Can computer science help physicists resolve the firewall paradox? Scott Aaronson (MIT) Papers and slides at www.scottaaronson.com

Me THEORETICAL PHYSICISTS 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 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

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)

The Xeroxing Problem The No-Cloning Theorem says there’s 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)?

Complementarity Susskind, ‘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 Remember, if anyone asks, I’ll be spinning up and you’ll be spinning down… Bell’s Theorem “Monogamy of entanglement”: Entanglement among 3 or more parties just reduces to classical correlation among any 2 of them

What Do “Generic” Many-Particle Entangled Pure States Look Like 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

No entanglement  No smooth vacuum In quantum field theory, the “vacuum” has huge amounts of short-range entanglement! No entanglement  No smooth vacuum

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!

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 anyway Complexity 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

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 classically—and had the idea to overcome that problem using computers that were themselves quantum

Exponential (inefficient) Polynomial (efficient)

Not Even a Quantum Computer Could Do Everything! Any hope for a speedup relies on the magic of quantum interference—amplitudes for wrong answers cancelling out Exponentially-many states, but you only get to observe one of them

BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993 Interesting Shor 1994: Factoring integers is in BQP NP NP-complete P Factoring BQP

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 functions—a central problem in cryptanalysis—as 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

Harlow and Hayden’s Theorem Let’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 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 Is the geometry of spacetime protected by an armor of computational complexity?

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 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!

Quantum circuit complexity Susskind’s Question: What function of the CFT state can we point to, that’s “dual” to wormhole length on the AdS side? 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 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. Time t Quantum circuit complexity 2n 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?

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 hard Cryptographic quantum money