NPcomplete Problems and Physical Reality

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NP-complete Problems and Physical Reality

• Scott Aaronson
• UC Berkeley ? IAS

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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 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
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NP Nondeterministic Polynomial Time
Does
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have a prime factor ending in 7?
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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)?
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NP-hard
NP-complete
NP
P
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Does PNP?
The (literally) 1,000,000 question
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But what if PNP, and the algorithm takes n10000
steps?
God will not be so cruel
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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
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Then why is it so hard to prove P?NP?
Algorithms can be very clever
Gödel/Turing-style self-reference arguments dont
seem powerful enough
Combinatorial arguments face the Razborov-Rudich
barrier
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But maybe theres some physical system that
solves an NP-complete problem just by reaching
its lowest energy state?
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• Dip two glass plates with pegs between them into
  soapy water

• Let the soap bubbles form a minimum Steiner tree
  connecting the pegs

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Other Physical Systems
Spin glasses Folding proteins ...
Well-known to admit metastable states
DNA computers Just highly parallel ordinary
computers
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Analog Computing
Schönhage 1979 If we could compute xy, 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!
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Quantum Computing
Shor 1994 Quantum computers can factor in
polynomial time
But can they solve NP-complete problems?
A. 2004 True even with quantum advice
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Quantum Adiabatic Algorithm (Farhi et al. 2000)
Hi
Hf
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
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Relativity Computing
DONE
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Topological Quantum Field Theories (TQFTs)
Freedman, Kitaev, Wang 2000 Equivalent to
ordinary quantum computers
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Nonlinear Quantum Mechanics (Weinberg 1989)
Abrams Lloyd 1998 Could use to solve
NP-complete and even harder problems in
polynomial time
1 solution to NP-complete problem
No solutions
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Time Travel Computing(Bacon 2003)
SupposePrx1 p,Pry1 q Then consistency
requires pq So Prx?y1 p(1-q) q(1-p)
2p(1-p)
x
x?y
Causalloop
Chronology-respecting bit
x
y
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Hidden Variables
Valentini 2001 Subquantum algorithm (violating
?2) to distinguish 0? from
Problem Valentinis 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 timebut again,
probably not NP-complete problems!
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Quantum Gravity
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Anthropic Computing
Guess a solution to an NP-complete problem. If
its wrong, kill yourself.
Doomsday alternativeIf solution is right,
destroy human race.If wrong, cause human race to
survive into far future.
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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 its done, then tell those copies the answer

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Second Law of Thermodynamics
Proposed Counterexamples
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No Superluminal Signalling
Proposed Counterexamples
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?
Intractability of NP-complete problems
Proposed Counterexamples