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We consider the problem of classical strong (amplitude-wise) simulation of $n$ -qubit quantum circuits, and identify a subclass of simulators we call monotone. This subclass encompasses almost all prominent simulation techniques. We prove an unconditional (i.e. without relying on any complexity-theoretic assumptions) and explicit $(n-2)(2^{n-3}-1)$ lower bound on the running time of simulators within this subclass. Assuming the Strong Exponential Time Hypothesis (SETH), we further remark that a universal simulator computing any amplitude to precision $2^{-n}/2$ must take at least $2^{n - o(n)}$ time. We then compare strong simulators to existing SAT solvers, and identify the time-complexity below which a strong simulator would improve on state-of-the-art general SAT solving. Finally, we investigate Clifford+ $T$ quantum circuits with $t~T$ -gates. Using the sparsification lemma, we identify a time complexity lower bound of $2^{2.2451\times 10^{-8}t}$ below which a strong simulator would improve on state-of-the-art 3-SAT solving. This also yields a conditional exponential lower bound on the growth of the stabilizer rank of magic states. [ABSTRACT FROM AUTHOR]

A function computation problem over a directed acyclic network has been considered in the literature, where a sink node is required to compute a target function correctly with the inputs arbitrarily generated at multiple source nodes. The network links are error free but capacity limited, and the intermediate nodes perform network coding. The computing rate of a network code is the average number of times that the target function is computed for one use of the network, i.e., each link in the network is used at most once. In the existing papers, two cut-set bounds were proposed on the computing rate. However, we in this paper show that these bounds are not valid for general network function computation problems. We analyze the reason of the invalidity and propose a general cut-set bound by using a new equivalence relation associated with the inputs of the target function. Moreover, some results in the existing papers were proved by applying the invalid upper bound. We also justify the validity of these results. [ABSTRACT FROM AUTHOR]