The Limits of Computation: Quantum Computers and Beyond

1. The Limits of Computation Quantum Computers and Beyond Scott Aaronson Computer Science, UT Austin AAAS Meeting, Feb. 19, 2017

2. Moore’s Law

3. Extrapolating: Robot uprising?

4. But even a killer robot would still be “merely” a Turing machine, operating on principles laid down in the 1930s… =

5. And it’s conjectured that thousands of interesting problems are inherently intractable for Turing machines… Is there any feasible way to solve these problems, consistent with the laws of physics?

6. Relativity Computer DONE

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

8. Quantum Computers What we’ve learned from quantum computers so far: 21 = 3 × 7(with high probability)

9. Quantum Mechanics in One Slide Probability Theory: Quantum Mechanics: Linear transformations that conserve 1-norm of probability vectors:Stochastic matrices Linear transformations that conserve 2-norm of amplitude vectors:Unitary matrices

10. Journalists Beware:A quantum computer is NOT like a massively-parallel classical computer! Exponentially-many basis states, but you only get to observe one of them Any hope for a speedup rides on the magic of quantum interference

11. Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Matrix permanentHalting problem… FactoringGraph isomorphism… Graph connectivityPrimality testingMatrix determinantLinear programming… NP-hardAll NP problems are efficiently reducible to these NP-complete NPEfficiently verifiable PEfficiently solvable

12. Interesting 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-complete NP Factoring BQP P

13. “QUANTUM SUPREMACY”: 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 can’t be sampled efficiently classically (unless P#P=BPPNP) Some of My Recent Research Experimentally demonstrated with 6 photons by O’Brien group at Bristol Random Quantum Circuit Sampling: Martinis group at Google is planning a system with 40-50 high-quality superconducting qubits in the near future; we’re thinking about what to do with it that’s classically hard

14. Complexity of Decoding Hawking Radiation Hawking famously asked in the 1970s how information can escape from a black hole, as it must if QM is universally valid His question led to the proposal of black hole complementarity (Susskind, ‘t Hooft 1990s) But then the “firewall paradox” (AMPS 2012) said that, by doing a suitable measurement on the Hawking radiation, you could destroy the spacetime geometry inside the black hole! Harlow and Hayden 2013:Yes, but that measurement would probably require performing an exponentially long quantum computation! (For a solar-mass black hole: ~210^67 years) I’ve improved Harlow and Hayden’s argument to base it on “standard” crypto assumptions (injective OWFs)

15. From a theoretical standpoint, modern computers are “all the same slop”: polynomial-time Turing machines • Quantum computing is interesting as the first serious proposal that would go beyond thi • Even going a bitbeyond the limits of today’s computers (say, with quantum supremacy experiments) is a challenge • Contrary to what you read, even quantum computers would face significant limitations • But those limitations could help protect the geometry of spacetime! Summary