Presentation on theme: "NP-complete Problems and Physical Reality"— Presentation transcript:
1 NP-complete Problems and Physical Reality Scott AaronsonUC Berkeley IAS
2 Computer Science 101 Problem: “Given a graph, is it connected?” Each particular graph is an instanceThe size of the instance, n, is the number of bits needed to specify itAn algorithm is polynomial-time if it uses at most knc steps, for some constants k,cP is the class of all problems that have polynomial-time algorithms
3 NP: Nondeterministic Polynomial Time Doeshave a prime factor ending in 7?
4 NP-hard: If you can solve it, you can solve everything in NP NP-complete: NP-hard and in NPIs there a Hamilton cycle (tour that visits each vertex exactly once)?
5 NP P NP-hard NP-complete Matrix permanent Halting problem …Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique …NP-completeNPFactoring Graph isomorphism Minimum circuit size …Graph connectivity Primality testing Matrix determinant Linear programming …P
6 The (literally) $1,000,000 question Does P=NP?The (literally) $1,000,000 question
7 But what if P=NP, and the algorithm takes n10000 steps? God will not be so cruel
8 What could we do if we could solve NP-complete problems? 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
9 Then why is it so hard to prove PNP? Algorithms can be very cleverGödel/Turing-style self-reference arguments don’t seem powerful enoughCombinatorial arguments face the “Razborov-Rudich barrier”
10 But maybe there’s some physical system that solves an NP-complete problem just by reaching its lowest energy state?
11 Dip two glass plates with pegs between them into soapy water Let the soap bubbles form a minimum Steiner tree connecting the pegs
12 Other Physical Systems Spin glassesFolding proteins...Well-known to admit “metastable” statesDNA computers: Just highly parallel ordinary computers
13 Analog Computing Schönhage 1979: If we could compute x+y, x-y, xy, x/y, xfor any real x,y in a single step, then we could solve NP-complete and even harder problems in polynomial timeProblem: The Planck scale!
14 Quantum ComputingShor 1994: Quantum computers can factor in polynomial timeBut can they solve NP-complete problems?Bennett, Bernstein, Brassard, Vazirani 1997: “Quantum magic” won’t be enough~2n/2 queries are needed to search a list of size 2n for a single marked itemA. 2004: True even with “quantum advice”
15 Quantum Adiabatic Algorithm (Farhi et al. 2000) HfHamiltonian with easily-prepared ground stateGround state encodes solution to NP-complete problemProblem (van Dam, Mosca, Vazirani 2001): Eigenvalue gap can be exponentially small
20 Hidden VariablesValentini 2001: “Subquantum” algorithm (violating ||2) to distinguish |0 fromProblem: Valentini’s algorithm still requires exponentially-precise measurements. But we probably could solve Graph Isomorphism subquantumlyA. 2002: Sampling the history of a hidden variable is another way to solve Graph Isomorphism in polynomial time—but again, probably not NP-complete problems!
22 “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.
23 “Transhuman Computing” Upload yourself onto a computerStart the computer working on a 10,000-year calculationProgram the computer to make 50 copies of you after it’s done, then tell those copies the answer
24 Second Law of Thermodynamics Proposed Counterexamples
25 No Superluminal Signalling Proposed Counterexamples
26 Intractability of NP-complete problems Proposed Counterexamples ?Intractability of NP-complete problemsProposed Counterexamples