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NP-complete Problems and Physics

This talk explores the connection between NP-complete problems and physics, discussing the possibility of solving arbitrary NP problems in polynomial time using physical means, especially quantum computing.

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NP-complete Problems and Physics

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  1. SAT NP-complete Problems and Physics Scott Aaronson (University of Texas at Austin) Hebrew University, June 11, 2019

  2. Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Matrix permanentHalting problem… FactoringGraph isomorphism… Graph connectivityPrimality testingMatrix determinantLinear programming… NP-hard NP-complete NPNondeterministic Polynomial Time PPolynomial Time

  3. What this talk won’t be about NP-hard problems that happen to involve physics QuantumNP (QMA) Completeness Ising spin minimization Best separable state Games with entangled provers … Quantum Cook-Levin Theorem (Kitaev): Quantum Gap-k-SAT is PromiseQMA-complete

  4. Scientific question that maximally ties together what I care about: Is there any physical means to solve arbitrary NP problems in polynomial time? Includes: 1. P = NP?2. Does energy minimization make NP easy?3. NP BQP? (BQP = Quantum Polynomial Time)4. Is BQP realizable?5. Is there anything beyond BQP?

  5. Why is NP-completeness important to this question? Without it, there might’ve been a thousand separate villages of hardness, and no good reason to focus on any one… NP-complete SZK NP SVP Factoring BQP P

  6. 1. Does P = NP? For more information: Or for masochists: 122 pages 12 pages

  7. 2. Does energy minimization make NP easy?

  8. Soap Bubble Computer

  9. Protein Folding etc. etc. “NP is trivial for Nature!” “Including proving the Riemann hypothesis? Bitcoin mining? If NP is so easy, why aren’t you rich?”

  10. 3. Is NPBQP?

  11. Quantum Computing In One Slide BPP: BQP: | 00  00 All quantum speedups come from interference:   | | 10 00 10 00 1 1 1 1  |  11 11 00 00 |

  12. The Fundamental QC Misconception “A QC just tries 2n possible answers in parallel!” The entire question is: how do you get a large probability of observing the answer you want?

  13. Black-Box Quantum Search Grover’s Algorithm: Searches a list of size N in O(N) steps MARKED ITEM BBBV’94: Without exploiting additional structure, this is optimal  There exists an oracle A such that NPABQPA

  14. Around the BBBV Theorem A. 2004: There’s even an oracle relative to which NPBQP/qpoly On the other hand, I recently got an oracle relative to which NPBQP and PH is infinite By building on the breakthrough of Raz-Tal 2018: an oracle relative to which BQP PH

  15. 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

  16. Landscapeology Adiabatic algorithm can find global minimum exponentially faster than simulated annealing (though maybe other classical algorithms do better) Simulated annealing can find global minimum exponentially faster than adiabatic algorithm (!) Simulated annealing and adiabatic algorithm both need exponential time to find global minimum

  17. 4. Is BQP Realizable? Conservative view: Yes. Radical speculation: No. One of Google’s superconducting chips currently in development Hopefully we’ll learn more soon Goal of near-term “quantum supremacy” demos: Sample distributions such that, if they could be efficiently sampled classically, then P#P=BPPNP. Of course they’ll only be sampled approximately…

  18. 5. Is there anything beyond BQP?

  19. Relativity Computer DONE

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

  21. Quantum Field Theory Jordan, Lee, Preskill 2014: Efficient simulation of nontrivial QFTs by “ordinary” quantum computers(not yet extended to the full Standard Model…) Freedman, Kitaev, Wang 2000: Topological QFTs yield exactly the power of BQP

  22. AdS/CFT has shown that, in toy models, even gravity can be subsumed into standard QM, which suggests (but doesn’t prove) that we’d get no more computational power than BQP Quantum Gravity Challenge: Show that, at least in AdS/CFT universes, the Church-Turing Thesis holds

  23. “To solve an NP-complete problem, why not just spawn 2n baby universes?” Cosmological Parallelism Alas: dark energy + Bekenstein bound   10122 qubits in any computation in our world[Bousso 2000]

  24. Nonlinear 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

  25. If x{0,1}n satisfies the formula , then send x backward in time. Otherwise, send back (x+1) mod 2n. Time Travel Computer A.-Watrous 2008:PCTC = BQPCTC = PSPACE. But CTCs would break many principles of physics!

  26. Postselected Final State(Yakir Aharonov’s proposal) A. 2004:PostBQP (quantum computing with postselected measurements) can solve all problems in PP—so, even more than a classical postselected computer, unless PH collapses!

  27. Suppose we want to find collisions in a 2-to-1 function f Non-Collapsing Measurements A., Bouland, Fitzsimons, Lee 2014 (following A. 2005): Could solve all problems in Statistical Zero Knowledge this way. But relative to an oracle, still not the NP-complete problems!

  28. Summary: A Bounded Universe? NO NP SUPER-SEARCH NO PERPETUAL MOTION NO SUPER-LUMINAL SIGNALS

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