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**NP-complete Problems and Physical Reality**

Scott Aaronson UC Berkeley IAS

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**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

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**NP: Nondeterministic Polynomial Time**

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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**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

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

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

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**But what if P=NP, and the algorithm takes n10000 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 ~Kn2), 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 PNP?**

Algorithms can be very clever Gödel/Turing-style self-reference arguments don’t seem powerful enough Combinatorial arguments face the “Razborov-Rudich barrier”

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But maybe there’s 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**

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

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**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

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**“Relativity Computing”**

DONE

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**Topological Quantum Field Theories (TQFT’s)**

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 1 solution to NP-complete problem No solutions

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**Time Travel Computing (Bacon 2003)**

Suppose Pr[x=1] = p, Pr[y=1] = q Then consistency requires p=q So Pr[xy=1] = p(1-q) + q(1-p) = 2p(1-p) xy x Causal loop Chronology-respecting bit x y

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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!

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Quantum Gravity

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**“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.

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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 it’s 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**

? Intractability of NP-complete problems Proposed Counterexamples

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