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With the advent of quantum processors exceeding $100$ qubits and the high engineering complexities involved, there is a need for holistically benchmarking the processor to have quality assurance. Linear cross-entropy benchmarking (XEB) has been used extensively for systems with $50$ or more qubits but is fundamentally limited in scale due to the exponentially large computational resources required for classical simulation. In this work we propose conducting linear XEB with Clifford circuits, a scheme we call Clifford XEB. Since Clifford circuits can be simulated in polynomial time, Clifford XEB can be scaled to much larger systems. To validate this claim, we run numerical simulations for particular classes of Clifford circuits with noise and observe exponential decays. When noise levels are low, the decay rates are well-correlated with the noise of each cycle assuming a digital error model. We perform simulations of systems up to 1,225 qubits, where the classical processing task can be easily dealt with by a workstation. Furthermore, using the theoretical guarantees in Chen et al. (arXiv:2203.12703), we prove that Clifford XEB with our proposed Clifford circuits must yield exponential decays under a general error model for sufficiently low errors. Our theoretical results explain some of the phenomena observed in the simulations and shed light on the behavior of general linear XEB experiments.