  Given black-box access to the input and output systems, we develop the first
efficient quantum causal order discovery algorithm with polynomial query
complexity with respect to the number of systems. We model the causal order
with quantum combs, and our algorithms output the order of inputs and outputs
that the given process is compatible with. Our algorithm searches for the last
input and the last output in the causal order, removes them, and iteratively
repeats the above procedure until we get the order of all inputs and outputs.
Our method guarantees a polynomial running time for quantum combs with a low
Kraus rank, namely processes with low noise and little information loss. For
special cases where the causal order can be inferred from local observations,
we also propose algorithms that have lower query complexity and only require
local state preparation and local measurements. Our algorithms will provide
efficient ways to detect and optimize available transmission paths in quantum
communication networks, as well as methods to verify quantum circuits and to
discover the latent structure of multipartite quantum systems.
