CAREER: Algebraic Approach to the Design of Novel Quantum Algorithms

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The investigator seeks to discover novel ways of harnessing quantum phenomena to advance the computational capabilities of information processing devices. This fundamental research leads to new efficient quantum algorithms for problems that cannot be solved efficiently by any known classical algorithm. These quantum algorithms make it possible to solve more efficiently larger instances of computationally hard real-life problems, such as those arising in optimization theory, machine learning, and signal processing. A second component of the research focuses on developing new quantum algorithms for algebraic problems which helps to design more secure cryptographic protocols that are immune to quantum attacks. This is important as today's encryption schemes have to withstand attacks employing more advanced computational devices in the future. More fundamentally, this research provides a deeper understanding of the computational capabilities of information processing devices operating in the quantum regime. As the end of scalability of conventional silicon-based devices approaches, it is essential to explore such novel paradigms of information technologies.The investigator finds new quantum algorithms for computing with large matrices. This research leads to new applications in machine learning, signal processing, and data analysis by achieving speed-ups for principal component analysis and kernel methods. These quantum algorithms also find applications in optimization theory and topology by providing better approximate solutions for NP-hard optimization and counting problems and more accurate evaluations of topological invariants, respectively. A second component of the research focuses on quantum algorithms for identifying group-theoretic and non-linear structures. The research goal is to extend them to a broad collection of structures arising in number theory and algebraic geometry, especially those that are relevant to cryptographic applications.