NP-complete Problems and Physical Reality

1. NP-complete Problems and Physical Reality Scott Aaronson Institute for Advanced Study

2. Shortest program that outputs works of Shakespeare in 107 steps Proof of Riemann hypothesis of length 100000? Circuit of size 100000 that does best at predicting stock market data What could we do if we could solve NP-complete problems?

3. If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude. That is to say, it would clearly indicate that, despite the unsolvability of the Entscheidungsproblem, the mental effort of the mathematician could be completely (apart from the postulation of axioms) replaced by machines.—Gödel to von Neumann, 1956

4. Current Situation Algorithms (GSAT, survey propagation, …) that work well on random 3SAT instances, but apparently not on “semantically hard” instances No proof of PNP in sight - Razborov-Rudich barrier - Depth-3 threshold circuits evade us - P vs. NP independent of set theory?

5. This Talk Is there a physical system that solves NP-complete problems in polynomial time? Classical? Quantum? Neither? • Argument: • This is a superb question to ask about physics • NP is special (along with NPcoNP, one-way functions, …) • Intractability as physical axiom?

6. Dip two glass plates with pegs between them into soapy water • Let the soap bubbles form a minimum Steiner tree connecting the pegs

7. Other Physical Systems Spin glasses: Well-known to admit “metastable” optima Folding proteins: Same (e.g. prions). But also, are local optima weeded out by evolution? DNA computers: Just highly parallel ordinary computers

8. Analog Computing Schönhage 1979: If we could compute x+y, x-y, xy, x/y, x for any real x,y in a single step, then we could solve NP- and even PSPACE-complete problems in polynomial time

9. Problem: The Planck Scale! 10-33 cm • Reasons to think spacetime is discrete • Past experience with matter, light, etc. • (2) Existence of a natural minimum length scale • (3) Infinities of quantum field theory • (4) Black hole entropy bounds (1.41069 bits/m2) • (5) Area quantization in loop quantum gravity • (6) Cosmic rays above GZK cutoff (~1020 eV) • (7) Independence of AC and CH?

10. Bennett, Bernstein, Brassard, Vazirani 1994: “Quantum magic” a la Grover won’t be enough Given a “black box” function f:{0,1}n{0,1}, a quantum computer needs (2n/2) queries to f to find an x such that f(x)=1 Thus NPA BQPA relative to some oracle A Quantum Computing Shor 1994: Quantum computers can factor in polynomial time But can they solve NP-complete problems?

11. To many quantum computing skeptics, |n is an “exponentially long vector.” So, could it encode the solutions to every SAT instance of length n? Quantum Advice BQP/qpoly: the class of problems solvable in bounded-error quantum polynomial time, given a polynomial-size “quantum advice state” |n that depends only on the input length n A. 2004: NPA BQPA/qpoly relative to some oracle A. Proof based on “direct product theorem” for quantum search

12. Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hf (1-s)Hi+sHf Ground state encodes solution to 3SAT instance Hamiltonian with easily-prepared ground state Quantum analogue of simulating annealing Numerical data suggested polynomial running time van Dam, Mosca, Vazirani 2001; Reichardt 2004: Takes exponential time on some 3SAT instances

13. Topological Quantum Field Theories (TQFT’s) Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

14. After we measure third register, first two registers will have the form if G  H, if not “Non-Collapsing Measurements” To solve Graph Isomorphism: Given G and H, prepare If only we could measure both ||0 and ||1 without collapsing, we’d solve the problem…(Generalizes to all problems in SZK)

15. A. 2002: Any quantum algorithm needs (N1/5) queries to decide w.h.p. whether a function f:{1,…,N}{1,…,N} is one-to-one or two-to-one Improved by Shi, Kutin, Ambainis, Midrijanis Yields oracle A such that SZKA BQPA A. 2004: On the other hand, if we could sample the entire history of a hidden variable (satisfying a reasonable axiom), we could solve anything in SZK But still not NP-complete problems, relative to an oracle!

16. To get a factor-k speedup: Exponentially close to c if k is exponentially large “Special Relativity Computing” DONE So need an exponential amount of energy. Where does it come from?

17. Nonlinear Quantum Mechanics Abrams & Lloyd 1998: Could use to solve NP-complete and even #P-complete problems in polynomial time 1 solution to NP-complete problem No solutions

18. C(x) Causalloop C x Time Travel Computing(Adapted from Brun 2003) Assumption (Deutsch): Probability distribution over x{0,1}n must be a fixpoint of polynomial-size circuit C Model: We choose C, then a fixpoint distribution D over x is chosen adversarially, then an xD is sampled To solve SAT: Let C(x)=x if x is a satisfying assignment, C(x)=x+1(mod 2n) otherwise To solve PSPACE-complete problems: Exercise for the audience…

19. Time Travel Computing with 1 Looping Bit(Adapted from Bacon 2003) SupposePr[x=1] = p,Pr[y=1] = q Then consistency requires p=q So Pr[xy=1]= p(1-q) + q(1-p)= 2p(1-p) x xy Causalloop Chronology-respecting bit x y

20. Quantum Gravity Spacetimes that have to be treated as identical if their metric structures are isomorphic? Probabilities that don’t sum to 1 unless they’re normalized by hand? Highly nonlocal unitaries implementable in polynomial time?

21. “Anthropic Computing” Guess a solution to an NP-complete problem. If it’s wrong, kill yourself. Doomsday alternative:If solution is right, destroy human race.If wrong, cause human race to survive into far future. Classically, anthropic computing lets us do exactly BPPpath (between MA and PP) A. 2003: Quantumly, it lets us do exactly PP

22. Second Law of Thermodynamics Proposed Counterexamples

23. No Superluminal Signalling Proposed Counterexamples

24. ? Intractability of NP-complete problems Proposed Counterexamples