Limitations of Noisy Quantum Devices in Computational and Entangling Power

Finding solid and practical quantum advantages via noisy quantum devices without error correction is a critical but challenging problem. Conversely, comprehending the fundamental limitations of the state-of-the-art is equally crucial. In this work, we consider the class of strictly contractive unital noise and derive its analytical representation by decomposition. Under such noise, we observe the polynomial-time indistinguishability of $n$-qubit devices from random coins when circuit depths exceed $\Omega(\log(n))$. Even with classical processing, we demonstrate the absence of computational advantage in polynomial-time algorithms with super-logarithmic noisy circuit depths. These results impact variational quantum algorithms, error mitigation, and quantum simulation with polynomial depth. Furthermore, we consider noisy quantum devices with a restricted gate topology. For one-dimensional noisy qubit circuits, we rule out super-polynomial quantum advantages in all-depth regimes. We also establish upper limits on entanglement generation: $O(\log(n))$ for one-dimensional circuits and $O(\sqrt{n} \log(n))$ for two-dimensional circuits. Our findings underscore the computational capacity and entanglement scalability constraints in noisy quantum devices.