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 ? Circuit of size that does best at predicting stock market data Shortest program that outputs works of Shakespeare in 10 7 steps

If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), 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 P NP 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 NP coNP, 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 ( bits/m 2 ) (5) Area quantization in loop quantum gravity (6) Cosmic rays above GZK cutoff (~10 20 eV) (7) Independence of AC and CH? cm

Quantum Computing Shor 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 wont be enough Given a black box function f:{0,1} n {0,1}, a quantum computer needs (2 n/2 ) queries to f to find an x such that f(x)=1 Thus NP A BQP A relative to some oracle A

Quantum Advice BQP/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: NP A BQP A /qpoly relative to some oracle A. Proof based on direct product theorem for quantum search

Quantum Adiabatic Algorithm (Farhi et al. 2000) HiHi Hamiltonian with easily-prepared ground state HfHf Ground state encodes solution to 3SAT instance van Dam, Mosca, Vazirani 2001; Reichardt 2004: Takes exponential time on some 3SAT instances (1-s)H i +sH f Quantum analogue of simulating annealing Numerical data suggested polynomial running time

Topological Quantum Field Theories (TQFTs) Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

Non-Collapsing Measurements To solve Graph Isomorphism: Given G and H, prepare If only we could measure both | |0 and | |1 without collapsing, wed solve the problem… (Generalizes to all problems in SZK) After we measure third register, first two registers will have the form if G H, if not

A. 2002: Any quantum algorithm needs (N 1/5 ) 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 SZK A BQP A 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

Nonlinear Quantum Mechanics Abrams & Lloyd 1998: Could use to solve NP-complete and even #P-complete problems in polynomial time No solutions 1 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 C Causal loop x C(x) To solve SAT: Let C(x)=x if x is a satisfying assignment, C(x)=x+1(mod 2 n ) otherwise Model: We choose C, then a fixpoint distribution D over x is chosen adversarially, then an x D is sampled To solve PSPACE-complete problems: Exercise for the audience…

Time Travel Computing with 1 Looping Bit (Adapted from Bacon 2003) x y x y x Chronology-respecting bit 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) Causal loop

Quantum Gravity Probabilities that dont sum to 1 unless theyre 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 its 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 BPP path (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 ?