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

?