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BQP PSPACE NP P PostBQP Computational Intractability As A Law of Physics Scott Aaronson University of Waterloo

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Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION Übercomputer Is the absence of these devices something physicists should think about? YES Goal of talk: Convince you to see the impossibility of übercomputers as a basic principle of physics

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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 …

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Does P=NP? The (literally) $1,000,000 question Q: What if P=NP, and the algorithm takes n 10000 steps? A: Then wed just change the question! Q: Why is it so hard to prove P NP? A: Mostly because algorithms can be so clever! No.

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What about quantum computers? Shor 1994: BQP contains integer factoring But factoring isnt believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? Bennett et al. 1997: Quantum magic wont be enough If we throw away the problem structure, and just consider a landscape of 2 n possible solutions, even a quantum computer needs ~2 n/2 steps to find a correct solution BQP: Bounded-Error Quantum Polynomial-Time

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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: Eigenvalue gap can be exponentially small

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Other Alleged Ways to Solve NP-complete Problems 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) Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: Just massively parallel classical computers!

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What would the world actually be like if we could solve NP-complete problems efficiently? 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 Proof of Riemann hypothesis with 10,000,000 symbols? Shortest efficient description of stock market data?

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Implies, but is stronger than, P NP As falsifiable as it gets Consistent with currently-known physical theory Scientifically fruitful? Alright, what can we say about this assumption? The NP Hardness Assumption There is no physical means to solve NP complete problems in polynomial time. Rest of talk: Show how complexity yields a new perspective on linearity of QM, anthropic postselection, closed timelike curves, and initial conditions

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1. 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 No solutions 1 solution to NP-complete problem Can take as an additional argument for why QM is linear

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2. 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 its wrong, kill yourself! Technicality: If there are no solutions, youre out of luck! Solution: With tiny probability dont do anything. Then, if you find yourself in a universe where you didnt do anything, there probably were no solutions, since otherwise you wouldve found one! NP Hardness Assumption yields a nontrivial constraint on anthropic theorizing: no use of the Anthropic Principle can be valid, if its validity would give us a way to solve NP-complete problems in polynomial time

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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) Certainly PostBQP contains NPbut is it even bigger than that?

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Some more animals from the complexity zoo… PSPACE: Class of problems solvable with a polynomial amount of memory PP: Class of problems of the form, out of 2 n possible solutions, are at least half of them correct? Adleman, DeMarrais, Huang 1998: BQP PP Proof: Feynman path integral Proof easily extends to show PostBQP PP

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BQP PP NP P PostBQP PSPACE A. 2004: PostBQP = PP In other words, quantum postselection gives exactly the power of PP Surprising part: This characterization yields a half-page proof of a celebrated result of Beigel, Reingold, and Spielman, that PP is closed under intersection

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Everyones 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 Doesnt take into account the computation youll have to do after getting the answer 3. Time Travel

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Deutschs 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 theres always a distribution D such that f(D)=D Probabilistic Resolution of the Grandfather Paradox - Youre born with ½ probability - If youre born, you back and kill your grandfather - Hence youre born with ½ probability

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Let P CTC 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: P CTC = PSPACE Proof: P CTC PSPACE is easy For PSPACE P CTC : Let s init, s acc, and s rej be the initial, accepting, and rejecting states of a PSPACE machine, and let (s) be the successor state of s. Then set The only fixed point is an infinite loop, with b set to its true value

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What if we perform a quantum computation around a CTC? Let BQP CTC 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 = P CTC BQP CTC A., Watrous 2006: BQP CTC = PSPACE If closed timelike curves exist, then quantum computers are no more powerful than classical ones

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BQP CTC PSPACE: Proof Sketch Furthermore, we can compute P exactly in PSPACE, using Csankys parallel algorithm for matrix inversion Solution: Let Then by Taylor expansion, Hence P projects onto the fixed points of M Let vec( ) be a vectorization of. We can reduce the problem to the following: given a 2 2n 2 2n matrix M, prepare a state such that

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4. Initial Conditions Useful? Normally we assume a quantum computer starts in an all-0 state, |0…0. But what if much better initial states were created in the Big Bang, and have been sitting around ever since? Leads to the concept of quantum advice… |

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Result #1: BQP/qpoly PostBQP/poly Any problem you can solve using short quantum advice, you can also solve using short classical advice, provided youre willing to use exponentially more computation time to extract what the advice is telling you. Result #2: There exists an oracle relative to which NP BQP/qpoly Limitations of Quantum Advice A., 2004 Evidence that NP-complete problems are still hard for quantum computers in the presence of quantum advice One can postulate bizarre, exponentially-hard-to-prepare initial states in Nature, without violating the NP Hardness Assumption

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Concluding Remarks Prediction: NP Hardness Assumption will eventually be seen as analogous to Second Law of Thermodynamics or impossibility of superluminal signaling COMPUTATIONAL COMPLEXITY PHYSICS THIS BRIDGE ALREADY EXISTS Open Question: What is polynomial time in quantum gravity? (First question: What is time in quantum gravity?)

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Links to papers, etc.: www.scottaaronson.com

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