NPcomplete Problems and Physics SAT Scott Aaronson University
NP-complete Problems and Physics SAT Scott Aaronson (University of Texas at Austin) Cook. Fest, University of Toronto, May 9, 2019
Why I’m Here
What this talk won’t be about NP-hard problems that happen to involve physics Completeness for Quantum NP (QMA) Ising spin minimization Quantum Cook-Levin Theorem (Kitaev): Quantum Gap-k-SAT is Promise. QMA-complete Best separable state Games with entangled provers …
Scientific question that maximally ties together what I care about: e lv o s o t s n a e m l a ic Is there any phys ? e im t l ia m o n ly o p in s m le b o r p P N y r a r it arb Includes: 1. P = NP? 2. Does energy minimization make NP easy? 3. NP BQP? 4. Is BQP realizable? 5. Is there anything beyond BQP?
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 NP BQP SVP Factoring P SZK
1. Does P = NP? For more information: Or for masochists: 12 pages 122 pages
2. Does energy minimization make NP easy?
Soap Bubble Computer
Protein Folding etc. “NP is trivial for Nature!” “Including proving RH? Mining Bitcoin? ”
Memcomputing P=NP ? “Again, if NP is so easy, why aren’t you rich? ”
3. Is NP BQP?
Quantum Computing In One Slide BQP: All. BPP: quantum speedups come 00 from interference: |00 00 10 1 11 |00 |10 1 |11
The Fundamental QC Misconception “A QC just tries 2 n possible answers in parallel!” The entire question is: how do you get a large probability of observing the answer you want?
Black-Box Quantum Search MARKED ITEM Grover’s Algorithm: Searches a list of size N in O( N) steps BBBV’ 94: Without exploiting additional structure, this is optimal There exists an oracle A such that NPA BQPA
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
Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hamiltonian with easilyprepared ground state Hf Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small
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
4. Is BQP Realizable? Conservative view: Yes. Radical speculation: No. Hopefully we’ll learn more soon Google’s 72 -qubit “Bristlecone” chip, currently in development 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…
5. Is there anything beyond BQP?
Relativity Computer DONE
Zeno’s Computer Time (seconds) STEP 1 STEP 2 STEP 3 STEP 4 STEP 5
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
Quantum Gravity Ad. S/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 Challenge: Show that, at least in Ad. S/CFT universes, the Church-Turing Thesis holds
Cosmological Parallelism “To solve an NP-complete problem, why not just spawn 2 n baby universes? ” Alas: dark energy + Bekenstein bound 10122 qubits in any computation in our world [Bousso 2000]
Nonlinear Schrödinger Equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NPcomplete problems in polynomial time 1 solution to NP-complete problem No solutions
Time Travel Computer If x {0, 1}n satisfies the formula , then send x backward in time. Otherwise, send back (x+1) mod 2 n. A. -Watrous 2008: PCTC = BQPCTC = PSPACE. But CTCs would break many principles of physics!
Postselected Final State (Yakir Aharonov’s proposal) A. 2004: Post. BQP = PP Even larger than Post. BPP, unless PH collapses!
Non-Collapsing Measurements Suppose we want to find collisions in a 2 -to-1 function f A, Bouland, Fitzsimons, Lee 2014 (following A. 2005): Could do all of SZK this way. But relative to an oracle, still not NP-complete problems!
Summary: A Bounded Universe? NO AL PERPETU MOTION UPERS P N O N SEARCH NO SU PE LUMIN RA SIGNA L LS Did anyone foresee in 1971 that NP-completeness would have this kind of reach? I don’t know. I wasn’t born yet.
- Slides: 29
