np-complete problems and physical reality scott aaronson uc berkeley ias
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NP-complete Problems and Physical Reality
Scott Aaronson
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
Computer Science 101
NP: Nondeterministic Polynomial Time
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Does
have a prime factor ending in 7?
NP-hard: If you can solve it, you can solve everything in NP
NP-complete: NP-hard and in NP
Is there a Hamilton cycle (tour that visits each vertex exactly once)?
P
NP
NP-complete
NP-hard
Graph connectivityPrimality testingMatrix determinantLinear programming…
Matrix permanentHalting problem…
Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Factoring
Graph isomorphismMinimum circuit size…
Does P=NP?The (literally) $1,000,000 question
But what if P=NP, and the algorithm takes n10000 steps?
God will not be so cruel
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
Then why is it so hard to prove PNP?
Algorithms can be very clever
Gödel/Turing-style self-reference arguments don’t seem powerful enough
Combinatorial arguments face the “Razborov-Rudich barrier”
But maybe there’s some physical system that solves
an NP-complete problem just by reaching its lowest
energy state?
- Dip two glass plates with pegs between them into soapy water
- Let the soap bubbles form a minimum Steiner tree connecting the pegs
Other Physical Systems
Well-known to admit “metastable” states
Spin glasses
Folding proteins
...
DNA computers: Just highly parallel ordinary computers
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-complete and even harder problems in polynomial time
Problem: The Planck scale!
Quantum ComputingShor 1994: Quantum computers can factor in polynomial time
But 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 item
A. 2004: True even with “quantum advice”
Quantum Adiabatic Algorithm (Farhi et al. 2000)
HiHamiltonian with easily-prepared
ground state
HfGround state encodes
solution to NP-complete problem
Problem (van Dam, Mosca, Vazirani 2001): Eigenvalue gap can be exponentially small
“Relativity Computing”
DONE
Topological Quantum Field Theories (TQFT’s)
Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers
Nonlinear Quantum Mechanics (Weinberg 1989)
Abrams & Lloyd 1998: Could use to solve NP-complete and even harder problems in polynomial time
No solutions1 solution to NP-complete problem
Time Travel Computing(Bacon 2003)
x y
xy x
Ch
ron
olo
gy-
resp
ecti
ng
bit
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)
Causalloop
Hidden VariablesValentini 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
1 2 0 2 1n n
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!
Quantum Gravity
“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.
“Transhuman Computing”
• Upload yourself onto a computer
• Start the computer working on a 10,000-year calculation
• Program the computer to make 50 copies of you after it’s done, then tell those copies the answer
Second Law of Thermodynamics
Proposed Counterexamples
No Superluminal Signalling
Proposed Counterexamples
Intractability of NP-complete
problems
Proposed Counterexamples
?