NP-complete Problems and Physical Reality. Scott Aaronson UC Berkeley IAS. Computer Science 101. Problem: “Given a graph, is it connected?” Each particular graph is an instance The size of the instance, n, is the number of bits needed to specify it
UC Berkeley IAS
Problem: “Given a graph, is it connected?”
Each particular graph is an instance
The size of the instance, n, is the number of bits needed to specify it
An algorithm is polynomial-time if it uses at most knc steps, for some constants k,c
P is the class of all problems that have polynomial-time algorithms
have a prime factor ending in 7?
NP-complete: NP-hard and in NP
Is there a Hamilton cycle (tour that visits each vertex exactly once)?
Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique…
Matrix permanentHalting problem…
FactoringGraph isomorphismMinimum circuit size…
Graph connectivityPrimality testingMatrix determinantLinear programming…
The (literally) $1,000,000 question
God will not be so cruel
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
Algorithms can be very clever
Gödel/Turing-style self-reference arguments don’t seem powerful enough
Combinatorial arguments face the “Razborov-Rudich barrier”
Well-known to admit “metastable” states
DNA computers: Just highly parallel ordinary computers
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-complete and even harder problems in polynomial time
Problem: The Planck scale!
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 itemQuantum Computing
Shor 1994: Quantum computers can factor in polynomial time
But can they solve NP-complete problems?
A. 2004: True even with “quantum advice”
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete problem
Problem (van Dam, Mosca, Vazirani 2001): Eigenvalue gap can be exponentially small
Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers
Abrams & Lloyd 1998: Could use to solve NP-complete and even harder problems in polynomial time
1 solution to NP-complete problem
SupposePr[x=1] = p,Pr[y=1] = q
Then consistency requires p=q
So Pr[xy=1]= p(1-q) + q(1-p)= 2p(1-p)
Valentini 2001: “Subquantum” algorithm (violating ||2) to distinguish |0 from
Problem: Valentini’s algorithm still requires exponentially-precise measurements.But we probably could solve Graph Isomorphism subquantumly
A. 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!
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.
Intractability of NP-complete problems