The Computational Complexity
of Decoding Hawking Radiation
Scott Aaronson
Hawking 1970s: What happens to quantum
information dropped into a black hole?
|
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!
MODEL SITUATION:

RBH

1
2
n 1
  x,0
x0,1n
R
0
B
f x 
H
 x,1 R 1 B g x 
H

R: “Old” Hawking photons
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
MODEL SITUATION (given a one-way function f):

RBH

1
2
2 n 1
 f x, s, a x  s   a
x , s0,1 , a0,1
n
R
B
x, s
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?
H