Estimating the frame potential of large-scale quantum circuit sampling using tensor networks up to 50 qubits

We develop numerical protocols for estimating the frame potential, the 2-norm distance between a given ensemble and the exact Haar randomness, using the \texttt{QTensor} platform. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high performing using CPU and GPU parallelism. We apply the above methods to two problems: the Brown-Susskind conjecture, with local and parallel random circuits in terms of the Haar distance and the approximate $k$-design properties of the hardware efficient ans{\"a}tze in quantum machine learning, which induce the barren plateau problem. We estimate frame potentials with these ensembles up to 50 qubits and $k=5$, examine the Haar distance of the hardware-efficient ans{\"a}tze, and verify the Brown-Susskind conjecture numerically. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.