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 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

NP: Nondeterministic Polynomial Time Does 37976595177176695379702491479374117272627593301950462688996367493665078453699421776635920409229841590432339850906962896040417072096197880513650802416494821602885927126968629464313047353426395204881920475456129163305093846968119683912232405433688051567862303785337149184281196967743805800830815442679903720933 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)?

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

The (literally) $1,000,000 question 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 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”

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 Spin glasses Folding proteins ... Well-known to admit “metastable” states 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 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”

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

“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 1 solution to NP-complete problem No solutions

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

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!

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 ? Intractability of NP-complete problems Proposed Counterexamples