Black Holes, Firewalls, and the Limits of Quantum Computers

1. Black Holes, Firewalls, and the Limits of Quantum Computers Scott Aaronson (UT Austin) Simons Theoretically Speaking Series, Oct. 18, 2017 Papers and slides at www.scottaaronson.com

2. GOLDBACH CONJECTURE: TRUE NEXT QUESTION Things we never see… Warp drive Übercomputer Perpetuum mobile The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively So what about the third one?

3. But Turing machines have fundamental limits—even more so, if you need the answer in a reasonable amount of time! P: Polynomial TimeClass of all “decision problems” (infinite sets of yes-or-no questions) solvable by a Turing machine, using a number of steps that scales at most like the size of the question raised to some fixed power Example: Is it mathematically possible to get between Berkeley and Palo Alto?

4. NP: Nondeterministic Polynomial TimeClass of all decision problems for which a “yes” answer can be verified in polynomial time, if you’re given a witness or proof for it Example: Does 37976595177176695379702491479374117272627593301950462688996367493665078453699421776635920409229841590432339850906962896040417072096197880513650802416494821602885927126968629464313047353426395204881920475456129163305093846968119683912232405433688051567862303785337149184281196967743805800830815442679903720933 have a divisor ending in 7?

5. NP-hard: If you can solve it, then you can solve every NP problem NP-complete: NP-hard and in NP Example: Is there a tour that visits each city once?

6. Does P=NP? The (literally) $1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude.—Gödel to von Neumann, 1956

7. Most computer scientists believe that PNP But if so, there’s a further question: is there any way to solve NP-complete problems in polynomial time, consistent with the laws of physics?

8. Old proposal: Dip two glass plates with pegs between them into soapy water. Let the soap bubbles form a minimum Steiner tree connecting the pegs—thereby solving a known NP-hard problem “instantaneously”

9. Relativity Computer DONE

10. Zeno’s Computer STEP 1 STEP 2 Time (seconds) STEP 3 STEP 4 STEP 5

11. Ah, but what about quantum computing?(you knew it was coming) Quantum mechanics: “Probability theory with minus signs” (Nature seems to prefer it that way)

12. The Famous Double-Slit Experiment Probability of landing in “dark patch” = |amplitude|2 = |amplitudeSlit1 + amplitudeSlit2|2 = 0 Yet if you close one of the slits, the photon can appear in that previously dark patch!

13. If we observe, we see |0 with probability |a|2 |1 with probability |b|2 Also, the object collapses to whichever outcome we see A bit more precisely: the key claim of quantum mechanics is that, if an object can be in two distinguishable states, call them |0 or |1, then it can also be in a superpositiona|0 + b|1 Here a and b are complex numbers called amplitudes satisfying |a|2+|b|2=1

14. Interesting Quantum Computing A general entangled state of n qubits requires ~2n amplitudes to specify: Where we are: A QC has now factored 21 into 37, with high probability (Martín-López et al. 2012) Scaling up is hard, because of decoherence! But unless QM is wrong, there doesn’t seem to be any fundamental obstacle Presents an obvious practical problem when using conventional computers to simulate quantum mechanics Feynman 1981: So then why not turn things around, and build computers that themselves exploit superposition? Shor 1994: Such a computer could do more than simulate QM—e.g., it could factor integers in polynomial time

15. Factoring is not believed to be NP-complete! And today, we don’t believe quantum computers can solve NP-complete problems in polynomial time in general(though not surprisingly, we can’t prove it) Bennett et al. 1997: “Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!) If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow

16. The “Adiabatic Optimization” Approach to Solving NP-Hard Problems with a Quantum Computer Hi Hf Operation with easily-prepared lowest energy state Operation whose lowest-energy state encodes solution to NP-hard problem

17. Hope: “Quantum tunneling” could give speedups over classical optimization methods for finding local optima Remains unclear whether you can get a practical speedup this way over the best classical algorithms. We might just have to build QCs and test it! Problem: “Eigenvalue gap” can be exponentially small

18. Getting a clear quantum speedup for some task—not necessarily a useful one BosonSampling (with Alex Arkhipov): A proposal for a simple optical quantum computer to sample a distribution that (we think) can’t be sampled efficiently classically “Quantum Supremacy” Experimentally demonstrated with 6 photons by group at Bristol Random Quantum Circuit Sampling: Martinis group at Google is building a system with 49 high-quality superconducting qubits this year. Lijie Chen and I studied the hardness of sampling its output distribution

19. Hawking 1970s: What happens to quantum information dropped into a black hole? | Stays in black hole forever  Violates quantum mechanics Comes out in Hawking radiation  if there’s also a copy inside the black hole, seems to violate 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

20. 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!

21. Harlow-Hayden (2013):Argued that the requisite computation would take exponential time (~210^70 years) even for a QC—by which time the black hole has already fully evaporated! Why? Because one can reduce the problem of finding collisions in a cryptographic hash function, to the problem of decoding the Hawking radiation. And I showed in my Berkeley PhD thesis that, in the “black-box setting,” the former takes exponential time for a quantum computer! Recently, I strengthened Harlow and Hayden’s argument, to show that performing the computation is generically at least as hard as inverting any injective one-way function with a quantum computer

22. A. 2017: If you had the technological capability to verify a Schrödinger cat state, then you’d also necessarily (i.e., with a similar-sized quantum circuit) have the capability to bring a dead cat back to life More Ways Computational Complexity Interacts with Physics Susskind 2013, many others: Quantum circuit complexity as a “dual” of wormhole volume in the AdS/CFT correspondence

23. Quantum computers are the most powerful kind of computer allowed by the currently-known laws of physics There’s a realistic prospect of building them Even quantum computers would have nontrivial limits—which might be the limits of what’s efficiently computable in reality But those limits might help protect the geometry of spacetime! Summary