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It has been recently proposed that the naive semiclassical prediction ofnon-unitary black hole evaporation can be understood in the fundamentaldescription of the black hole as a consequence of ignorance of high-complexityinformation. Validity of this conjecture implies that any algorithm which ispolynomially bounded in computational complexity cannot accurately reconstructthe black hole dynamics. In this work, we prove that such bounded quantumalgorithms cannot accurately predict (pseudo)random unitary dynamics, even ifthey are given access to an arbitrary set of polynomially complex observablesunder this time evolution; this shows that "learning" a (pseudo)random unitaryis computationally hard. We use the common simplification of modeling blackholes and more generally chaotic systems via (pseudo)random dynamics. Thequantum algorithms that we consider are completely general, and their attemptedguess for the time evolution of black holes is likewise unconstrained: it neednot be a linear operator, and may be as general as an arbitrary (e.g.decohering) quantum channel.

The Complexity of Learning (Pseudo)random Dynamics of Black Holes and Other Chaotic Systems