Challenges and opportunities in the supervised learning of quantum circuit expectation values

Recently, deep neural networks have been proven capable of predicting output expectation values of certain random quantum circuits via a supervised learning approach. Here we investigate the potential of this possible approach to the emulation of quantum circuits, testing both its limitations and the successful applications where it might outperform conventional direct simulation methods. The testbeds we consider are circuits often employed in variational quantum algorithms, featuring layers of cnot gates alternated with single-qubit random rotations. On the one hand, we find that the computational cost of supervised learning scales exponentially with the interlayer variance of the random angles. This allows for entering a promising regime for quantum advantage, where quantum computers could easily outperform classical neural networks. On the other hand, circuits featuring only interqubit angle variations are easily emulated. In fact, thanks to a suitable scalable design, the trained networks accurately predict the expectation values of larger and deeper circuits than those used for training, even reaching circuit sizes, which, as we numerically show, are computationally intractable for the most common simulation libraries of state-vector and tensor-network algorithms. A repository of test data in the intractable regime is provided. We also analyze the most common metrics for the entanglement content, the expressibility, and the classical computational cost of quantum circuits, finding that they do not distinguish the easy from the hard circuit configurations.