Quantum Computing and the Limits of the Efficiently Computable

1. Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT

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. Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Matrix permanentHalting problem… FactoringGraph isomorphism… Graph connectivityPrimality testingMatrix determinantLinear programming… NP-hard NP-complete NP P

4. 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

5. Extended Church-Turing Thesis An important presupposition underlying P vs. NP is the... “Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory” So how sure are we of this thesis? Have there been serious challenges to it?

6. 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”

7. Other Approaches Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: Just massively parallel classical computers

8. Ah, but what about quantum computing? 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—and had the amazing idea of building a “quantum computer” to overcome that problem Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

9. Science PopularizersBeware:A quantum computer is NOTa 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

10. 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

11. Remember: factoring isn’t thought to be NP-complete! Today, we don’t believe BQP contains all of NP(though not surprisingly, we can’t prove that it doesn’t) 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?

12. 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

13. Nonlinear variants of the Schrödinger Equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time 1 solution to NP-complete problem No solutions

14. Relativity Computer DONE

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

16. Here’s a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel): Read an integer x{0,…,2n-1} from the future If x encodes a valid solution, then output x Otherwise, output (x+1) mod 2n Closed Timelike Curves (CTCs) If valid solutions exist, then the only fixed-points of the above program input and output them Building on work of Deutsch, [A.-Watrous 2008] defined a formal model of CTC computation, and showed that in both the classical and quantum cases, it has exactly the power of PSPACE (believed to be even larger than NP)

17. “The No-SuperSearch Postulate”There is no physical means to solve NP-complete problems in polynomial time. Includes PNP as a special case, but is stronger No longer a purely mathematical conjecture, but also a claim about the laws of physics If true, would “explain” why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, CTCs don’t exist...

18. Question: What exactly does it mean to “solve” an NP-complete problem? Example: It’s been known for decades that, if you send n identical photons through a network of beamsplitters, the amplitude for the photons to reach some final state is given by the permanent of an nn matrix of complex numbers Lesson: If you can’t observe the answer, it doesn’t count! Recently, Alex Arkhipov and I gave the first evidence that even the observed output distribution of such a linear-optical network would be hard to simulate on a classical computer—but the argument was necessarily more subtle But the permanent is #P-complete (believed even harder than NP-complete)! So how can Nature do such a thing? Resolution: The amplitudes aren’t directly observable, and require exponentially-many probabilistic trials to estimate

19. One could imagine worse research agendas than the following: Prove P≠NP(better yet, prove factoring is classically hard, implying P≠BQP) Prove NPBQP—i.e., that not even quantum computers can solve NP-complete problems Build a scalable quantum computer(or even more interesting, show that it’s impossible) Determine whether all of physics can be simulated by a quantum computer “Derive” as much physics as one can from No-SuperSearch and other impossibility principles Conclusion

20. Papers, talk slides, blog: www.scottaaronson.com