14 July 2023 Fri 4 pm

The randomness of a quantum system is often measured by the emergence of physical properties such as thermalization, large entanglements, and unitary designs. In this talk, we discuss relations between those properties using parameterized quantum circuits (PQCs). We first introduce the instantaneous quantum polynomial-time (IQP) circuit, widely studied for quantum advantage in a sampling task. Using the model with a varying density of 2-qubit gates, we show that a parameter condition for anticoncentration, a necessary condition for a unitary 2-design and sampling complexity, is distinguished from classical simulability or the Porter-Thomas distribution. We next study randomness in the Hamiltonian variational Ansatz (HVA), which is constructed using a local Hamiltonian. Randomly initialized PQCs often have barren plateaus, a consequence of unitary 2-design and characterized by exponentially small gradients, limiting the trainability of the model beyond tens of qubits. After observing that the thermalization does not imply barren plateaus, we find a parameter condition that the HVA is well approximated by a local Hamiltonian evolution. Thus the HVA within this parameter condition is free from exponentially small gradients. Our two examples show that different quantum randomness measures do not coincide, and a quantum system can show different fine-grained complexity phenomena.

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