Can computer science help physicists resolve the firewall

paradox?

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

THEORETICALPHYSICISTS

Me

But in this talk, I’ll tell you about a developing story, centered around the black hole information problem, that’s been bringing computer science and physics together in a remarkable and unexpected way—going beyond the connection established in the 1990s by quantum computing

Black Holes in Classical GR

No hair: just mass, charge, and angular momentum

Black Holes in Quantum Mechanics

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: That’s absurd! If true, it would imply that black holes have a temperature and radiate … no, wait …

Modern PictureBlack 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

Information Problem

Yet all known laws of fundamental physics, from Galileo through

quantum field theory, are perfectly reversible (information-preserving)

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 holeSo why not just assume the information

somehow gets out in the Hawking radiation?

The Xeroxing Problem

10 1010

The No-Cloning Theorem says there’s no procedure to copy an unknown quantum state

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)?

VIOLATES LINEARITY OF QM

ComplementaritySusskind, ‘t Hooft 1990s

“It’s 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

The AMPS Firewall Argument (2012)

“When people much more expert than me admitted that they also didn’t 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.

Digression: Quantum Entanglement

2

111000 “Monogamy of entanglement”:

Entanglement among 3 or more parties just reduces

to classical correlation among any 2 of them

Remember, if anyone asks, I’ll be spinning up and you’ll

be spinning down…

Bell’s Theorem

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

In quantum field theory, the “vacuum” has huge amounts of

short-range entanglement!

No entanglement No smooth vacuum

Violates monogamy of entanglement!

The Firewall Paradox (AMPS 2012)

H = Interior of “Old” Black Hole

(with known pure starting state)

R = Faraway Hawking Radiation

B = Just-Emitted Hawking Radiation

Near-maximal entanglement

Also near-maximal entanglement

Harlow-Hayden Argument

Striking argument that Alice’s 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 would’ve 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

67102

Digression About Quantum Computers

In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2n time to simulate classically—and had the idea to overcome that problem using computers that were themselves quantum

Quantum mechanics: “Probability theory with minus signs”(Nature seems to prefer it that way)

Exponential (inefficient)

Polynomial (efficient)

Not Even a Quantum Computer Could Do Everything!

Exponentially-many states, but you only get to

observe one of them

nxx x

2,,1

Any hope for a speedup relies on the magic of

quantum interference—amplitudes for wrong

answers cancelling out

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

NP

NP-complete

P

FactoringBQP

Interesting

The Collision Lower BoundProblem: Decide whether a function f is one-to-one or two-to-one, promised that one of those is the case

Models the breaking of collision-resistant hash functions—a central problem in cryptanalysis—as well as graph isomorphism

10 4 1 8 7 9 11 5 6 4 2 10 3 2 7 9 11 5 1 6 3 8

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

Harlow and Hayden’s TheoremLet’s 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 there’d also be a polynomial-time quantum algorithm for the collision problem!

My Improvement to Harlow-Hayden

Is the geometry of spacetime protected by an armor of computational complexity?

Decoding entanglement between R and B is generically hard, assuming only that there exists a one-way function that’s hard to invert using a quantum computer

Indeed, even decoding classical correlation is hard

Computational Complexity and AdS/CFT

AdS/CFT correspondence:A 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 years—”a place where quantum gravity works”

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 one’s coordinates)

What’s 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

nx

nxx

1,02

1

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!

TIM

E

His Question for Me: But does the circuit complexity actually increase like this—at least for “natural” scrambling dynamics, and under some plausible hardness assumption?

His Proposal: The quantum circuit complexity—that is, the number of quantum logic gates in the smallest circuit that prepares the state from a simple initial state

Susskind’s Question: What function of the CFT state can we point to, that’s “dual” to wormhole length on the AdS side?

Time t

Quantum circuit

complexity

2n00

2n

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.

A Favorite Research Direction“Not 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 |, there’s 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 Hayden’s logic, and give a sufficient condition for the firewall experiment to be doable in polynomial time

Now, to end this talk with something crazy

Wiesner 1969: Because of the No-Cloning Theorem, in principle it’s 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 doesn’t need to remember stuff about every bill in circulation

Quantum resistant one-way functions

Firewall experiment

is hardCryptographic quantum money