Black Holes, Firewalls, and the Complexity of States and Unitaries Scott Aaronson (MIT) Papers and slides at .

Black Holes, Firewalls, and the Complexity of States and Unitaries

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

My Starting Point

PHYSICS

COMPUTER SCIENCE

QUANTUM COMPUTING

BOSONS & FERMIONS(E.g., BosonSampling)

BLACK HOLES, AdS/CFT

(Today’s talk)

AND MORE!

Black Holes and Computational Complexity??

YES!Amazing connection made in 2013 by Harlow & Hayden

But first, let’s review 40 years of black hole history

SZK

QSZK

BPPBQP

AMQAM

Bekenstein, Hawking 1970s: Black holes have entropy and temperature! They emit radiation

The Information Loss Problem: Calculations suggest that Hawking radiation is thermal—uncorrelated with whatever fell in. So, is infalling information lost forever? Would seem to violate the unitarity / reversibility of QM

OK then, assume the information somehow gets out!

The Xeroxing Problem: How could the same qubit | fall inexorably toward the singularity, and emerge in Hawking radiation? Would violate the No-Cloning Theorem

Black Hole Complementarity (Susskind, ‘t Hooft): An external observer can describe everything unitarily without including the interior at all! Interior should be seen as “just a scrambled re-encoding” of the exterior degrees of freedom

Violates monogamy of entanglement! The same qubit can’t be maximally entangled with 2 things

The Firewall Paradox (AMPS 2012)

H = Interior of “Old”

Black Hole

R = Faraway Hawking Radiation

B = Just-Emitted Hawking Radiation

Near-maximal entanglement

Also near-maximal entanglement

Harlow-Hayden 2013 (arXiv:1301.4504): Striking argument that Alice’s first task, decoding the entanglement between R and B, would take exponential time—by which point, the black hole would’ve long ago evaporated anywayComplexity theory to the rescue of quantum field theory?

Are we 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 field theories; question is in what regimes they break down

Caveats of Complexity Arguments1. Asymptotic

E.g., 88 chess takes O(1) time! Only for nn chess can we give evidence of hardness. But for black holes, n1070…

2. (Usually) ConjecturalRight now, we can’t even prove P≠NP! To get where we want, we almost always need to make assumptions. Question is, which assumptions?

3. Worst-CaseWe can argue that a natural formalization of Alice’s decoding task is “generically” hard. We can’t rule out that a future quantum gravity theory would make her task easy, for deep reasons not captured by our formalization.

Quantum Circuits

Given a description of a quantum circuit C, such that

Promised that, by acting only on R (the “Hawking radiation part”), it’s possible to distill an EPR pair

between R and B

Problem: Distill such an EPR pair, by applying a unitary transformation UR to the qubits in R

The HH Decoding Problem

21100

RBH

nC 0

Problem: That would require waiting until the black hole was fully evaporated ( no more firewall problem)

When the BH is “merely” >50% evaporated, we know from a dimension-counting argument that “generically,” there will exist a UR that distills an EPR pair between R and B

But interestingly, this argument doesn’t suggest any efficient procedure to find UR or apply it!

Isn’t the Decoding Task Trivial?Just invert C!

Set Equality: Given two efficiently-computable injective functions f,g:{0,1}n{0,1}p(n). Promised that Range(f) and Range(g) are either equal or disjoint. Decide which.

In the “black-box” setting, this problem takes (2n/7) time even with a quantum computer (slight variant of the “collision lower bound” I proved in 2002). Even in non-black-box setting, would let us solve e.g. Graph Isomorphism

Theorem (Harlow-Hayden): Suppose there’s a polynomial-time quantum algorithm for HH decoding. Then there’s also a polynomial-time quantum algorithm for Set Equality!

The HH Hardness Result

Intuition: If Range(f) and Range(g) are disjoint, then the H register decoheres all entanglement between R and B, leaving only classical correlation

If, on the other hand, Range(f)=Range(g), then there’s some permutation of the |x,1R states that puts the last qubit of R into an EPR pair with B

Thus, if we had a reliable way to distill EPR pairs whenever possible, then we could also decide Set Equality

The HH Construction

nx

HBRHBRnRBHxgxxfx

1,01

11,00,2

1

(easy to prepare in poly(n) time given f,g)

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

1,0,1,0,

12,,,

2

1

asxHBRnRBH

n

sxasxasxf

B is maximally entangled with R. But in order to see that B and R are even classically correlated, one would need to learn xs (a “hardcore bit” of f), and therefore invert f

Is computational intractability the only “armor” protecting the geometry of

spacetime inside the black hole?

R: “old” Hawking photons / B: photons just coming out / H: still in black hole

Quantum Circuit Complexity and Wormholes[A.-Susskind, in progress]

The AdS/CFT correspondence relates anti-deSitter quantum gravity in D spacetime dimensions to conformal field theories (without gravity) in D-1 dimensions

But the mapping is extremely nonlocal!

It was recently found that an expanding wormhole, on the AdS side, maps to a collection of qubits on the CFT side that just seems to get more and more “complex”:

n

tt UI

21100

But does C(|t) actually increase like this, for natural scrambling dynamics U?

Susskind’s Proposal: The quantum circuit complexity C(|t)—that is, the number of gates in the smallest circuit that prepares |t from |0n

(Not clear if it’s right, but has survived some nontrivial tests)

Question: What function of |t can we point to on the CFT side, that’s “dual” to wormhole length on the AdS side?

Time t

C(|t)

2n00

2n

Theorem: Suppose U implements (say) a computationally-universal, reversible cellular automaton. Then after t=exp(n) iterations, C(|t) is superpolynomial in n, unless something very unlikely happens with complexity classes (PSPACEPP/poly)

Proof Sketch: I proved in 2004 that PP=PostBQP

Suppose C(|t)=nO(1). Then we could give a description of C as advice to a PostBQP machine, and the machine could efficiently prepare

xUx t

xnt

n

1,0

2/21

The machine could then measure the first register, postselect on some |x of interest, then measure the second register to learn Ut|x—thereby solving a PSPACE-complete problem!

Also have results for approximate circuit complexity, C(|t)exp(n), and more

Note that some complexity assumption must be made to lower-bound C(|t)

Public-Key Quantum MoneyAnother area where one needs to argue certain quantum states are hard to

prepare

A

A

In a scheme proposed by A.-Christiano 2012, each bill

contains an equal superposition over a “hidden subspace,”

which anyone can measure in two complementary bases to

check the bill’s veracity

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 counterfeit. But how to verify a bill?

Bizarre strengthening of my result on Harlow-Hayden decoding:

HH decoding is hard if there exists a secure quantum money scheme

(even private-key)

A Favorite Research DirectionUnderstand, more systematically, the quantum circuit complexity of preparing n-qubit states and applying unitary transformations (“not just for quantum gravity! also for quantum algorithms, quantum money, and so much more”)

Example question: For every n-qubit unitary U, is there a Boolean function f such that U can be realized by a polynomial-time quantum algorithm with an oracle for f?

(I’m giving you any computational capability f you could possibly want—but it’s still far from obvious how to get the physical capability U!)

Easy to show: For every n-qubit state |, there’s a Boolean function f such that | can be prepared by a polynomial-time quantum algorithm with an oracle for f

SummaryA major challenge is to understand which quantum states can be efficiently prepared, and which unitaries can be efficiently applied

Applications range from quantum money to quantum gravity!