SIMULATING QUANTUM CIRCUITS---UNDERSTANDING THE ROLES OF WIGNER NEGATIVITY, SYMMETRY, AND SYMMETRY BREAKING

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This proposal is on the theory of quantum computational advantage; and the central question it asks is ``How can we harness the quantum for computation?''. The idea of the algorithm dates back to the Babylonian mathematicians, more than 4,000 years ago. The idea of quantum algorithms, i.e., using quantum mechanics to solve complex mathematical problems, is very recent in comparison. The first steps were made by Deutsch, Feynman and Shor, only about 30 years ago. Correspondingly, we don't understand quantum algorithms very well yet. Powerful examples such as Shor's and Grover's algorithms are known, but they are few and far between.In this proposal, we seek to better understand the quantum mechanical foundation of quantum algorithms, and to extend the quantum computational tool set. To this end, we scrutinize various known core quantum mechanical properties for their role in quantum speedup, such as Wigner function negativity and contextuality. We also investigate areas of phenomenology where novel origins of quantum speedup may arise, such as the Lambda polytopes and computational phases of quantum matter.