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Quantum computing devices have been rapidly developed in the past decade.Tremendous efforts have been devoted to finding quantum advantages for usefulbut classically intractable problems via current noisy quantum devices withouterror correction. It is important to know the fundamental limitations of noisyquantum devices with the help of classical computers. For computation withgeneral classical processing, we show that noisy quantum devices with a circuitdepth of more than $O(\log n)$ provide no advantages in any quantum algorithms.This rigorously rules out the possibility of implementing well-known quantumalgorithms, including Shor's, Grover's, Harrow-Hassidim-Lloyd, and linear-depthvariational algorithms. Then, we study the maximal entanglement that noisyquantum devices can produce under one- and two-dimensional qubit connections.In particular, for a one-dimensional qubit chain, we show an upper bound of$O(\log n)$. This finding highlights the restraints for quantum simulation andscalability regarding entanglement growth. Additionally, our result sheds lighton the classical simulatability in practical cases.

Limitations of Noisy Quantum Devices in Computational and Entangling Power