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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

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



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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Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique…

Matrix permanentHalting problem…

FactoringGraph isomorphismMinimum circuit size…

Graph connectivityPrimality testingMatrix determinantLinear programming…





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Does P=NP?

The (literally) $1,000,000 question

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But what if P=NP, 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 don’t seem powerful enough

Combinatorial arguments face the “Razborov-Rudich barrier”

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But maybe there’s some physical system that solves an NP-complete problem just by reaching its lowest energy state?

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  • 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

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!

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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”

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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

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Topological Quantum Field Theories (TQFT’s)

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)

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



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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!

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“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.

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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 it’s 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