Scott Aaronson Associate Professor, EECS

1. The Limits of Computation Quantum Computers and Beyond Scott Aaronson Associate Professor, EECS

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. 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.

9. Quantum Computers

10. Quantum Mechanics in 1 Slide “Like probability theory, but over the complex numbers” 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

11. Interference “The source of all quantum weirdness” Possible states of a single quantum bit, or qubit:

12. Interesting Quantum Computing“Quantum Mechanics on Steroids” 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 A general entangled state of n qubits requires ~2n amplitudes to specify: 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

13. But 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!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

14. Quantum Adiabatic Algorithm(Farhi et al. 2000) Hi Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small

15. BosonSampling (with Alex Arkhipov): A proposal for a rudimentary optical quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers Some of My Recent Research Computational Complexity of Decoding Hawking Radiation: Building on a striking recent proposal by Harlow and Hayden—that part of the resolution of the black hole information problem might be that reconstructing the infalling information from the Hawking radiation would require an exponentially long computation