Estimating the randomness of quantum circuit ensembles up to 50 qubits

Abstract

We develop numerical protocols for estimating the frame potential, the 2-norm distance between a given ensemble and the exact Haar randomness, using the 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 ansätze 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 ansätze, 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.

Random Quantum Circuits

“…have also been introduced in order to uncover the precise sense in which RMT can describe the far-from-equilibrium dynamics of a quantum many-body system (223,224). Turning to the case in which W describes a quantum circuit composed of local unitary gates, K(t) becomes a two-layer circuit with periodic boundary conditions in the time direction and can be written as a double sum over periodic Feynman trajectories, one in each layer.…”