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

2. 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

3. NP: Nondeterministic Polynomial Time
Does
have a prime factor ending in 7?

4. 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)?

5. NP P NP-hard NP-complete
Matrix permanent Halting problem …
Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique …
NP-complete NP
Factoring Graph isomorphism Minimum circuit size …
Graph connectivity Primality testing Matrix determinant Linear programming … P

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

7. But what if P=NP, and the algorithm takes n10000 steps?
God will not be so cruel

8. 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

9. 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”

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

11. Dip two glass plates with pegs between them into soapy water
Let the soap bubbles form a minimum Steiner tree connecting the pegs

12. Other Physical Systems
Spin glasses
Folding proteins
...
Well-known to admit “metastable” states
DNA computers: Just highly parallel ordinary computers

13. 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!

14. Quantum Computing
Shor 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”

15. Quantum Adiabatic Algorithm (Farhi et al. 2000)Hf
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

16. “Relativity Computing”
DONE

17. Topological Quantum Field Theories (TQFT’s)
Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

18. 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

19. Time Travel Computing (Bacon 2003)
Suppose Pr[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)
xy x
Causal loop
Chronology-respecting bit x y

20. 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!

21. Quantum Gravity

22. “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.

23. “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

24. Second Law of Thermodynamics Proposed Counterexamples

25. No Superluminal Signalling Proposed Counterexamples

26. Intractability of NP-complete problems Proposed Counterexamples?
Intractability of NP-complete problems
Proposed Counterexamples