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Limits on Efficient Computation in the Physical World Scott Aaronson MIT Things we never see… GOLDBACH CONJECTURE: TRUE NEXT QUESTION Warp drive Perpetuum mobile Übercomputer $3 billion But does the absence of these devices have any scientific importance? Goal of talk: Explain why the impossibility of übercomputers is a great question for 21st-century science CS Theory 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 37976595177176695379702491479374117272627593 30195046268899636749366507845369942177663592 04092298415904323398509069628960404170720961 97880513650802416494821602885927126968629464 31304735342639520488192047545612916330509384 69681196839122324054336880515678623037853371 49184281196967743805800830815442679903720933 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-hard Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … NPcomplete NP Graph connectivity Primality testing Matrix determinant Linear programming … P Matrix permanent Halting problem … Factoring Graph isomorphism … Does P=NP? The (literally) $1,000,000 question Q: What if P=NP, and the algorithm takes n10000 steps? A: Then we’d just change the question! Q: Why is it so hard to prove PNP? A: Because polynomial-time algorithms are so rich What about quantum computers? BQP: Bounded-Error Quantum Polynomial-Time Shor 1994: BQP contains integer factoring But factoring isn’t believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? Bennett et al. 1997: “Quantum magic” won’t be enough If we throw away the problem structure, and just consider a “landscape” of 2n possible solutions, even a quantum computer needs ~2n/2 steps to find a correct solution Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hamiltonian with easily-prepared ground state Hf Ground state encodes solution to NPcomplete problem Problem: Eigenvalue gap can be exponentially small Other Alleged Ways to Solve NP-complete Problems Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: A proposal for massively parallel classical computing The cognitive science approach: Think about it really hard My Personal Favorite Dip two glass plates with pegs between them into soapy water; let the soap bubbles form a minimum “Steiner tree” connecting the pegs (thereby solving a known NP-complete problem) What would the world actually be like if we could solve NP-complete problems efficiently? Proof of Riemann hypothesis with 10,000,000 symbols? Shortest efficient description of stock market data? 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 The NP Hardness Assumption There is no physical means to solve NP complete problems in polynomial time. Alright, what can we say about this assumption? • Implies, but is stronger than, PNP Rest of talk: Try to give indications • As falsifiable as it getsthat it is • Consistent with currently-known physical theory • Scientifically fruitful? 1. “Relativity Computing” DONE 2. Topological Quantum Field Theories (TQFT’s) Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers 3. Nonlinear variants of the Schrödinger Equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time Can take as an additional argument for why QM is linear 1 solution to NP-complete problem No solutions 4. Anthropic Principle Foolproof way to solve NP-complete problems in polynomial time (at least in the Many-Worlds Interpretation): First guess a random solution. Then, if it’s wrong, kill yourself Technicality: If there are no solutions, you’d seem to be out of luck! Solution: With tiny probability don’t do anything. Then, if you find yourself in a universe where you didn’t do anything, there probably were no solutions, since otherwise you would’ve found one What if we combine quantum computing with the Anthropic Principle? I.e. perform a polynomial-time quantum computation, but where we can measure a qubit and assume the outcome will be |1 Leads to a new complexity class: PostBQP (Postselected BQP) A. 2005: PostBQP=PP—and this yields a 1page proof of the Beigel-Reingold-Spielman theorem, that PP is closed under intersection 5. Time Travel Everyone’s first idea for a time travel computer: Do an arbitrarily long computation, then send the answer back in time to before you started THIS DOES NOT WORK Why not? • Ignores the Grandfather Paradox • Doesn’t take into account the computation you’ll have to do after getting the answer Deutsch’s Model A closed timelike curve (CTC) is a computational resource that, given an efficiently computable function f:{0,1}n{0,1}n, immediately finds a fixed point of f— that is, an x such that f(x)=x Admittedly, not every f has a fixed point But there’s always a distribution D such that f(D)=D Probabilistic Resolution of the Grandfather Paradox - You’re born with ½ probability - If you’re born, you back and kill your grandfather - Hence you’re born with ½ probability Let PCTC be the class of problems solvable in polynomial time, if for any function f:{0,1}n{0,1}n described by a poly-size circuit, we can immediately get an x{0,1}n such that f(m)(x)=x for some m Theorem: PCTC = PSPACE What if we perform a quantum computation around a CTC? Let BQPCTC be the class of problems solvable in quantum polynomial time, if for any superoperator E described by a quantum circuit, we can immediately get a mixed state such that E() = Clearly PSPACE = PCTC BQPCTC A., Watrous 2008: BQPCTC = PSPACE If closed timelike curves exist, then quantum computers are no more powerful than classical ones Concluding Remarks Are NP-complete problems intractable in the physical universe? I conjecture that they are, but fully understanding why will bring in: • Math and computer science (duh): The P vs. NP question Prediction: The “NP Hardness Assumption” will eventually be seen as analogous to Second Law • Quantum mechanics: The NP vs. BQP question of Thermodynamics or the impossibility of • Other physics: superluminal Quantum fieldsignaling theory, quantum gravity, closed timelike curves… • Biology, cognitive science, economics? Open Question: What is “polynomial time” in quantum gravity? Scientific American, March 2008: www.scottaaronson.com