Validating quantum-supremacy experiments with exact and fast tensor network contraction

The quantum circuits that declare quantum supremacy, such as Google Sycamore [Nature \textbf{574}, 505 (2019)], raises a paradox in building reliable result references. While simulation on traditional computers seems the sole way to provide reliable verification, the required run time is doomed with an exponentially-increasing compute complexity. To find a way to validate current ``quantum-supremacy" circuits with more than $50$ qubits, we propose a simulation method that exploits the ``classical advantage" (the inherent ``store-and-compute" operation mode of von Neumann machines) of current supercomputers, and computes uncorrelated amplitudes of a random quantum circuit with an optimal reuse of the intermediate results and a minimal memory overhead throughout the process. Such a reuse strategy reduces the original linear scaling of the total compute cost against the number of amplitudes to a sublinear pattern, with greater reduction for more amplitudes. Based on a well-optimized implementation of this method on a new-generation Sunway supercomputer, we directly verify Sycamore by computing three million exact amplitudes for the experimentally generated bitstrings, obtaining an XEB fidelity of $0.191\%$ which closely matches the estimated value of $0.224\%$. Our computation scales up to $41,932,800$ cores with a sustained single-precision performance of $84.8$ Pflops, which is accomplished within $8.5$ days. Our method has a far-reaching impact in solving quantum many-body problems, statistical problems as well as combinatorial optimization problems where one often needs to contract many tensor networks which share a significant portion of tensors in common.