Presentation on theme: "NP-complete Problems and Physical Reality"— Presentation transcript:
1NP-complete Problems and Physical Reality Scott AaronsonUC Berkeley IAS
2Computer 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
3NP: Nondeterministic Polynomial Time Doeshave a prime factor ending in 7?
4NP-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)?
5NP 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
6The (literally) $1,000,000 question Does P=NP?The (literally) $1,000,000 question
7But what if P=NP, and the algorithm takes n10000 steps? God will not be so cruel
8What 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
9Then 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”
10But maybe there’s some physical system that solves an NP-complete problem just by reaching its lowest energy state?
11Dip two glass plates with pegs between them into soapy water Let the soap bubbles form a minimum Steiner tree connecting the pegs
12Other Physical Systems Spin glassesFolding proteins...Well-known to admit “metastable” statesDNA computers: Just highly parallel ordinary computers
13Analog 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!
14Quantum 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”
15Quantum 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
20Hidden 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
24Second Law of Thermodynamics Proposed Counterexamples