions for protecting a finite-dimensional system through DD are reviewed, and a numerical analysis of the DD conditions for simple systems is conducted. Using bang-bang controls, a framework for decoupling decoherence-inducing components from a general finite-dimensional system is studied. Later, following an overview of schemes that amplify the strength of a quantum signal through reversible squeezing, a theoretical study of Hamiltonian Amplification (HA) for quantum harmonic oscillators is presented. By implementing the DD framework with squeezing operations, HA achieves speed-up in the dynamics of quantum harmonic oscillators, which translates into the strengthening of interactions between harmonic oscillators. Finally, the application of HA in amplifying the third-order nonlinearity in a Kerr medium is proposed to obtain a speed-up in the implementation of controlled phase gates for optical quantum computations. Numerically simulated results show that large amplification in nonlinearity is feasible with sufficient squeezing resources, completing the set of universal quantum gates in optical quantum computing.

Quantum computing is becoming more accessible through modern noisy intermediate scale quantum (NISQ) devices. These devices require substantial error correction and scaling before they become capable of fulfilling many of the promises that quantum computing algorithms make. This work investigates the current state of NISQ devices by implementing multiple classical computing scenarios with a quantum analog to observe how current quantum technology can be leveraged to achieve different tasks. First, quantum homomorphic encryption (QHE) is applied to the quantum teleportation protocol to show that this form of algorithm security is possible to implement with modern quantum computing simulators. QHE is capable of completely obscuring a teleported state with a liner increase in the number of qubit gates O(n). Additionally, the circuit depth increases minimally by only a constant factor O(c) when using only stabilizer circuits. Quantum machine learning (QML) is another potential application of NISQ technology that can be used to modify classical AI. QML is investigated using quantum hybrid neural networks for the classification of spoken commands on live audio data. Additionally, an edge computing scenario is examined to profile the interactions between a quantum simulator acting as a cloud server and an embedded processor board at the network edge. It is not practical to embed NISQ processors at a network edge, so this paradigm is important to study for practical quantum computing systems. The quantum hybrid neural network (QNN) learned to classify audio with equivalent accuracy (~94%) to a classical recurrent neural network. Introducing quantum simulation slows the systems responsiveness because it takes significantly longer to process quantum simulations than a classical neural network. This work shows that it is viable to implement classical computing techniques with quantum algorithms, but that current NISQ processing is sub-optimal when compared to classical methods.