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

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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 kn c steps, for some constants k,c P is the class of all problems that have polynomial-time algorithms Computer Science 101

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NP: Nondeterministic Polynomial Time 37976595177176695379702491479374117272627593 30195046268899636749366507845369942177663592 04092298415904323398509069628960404170720961 97880513650802416494821602885927126968629464 31304735342639520488192047545612916330509384 69681196839122324054336880515678623037853371 49184281196967743805800830815442679903720933 Does have a prime factor ending in 7?

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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)?

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

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

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But what if P=NP, and the algorithm takes n 10000 steps? God will not be so cruel

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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 ~Kn 2 ), this would have consequences of the greatest magnitude. Gödel to von Neumann, 1956

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Then why is it so hard to prove P NP? Algorithms can be very clever Gödel/Turing-style self-reference arguments dont seem powerful enough Combinatorial arguments face the Razborov-Rudich barrier

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But maybe theres some physical system that solves an NP-complete problem just by reaching its lowest energy state?

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

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

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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!

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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 wont be enough ~2 n/2 queries are needed to search a list of size 2 n for a single marked item A. 2004: True even with quantum advice

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Quantum Adiabatic Algorithm (Farhi et al. 2000) HiHi Hamiltonian with easily-prepared ground state HfHf Ground state encodes solution to NP- complete problem Problem (van Dam, Mosca, Vazirani 2001): Eigenvalue gap can be exponentially small

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Relativity Computing DONE

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

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Nonlinear Quantum Mechanics (Weinberg 1989) Abrams & Lloyd 1998: Could use to solve NP-complete and even harder problems in polynomial time No solutions 1 solution to NP-complete problem

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Time Travel Computing (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

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Hidden Variables Valentini 2001: Subquantum algorithm (violating | | 2 ) to distinguish |0 from Problem: Valentinis 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 timebut again, probably not NP-complete problems!

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Quantum Gravity

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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.

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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 its done, then tell those copies the answer

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Second Law of Thermodynamics Proposed Counterexamples

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No Superluminal Signalling Proposed Counterexamples

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Intractability of NP-complete problems Proposed Counterexamples ?

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