The paper discusses the characterization of quantum supremacy in near-term devices, focusing on the task of sampling from the output distributions of (pseudo-)random quantum circuits. The authors argue that classical simulation of these circuits is exponentially costly, making it a challenging computational task. They extend previous results in computational complexity to formally show that this sampling task must take exponential time on a classical computer. Using extensive supercomputer simulations, they study the convergence to the chaotic regime, modeling circuits with up to 42 qubits. They propose cross entropy as a benchmark for quantum circuits, which approximates the circuit fidelity. The paper also discusses the robust conditions needed to confidently claim quantum supremacy, including the estimation of the cross entropy difference between experimental and theoretical distributions. The authors conclude that while chaotic states are highly sensitive to errors, quantum supremacy can be achieved with approximately fifty superconducting qubits.The paper discusses the characterization of quantum supremacy in near-term devices, focusing on the task of sampling from the output distributions of (pseudo-)random quantum circuits. The authors argue that classical simulation of these circuits is exponentially costly, making it a challenging computational task. They extend previous results in computational complexity to formally show that this sampling task must take exponential time on a classical computer. Using extensive supercomputer simulations, they study the convergence to the chaotic regime, modeling circuits with up to 42 qubits. They propose cross entropy as a benchmark for quantum circuits, which approximates the circuit fidelity. The paper also discusses the robust conditions needed to confidently claim quantum supremacy, including the estimation of the cross entropy difference between experimental and theoretical distributions. The authors conclude that while chaotic states are highly sensitive to errors, quantum supremacy can be achieved with approximately fifty superconducting qubits.