1. Scott Aaronson (MIT) The Limits of Computation: Quantum Computers and Beyond

2. Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION Übercomputer The (seeming) impossibility of the first two machines reflects fundamental principles of physicsSpecial Relativity and the Second Law respectively Does physics also put limits on computation?

3. Moores Law

4. Extrapolating: Robot uprising?

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

6. Is there any feasible way to solve NP-complete problems, consistent with the laws of physics? And its conjectured that thousands of interesting problems are inherently intractable for Turing machines… (Why is it so hard to prove P NP? We know a lot about that today, most recently from algebrization [A.-Wigderson 2007])

7. Relativity Computer DONE

8. Zenos Computer STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time (seconds)

9. Time Travel Computer S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669.

10. A quantum state of n qubits takes 2 n complex numbers to describe: Chemists and physicists knew that for decades, as a major practical problem! In the 1980s, Feynman, Deutsch, and others had the amazing idea of building a new type of computer that could overcome the problem, by itself exploiting the exponentiality inherent in QM Shor 1994: Such a machine could also factor integers Interesting

11. The practical problem: decoherence. What weve learned from quantum computers so far: 21 = 3 × 7 (with high probability) A few people think scalable QC is fundamentally impossible... but that would be even more interesting than if its possible! [A. 2004]: Theory of Sure/Shor separators

12. Limitations of Quantum Computers [BBBV 1994] explained why quantum computers probably dont offer exponential speedups for the NP-complete problems [A. 2002] proved the first lower bound (~N 1/5 ) on the time needed for a quantum computer to find collisions in a long list of numbers from 1 to Nthereby giving evidence that secure cryptography should still be possible even in a world with QCs 4 2 1 3 2 5 4 5 1 3

13. BosonSampling [A.-Arkhipov 2011] Recent experimental proposal, which involves generating n identical photons, passing them through a network of beamsplitters, then measuring where they end up Almost certainly wouldnt yield a universal quantum computerand indeed, it seems easier to implement Nevertheless, our experiment would sample a certain probability distribution, which we give strong evidence is hard to sample with a classical computer Jeremy OBriens group at the University of Bristol has built our experiment with 4 photons and 16 optical modes on-chip

14. 10 Years of My Other Research in 1 Slide Using quantum techniques to understand classical computing better [A. 2004] [A. 2005] [A. 2011] Quantum Money that anyone can verify, but thats physically impossible to counterfeit [A.-Christiano 2012] Quantum Generosity … Giving back because we care TM The Information Content of Quantum States For many practical purposes, the exponentiality of quantum states doesnt actually mattertheres a shorter classical description that works fine Describing quantum states on efficient measurements only [A. 2004], pretty-good tomography [A. 2006]

15. Thank you for your support! NP NP-complete P Factoring BQP Boson Sampling