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

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 So what about the third one?

3. Problem: Given a flight map, is every airport reachable from every other in 5 flights or less? Any specific map is an instance of the problem The size of an instance, n, is the number of bits used to specify it An algorithm is polynomial-time if it uses at most kn c steps, for some constants k,c P is the class of all problems for which theres a deterministic, polynomial-time algorithm that correctly solves every instance Complexity Theory 101

4. NP: Nondeterministic Polynomial Time 379765951771766953797024914793741172726275933019 504626889963674936650784536994217766359204092298 415904323398509069628960404170720961978805136508 024164948216028859271269686294643130473534263952 048819204754561291633050938469681196839122324054 336880515678623037853371491842811969677438058008 30815442679903720933 Does have a factor 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 Is there a tour that visits each city once?

6. P NP NP- complete NP-hard Graph connectivity Primality testing Matrix determinant Linear programming … Matrix permanent Halting problem … Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … Factoring Graph isomorphism …

7. Does P=NP? The (literally) $1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), this would have consequences of the greatest magnitude. Gödel to von Neumann, 1956

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

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

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

11. Ah, but what about quantum computing? (you knew it was coming) Quantum computing: The power of 2 n complex numbers working for YOU In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2 n time to simulateand had the amazing idea of building a quantum computer to overcome that problem Quantum mechanics: Probability theory with minus signs (Nature seems to prefer it that way) Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

12. 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

13. 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 NP-complete P Factoring BQP Interesting

14. Remember: factoring isnt thought to be NP-complete! Today, we dont believe BQP contains all of NP (though not surprisingly, we cant prove that it doesnt) Bennett et al. 1997: Quantum magic wont be enough If you throw away the problem structure, and just consider an abstract landscape of 2 n possible solutions, then even a quantum computer needs ~2 n/2 steps to find the correct one (That bound is actually achievable, using Grovers algorithm!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

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

16. 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 No solutions 1 solution to NP-complete problem

17. Relativity Computer DONE

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

19. Heres a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel): Read an integer x {0,…,2 n -1} from the future If x encodes a valid solution, then output x Otherwise, output (x+1) mod 2 n 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)

20. 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 dont exist... The No-SuperSearch Postulate There is no physical means to solve NP-complete problems in polynomial time.

21. Question: What exactly does it mean to solve an NP- complete problem? Example: Its 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 But the permanent is #P-complete (believed even harder than NP- complete)! So how can Nature do such a thing? Resolution: The amplitudes arent directly observable, and require exponentially-many probabilistic trials to estimate Lesson: If you cant observe the answer, it doesnt 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

22. One could imagine worse research agendas than the following: Prove PNP (better yet, prove factoring is classically hard, implying PBQP) Prove NP BQPi.e., that not even quantum computers can solve NP-complete problems Build a scalable quantum computer (or even more interesting, show that its 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

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