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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.

Mar 26, 2015

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Page 1: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

NP-complete Problems and Physical Reality

Scott Aaronson

UC Berkeley IAS

Page 2: 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

Page 3: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

NP: Nondeterministic Polynomial Time

37976595177176695379702491479374117272627593301950462688996367493665078453699421776635920409229841590432339850906962896040417072096197880513650802416494821602885927126968629464313047353426395204881920475456129163305093846968119683912232405433688051567862303785337149184281196967743805800830815442679903720933

Does

have a prime factor ending in 7?

Page 4: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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)?

Page 5: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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…

Page 6: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Does P=NP?The (literally) $1,000,000 question

Page 7: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

But what if P=NP, and the algorithm takes n10000 steps?

God will not be so cruel

Page 8: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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

Page 9: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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”

Page 10: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

But maybe there’s some physical system that solves

an NP-complete problem just by reaching its lowest

energy state?

Page 11: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

- Dip two glass plates with pegs between them into soapy water

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

Page 12: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Other Physical Systems

Well-known to admit “metastable” states

Spin glasses

Folding proteins

...

DNA computers: Just highly parallel ordinary computers

Page 13: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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!

Page 14: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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”

Page 15: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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

Page 16: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

“Relativity Computing”

DONE

Page 17: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Topological Quantum Field Theories (TQFT’s)

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

Page 18: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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

Page 19: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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

Page 20: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

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!

Page 21: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Quantum Gravity

Page 22: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

“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.

Page 23: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

“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

Page 24: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Second Law of Thermodynamics

Proposed Counterexamples

Page 25: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

No Superluminal Signalling

Proposed Counterexamples

Page 26: NP-complete Problems and Physical Reality Scott Aaronson UC Berkeley IAS.

Intractability of NP-complete

problems

Proposed Counterexamples

?