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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

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np complete problems and physical reality

NP-complete Problems and Physical Reality

Scott Aaronson

UC Berkeley  IAS

computer science 101
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

np nondeterministic polynomial time
NP: Nondeterministic Polynomial Time



have a prime factor ending in 7?

np hard if you can solve it you can solve everything in np
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)?


Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique…

Matrix permanentHalting problem…

FactoringGraph isomorphismMinimum circuit size…

Graph connectivityPrimality testingMatrix determinantLinear programming…





does p np
Does P=NP?

The (literally) $1,000,000 question

what could we do if we could solve np complete problems
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 p np
Then why is it so hard to prove PNP?

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
Other Physical Systems

Spin glasses

Folding proteins


Well-known to admit “metastable” states

DNA computers: Just highly parallel ordinary computers

analog computing
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 computing

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

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”

quantum adiabatic algorithm farhi et al 2000
Quantum Adiabatic Algorithm (Farhi et al. 2000)



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

topological quantum field theories tqft s
Topological Quantum Field Theories (TQFT’s)

Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

nonlinear quantum mechanics weinberg 1989
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

time travel computing bacon 2003
Time Travel Computing(Bacon 2003)

SupposePr[x=1] = p,Pr[y=1] = q

Then consistency requires p=q

So Pr[xy=1]= p(1-q) + q(1-p)= 2p(1-p)




Chronology-respecting bit



hidden variables
Hidden Variables

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!

anthropic computing
“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
“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