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Explore the intricate relationships between various measures of quantum randomness in parameterized quantum circuits (PQCs) through this comprehensive lecture. Delve into the connections between thermalization, large entanglements, and unitary designs in quantum systems. Examine the instantaneous quantum polynomial-time (IQP) circuit and its role in demonstrating quantum advantage in sampling tasks. Investigate how varying the density of 2-qubit gates affects anticoncentration, classical simulability, and the Porter-Thomas distribution. Analyze the Hamiltonian variational Ansatz (HVA) and its implications for randomness in quantum systems. Discover the conditions under which HVA can avoid barren plateaus and maintain trainability beyond tens of qubits. Gain insights into the fine-grained complexity phenomena in quantum systems and understand why different quantum randomness measures may not always align.
