What Google Won’t Find: The Ultimate Physical Limits of Search Scott Aaronson University of Waterloo

—Thomas Friedman, NYT, 6/29/2003 Why Am I Speaking Here? “Is Google God?” —Thomas Friedman, NYT, 6/29/2003 My field—theoretical computer science—is directly concerned with the question of how to distinguish God from mortal impostors.

CS Theory in One Slide Problem: “Given the Internet, are at least 50% of web pages all reachable from one another?” Each particular Internet 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 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 Halting problem Counting problems … Hamilton cycle Graph 3-coloring Satisfiability Maximum clique … NP-complete NP Factoring Graph isomorphism … Graph connectivity Primality testing Linear programming … P

Extra credit: Prove it. (You’ll win at least $1,000,000 if you do) Audience Exam Does P=NP? Answer: No. Extra credit: Prove it. (You’ll win at least $1,000,000 if you do)

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 diagonalization 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 “relaxing” to 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

It’s Quantum Time If an object can be in two states |0 or |1, then it can also be in a superposition |0 + |1 Here  and  are complex amplitudes satisfying ||2+||2=1 If we observe, we see |0 with probability ||2 |1 with probability ||2 Also, the object collapses to whichever outcome we see

To modify a state we can multiply the vector of amplitudes by a unitary matrix—one that preserves

We’re seeing interference between positive and negative amplitudes—the source of all “quantum weirdness”

Quantum Computing A quantum state of n “qubits” takes 2n complex numbers to describe: The goal of quantum computing is to exploit this exponentiality in Nature.

Shor 1994: Quantum computers can factor integers in polynomial time Interesting But what about NP-complete problems? Bennett, Bernstein, Brassard, Vazirani 1994: “Quantum magic” won’t be enough Even a quantum computer would need ~2n/2 queries to search an unsorted array of size 2n for a single “marked” item

“Relativity Computing” DONE

Analog Computing Do the first step of a computation in 1 sec, the second in ½ sec, the third in ¼ sec, … Possible in “Malament-Hogarth spacetimes,” which have naked singularities Problem: The Planck scale (10-33 cm, 10-43 sec) seems to impose a fundamental limit!

Time Travel Computing Naïve idea: Do the first step of a computation, then go back in time and do the next step, etc. Problem: Grandfather paradoxes Resolution (Deutsch 1991): Use probability or quantum theory. E.g. you’re born with ½ probability, and if you’re born you go back and kill your grandfather, ergo you’re born with ½ probability, etc. Immediately suggests a model of computation, which can be shown to be exactly as powerful as the class PSPACE (A. 2005)

Quantum Gravity Freedman, Kitaev, Wang 2000: “Topological quantum field theories,” a particular class of (2+1)-dimensional quantum gravity theories, yield no more power than ordinary quantum computers String theory? Loop quantum gravity? It’d help if the physicists themselves understood these things better!

“Anthropic Computing” Guess a solution to an NP-complete problem. If it’s wrong, kill yourself. Suppose you could kill yourself in all universes where a quantum computer fails, then condition on remaining alive. What’s the class of problems you could then solve in polynomial time? A. 2005: It’s exactly the classical complexity class PP (Probabilistic Polynomial-Time), which is believed to be strictly larger than NP

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

Concluding Remark I know this talk seemed pessimistic… But I’m an optimist