The Computational Complexity of Decoding Hawking Radiation

Scott Aaronson

Hawking 1970s: What happens to quantum information dropped into a black hole?

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 Stays in black hole forever  Violates unitarity of QM Comes out in Hawking radiation  if there’s also a copy inside the black hole, violates the No-Cloning Theorem

Complementarity

(modern view): Inside is just a “re-encoding” of exterior, so no cloning is needed to have |  in both places

The Firewall Paradox (Almheiri et al. 2012):

Refinement of Hawking’s information paradox that challenges complementarity

If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits, then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior?

Entanglement among Hawking photons detected!

Harlow-Hayden 2013: To create the firewall, you’d need to process the Hawking radiation in a way that probably requires exponential computation time!



RBH

 1 2

n

 1 

x

  

n



MODEL SITUATION:

x

, 0

R

0

B f H

 R: “Old” Hawking photons

x

, 1

R

1

B g H

 B: Hawking photon just now coming out H: Degrees of freedom still inside black hole f,g: Two functions such that it’s hard to tell whether their ranges are equal or disjoint [A. 2002: quantum lower bound for this problem]

Idea:

If Range(f)=Range(g), then R and B are entangled, but acting on R to reveal the entanglement (as in AMPS experiment) requires proving that Range(f)=Range(g), hence solving the hard problem

My result: Harlow-Hayden decoding is as hard as inverting an arbitrary one-way function



RBH



MODEL SITUATION (given a one-way function f):

1 2 2

n

 1

x

, 

s

  

n

,

a



f

, 1   ,

s

,

a R x s



a B x

,

s H

B is maximally entangled with the last qubit of R. But in order to see that B and R are even

classically correlated

, one would need to learn x  s (a “hardcore bit” of f), and therefore invert f With realistic dynamics, the decoding task seems like it should only be “harder” than in this model case (though unclear how to formalize that) Is the geometry of spacetime protected by an armor of computational complexity?