Can computer science help physicists resolve the firewall paradox?

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Can computer science help physicists resolve the
firewall paradox?

• Scott Aaronson (MIT)
• Papers and slides at www.scottaaronson.com

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Me
THEORETICAL PHYSICISTS
But in this talk, Ill tell you about a
developing story, centered around the black hole
information problem, thats been bringing
computer science and physics together in a
remarkable and unexpected waygoing beyond the
connection established in the 1990s by quantum
computing
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Black Holes in Classical GR
No hair just mass, charge, and angular momentum
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Black Holes in Quantum Mechanics
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Jacob Bekenstein Classical black holes seem to
violate the Second Law of Thermodynamics! To fix,
assume they somehow have an entropy proportional
to the square of the surface area of the event
horizon
Stephen Hawking Thats absurd! If true, it
would imply that black holes have a temperature
and radiate no, wait
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Modern Picture
Black holes are the most efficient hard disks in
the universe they store 1069 bits per square
meter of surface area (any denser arrangement
will just collapse to a black hole)
If you try to do more than 1043 computation steps
per second, that will also trigger collapse to a
black hole
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Information Problem
The QFT calculation that says in the first place
that the Hawking radiation exists, also predicts
that it should be thermal that is, completely
uncorrelated with whatever information fell into
the black hole
So why not just assume the information somehow
gets out in the Hawking radiation?
Yet all known laws of fundamental physics, from
Galileo through quantum field theory, are
perfectly reversible (information-preserving)
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The Xeroxing Problem
The No-Cloning Theorem says theres no procedure
to copy an unknown quantum state
VIOLATES LINEARITY OF QM
So then how could the same state ?? both be
permanently in the hole (as seen by the infalling
observer), and out in the Hawking radiation (as
seen by the external observer)?
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ComplementaritySusskind, t Hooft 1990s
Its OK, as long as the same observer never
measures both copies of ?? !
Jumping into a black hole just a convoluted way
of measuring the same quantum states that were
already there outside the black hole, and on the
event horizon
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The AMPS Firewall Argument (2012)
When people much more expert than me admitted
that they also didnt understand black hole
complementarity
No longer a dispute about formalism now an
actual (zany) thought experiment, such that if
you claim to understand black holes, then you
must be able to say what the infalling observer
would experience if this experiment were done.
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Digression Quantum Entanglement
Remember, if anyone asks, Ill be spinning up and
youll be spinning down
Bells Theorem
Monogamy of entanglement Entanglement among 3
or more parties just reduces to classical
correlation among any 2 of them
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What Do Generic Many-Particle Entangled Pure
States Look Like?(Again, pure quantum
information theory, nothing to do with black
holes)
Subset of fewer than half of the particles In a
completely random (maximally mixed) state
Subset of more than half of the particles Not
maximally mixed. Any one particle in the subset
is entangled with the remaining ones
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In quantum field theory, the vacuum has huge
amounts of short-range entanglement!
No entanglement ? No smooth vacuum
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The Firewall Paradox (AMPS 2012)
R Faraway Hawking Radiation
B Just-Emitted Hawking Radiation
H Interior of Old Black Hole (with known pure
starting state)
Near-maximal entanglement
Also near-maximal entanglement
Violates monogamy of entanglement!
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Harlow-Hayden Argument
Striking argument that Alices first task,
decoding the entanglement between R and B, would
take time exponential in the number of qubits of
the black hole (so not 1067 years but
)by which point, the black hole wouldve long
ago evaporated anywayComplexity to the rescue of
quantum field theory?
Are they saying that an inconsistency in the laws
of physics is OK, as long as it takes exponential
time to discover it? NO! Inconsistency is
only in low-energy effective theories question
is where they break down
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Digression About Quantum Computers
Quantum mechanics Probability theory with minus
signs (Nature seems to prefer it that way)
In the 1980s, Feynman, Deutsch, and others
noticed that quantum systems with n particles
seemed to take 2n time to simulate
classicallyand had the idea to overcome that
problem using computers that were themselves
quantum
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Exponential (inefficient)
Polynomial (efficient)
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Not Even a Quantum Computer Could Do Everything!
Any hope for a speedup relies on the magic of
quantum interferenceamplitudes for wrong answers
cancelling out
Exponentially-many states, but you only get to
observe one of them
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BQP (Bounded-Error Quantum Polynomial-Time) The
class of problems solvable efficiently by a
quantum computer, defined by Bernstein and
Vazirani in 1993
Shor 1994 Factoring integers is in BQP
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The Collision Lower Bound
Problem Decide whether a function f is
one-to-one or two-to-one, promised that one of
those is the case
10 4 1 8 7 9 11 5 6 4 2 10 3 2 7 9
11 5 1 6 3 8
Models the breaking of collision-resistant hash
functionsa central problem in cryptanalysisas
well as graph isomorphism
Aaronson 2001 If f has 2n inputs, and is only
accessible as a black box, then any quantum
algorithm to solve the collision problem takes at
least 2n/5 steps (improved to 2n/3 by Yaoyun
Shi, which is optimal) Evidence that problems of
this kind are not in BQP
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Harlow and Haydens Theorem
Lets model a black hole by a set of qubits that
start in a known state, and the physics of a
black hole by a known polynomial-size quantum
circuit acting on those qubits.
Suppose that, for any circuit C, there were
another polynomial-size quantum circuit to solve
the Harlow-Hayden decoding problem, of acting
on R to produce an entangled pair with B. Then
thered also be a polynomial-time quantum
algorithm for the collision problem!
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My Improvement to Harlow-Hayden
Decoding entanglement between R and B is
generically hard, assuming only that there exists
a one-way function thats hard to invert using a
quantum computer Indeed, even decoding classical
correlation is hard
Is the geometry of spacetime protected by an
armor of computational complexity?
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Computational Complexity and AdS/CFT
AdS/CFT correspondenceA duality between anti
de-Sitter space in D dimensions, and conformal
field theory in D-1 dimensions. Considered one of
the main achievements of theoretical physics of
the past 30 yearsa place where quantum gravity
works
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Thermofield Double State
A state in AdS involving two regions of spacetime
connected only by a wormhole. The wormhole is
non-traversable, because it expands faster than
light, before pinching off in a singularity
(after either finite or infinite time, depending
on ones coordinates)
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Whats the CFT dual of the thermofield double
state?
Just a bunch of qubits that start out in a simple
state, and get more and more scrambled as time
goes on
TIME
Problem Something being scrambled quickly
reaches a state of maximum scrambling (as
measured in the usual ways). Yet the wormhole
continues to get longer for exponential time!
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Susskinds Question What function of the CFT
state can we point to, thats dual to wormhole
length on the AdS side?
His Proposal The quantum circuit complexitythat
is, the number of quantum logic gates in the
smallest circuit that prepares the state from a
simple initial state
Theorem (Aaronson-Susskind) Suppose the
scrambling transformation is complicated enough
to encode universal computation. Then after
exponential time, the circuit complexity of the
state will be more than polynomial, unless PSPACE
? PP/poly.
His Question for Me But does the circuit
complexity actually increase like thisat least
for natural scrambling dynamics, and under some
plausible hardness assumption?
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A Favorite Research DirectionNot just for black
holes and quantum gravity, for lots of things
Understand the sizes of the smallest quantum
circuits needed to prepare states and apply
transformations. Relate this to the quantum
circuit complexity of solving traditional
problems with yes-or-no answers Example question
(Aaronson-Kuperberg 2006) For every
transformation T of n-qubit quantum states, is
there a decision problem such that a magic box
for solving it would let you apply T in only
poly(n) steps? Easy to show for every n-qubit
state ??, theres a decision problem such that a
magic box for solving it would let you prepare
?? in only poly(n) steps
Relevant to whether one can reverse Harlow and
Haydens logic, and give a sufficient condition
for the firewall experiment to be doable in
polynomial time
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Now, to end this talk with something crazy
Wiesner 1969 Because of the No-Cloning Theorem,
in principle its possible to have quantum
money, where each bill includes qubits that are
physically impossible to duplicate. Bennett et
al. 1982 Can even combine with cryptography so
the bank doesnt need to remember stuff about
every bill in circulation
Quantum resistant one-way functions
Firewall experiment is hard
Cryptographic quantum money