NP-complete Problems and Physical Reality

Scott Aaronson

Institute for Advanced Study

What could we do if we could solve NP-complete problems?

Proof of Riemann

hypothesis of length

100000?

Circuit of size 100000 that does best at

predicting stock market data

Shortest program that outputs works

of Shakespeare in 107 steps

If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude. That is to say, it would clearly indicate that, despite the unsolvability of the Entscheidungsproblem, the mental effort of the mathematician could be completely (apart from the postulation of axioms) replaced by machines.—Gödel to von Neumann, 1956

Current Situation

Algorithms (GSAT, survey propagation, …) that work well on random 3SAT instances, but apparently not on “semantically hard” instances

No proof of PNP in sight

- Razborov-Rudich barrier

- Depth-3 threshold circuits evade us

- P vs. NP independent of set theory?

This Talk

Is there a physical system that solves NP-complete problems in polynomial time? Classical? Quantum? Neither?

Argument:

- This is a superb question to ask about physics

- NP is special (along with NPcoNP, one-way functions, …)

- Intractability as physical axiom?

- 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

Spin glasses: Well-known to admit “metastable” optima

DNA computers: Just highly parallel ordinary computers

Folding proteins: Same (e.g. prions). But also, are local optima weeded out by evolution?

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- and even PSPACE-complete problems in polynomial time

Problem: The Planck Scale!

Reasons to think spacetime is discrete(1) Past experience with matter, light, etc.(2) Existence of a natural minimum length scale(3) Infinities of quantum field theory(4) Black hole entropy bounds (1.41069 bits/m2)(5) Area quantization in loop quantum gravity(6) Cosmic rays above GZK cutoff (~1020 eV)(7) Independence of AC and CH?

10-33 cm

Quantum ComputingShor 1994: Quantum computers can factor in polynomial time

But can they solve NP-complete problems?

Bennett, Bernstein, Brassard, Vazirani 1994: “Quantum magic” a la Grover won’t be enough

Given a “black box” function f:{0,1}n{0,1}, a quantum computer needs (2n/2) queries to f to find an x such that f(x)=1

Thus NPA BQPA relative to some oracle A

Quantum AdviceBQP/qpoly: the class of problems solvable in bounded-error quantum polynomial time, given a polynomial-size “quantum advice state” |n that depends only on the input length n

To many quantum computing skeptics, |n is an “exponentially long vector.” So, could it encode the solutions to every SAT instance of length n?

A. 2004: NPA BQPA/qpoly relative to some oracle A. Proof based on “direct product theorem” for quantum search

Quantum Adiabatic Algorithm (Farhi et al. 2000)

HiHamiltonian with easily-prepared

ground state

HfGround state

encodes solution to 3SAT instance

van Dam, Mosca, Vazirani 2001; Reichardt 2004: Takes exponential time on some 3SAT instances

(1-s)Hi+sHf

Quantum analogue of simulating annealing

Numerical data suggested polynomial running time

Topological Quantum Field Theories (TQFT’s)

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

“Non-Collapsing Measurements”To solve Graph Isomorphism: Given G and H, prepare

10 1

2 !nS

G Hn

If only we could measure both ||0 and ||1 without collapsing, we’d solve the problem…(Generalizes to all problems in SZK)

After we measure third register, first two registers will have the form

0 1

2

if G H, b if not

A. 2002: Any quantum algorithm needs (N1/5) queries to decide w.h.p. whether a function f:{1,…,N}{1,…,N} is one-to-one or two-to-one

Improved by Shi, Kutin, Ambainis, Midrijanis

Yields oracle A such that SZKA BQPA

But still not NP-complete problems, relative to an oracle!

A. 2004: On the other hand, if we could sample the entire history of a hidden variable (satisfying a reasonable axiom), we could solve anything in SZK

“Special Relativity Computing”

DONE

So need an exponential amount of energy.

Where does it come from?

To get a factor-k speedup:

Exponentially close to c if k is exponentially large

21

1v

c k

Nonlinear Quantum Mechanics

Abrams & Lloyd 1998: Could use to solve NP-complete and even #P-complete problems in polynomial time

No solutions1 solution to NP-complete problem

Time Travel Computing(Adapted from Brun 2003)

Assumption (Deutsch): Probability distribution over x{0,1}n must be a fixpoint of polynomial-size circuit C

CCausal

loopx

C(x)

To solve SAT: Let C(x)=x if x is a satisfying assignment, C(x)=x+1(mod 2n) otherwise

Model: We choose C, then a fixpoint distribution D over x is chosen adversarially, then an xD is sampled

To solve PSPACE-complete problems: Exercise for the audience…

Time Travel Computing with 1 Looping Bit(Adapted from 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

Quantum Gravity

Probabilities that don’t sum to 1 unless they’re normalized by hand?

Spacetimes that have to be treated as identical if their metric structures are isomorphic?

Highly nonlocal unitaries implementable in polynomial time?

“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.

Classically, anthropic computing lets us do exactly BPPpath (between MA and PP)

A. 2003: Quantumly, it lets us do exactly PP

Second Law of Thermodynamics

Proposed Counterexamples

No Superluminal Signalling

Proposed Counterexamples

Intractability of NP-complete

problems

Proposed Counterexamples

?