Black Hole Complementarity and the Harlow-Hayden Conjecture

Abstract

Black hole complementarity, as originally formulated in the 1990's by Preskill, 't Hooft, and myself is now being challenged by the Almheiri-Marolf-Polchinski-Sully firewall argument. The AMPS argument relies on an implicit assumption---the ``proximity postulate---which says that the interior of a black hole must be constructed from degrees of freedom that are physically near the black hole. The proximity postulate manifestly contradicts the idea that interior information is redundant with information in Hawking radiation, which is very far from the black hole. AMPS argue that a violation of the proximity postulate would lead to a contradiction in a thought-experiment in which Alice distills the Hawking radiation and brings a bit back to the black hole. According to AMPS the only way to protect against the contradiction is for a firewall to form at the Page time. But the measurement that Alice must make, is of such a fine-grained nature that carrying it out before the black hole evaporates may be impossible. Harlow and Hayden have found evidence that the limits of quantum computation do in fact prevent Alice from carrying out her experiment in less than exponential time. If their conjecture is correct then black hole complementarity may be alive and well. My aim here is to give an overview of the firewall argument, and its basis in the proximity postulate; as well as the counterargument based on computational complexity, as conjectured by Harlow and Hayden.