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Quantum Polynomial Time and the Human Condition / 2 Scott Aaronson (UC Berkeley) ―Computers are useless. They only give you answers.‖ –Pablo Picasso Not merely a false statement, but a pompous and asinine one Computers CAN ask questions Answers are not always useless (Picasso would say I‘m not addressing the meat of his objection) Computers led to some of the deepest questions ever asked If you can recognize good ideas, can Can quantum parallelism be harnessed Could a machine be (Does P=NP?) you also have them? conscious? to solve astronomically hard problems? Goal of Talk: Show that this question is ‗useful‘ in Picasso‘s sense These movies don‘t take their premise to its logical conclusion. Why can‘t you learn from 2n-1 alternate realities instead of just one? “Let a computer smear—with the right kind of quantum randomness —and you create, in effect, a „parallel‟ machine with an astronomical number of processors … All you have to do is be sure that when you collapse the system, you choose the version that happened to find the needle in the mathematical haystack.” From a scene in which the protagonist causes a computer to factor a huge number, by using his newfound ability to postselect quantum measurement outcomes The Popularizers Have Spoken Ridiculous! Nature couldn’t possibly allow this! QC of the sort that factors long numbers seems firmly rooted in science fiction … The present attitude would be analogous to, say, Maxwell selling the Daemon of his famous thought experiment as a path to cheaper electricity from heat. –Leonid Levin [T]he cost of such an operation or of maintaining such vectors should be linearly related to the amount of ―non-degeneracy‖ of these vectors, where the ―non-degeneracy‖ may vary from a constant to linear in the length of the vector. –Oded Goldreich Ridiculous! Nature couldn’t possibly allow this! [i]ndeed within the usual formalism one can construct quantum computers that may be able to solve at least a few specific problems exponentially faster than ordinary Turing machines. But particularly after my discoveries … I strongly suspect that even if this is formally the case, it will still not turn out to be a true representation of ultimate physical reality… –Stephen Wolfram It will never be possible to construct a ‗quantum computer‘ that can factor a large number faster, and within a smaller region of space, than a classical machine would do, if the latter could be built out of parts at least as large and as slow as the Planckian dimensions. –Gerard ‗t Hooft Crucial Question for Me Exactly what property separates the Sure States we know we can prepare, from the Shor States that suffice for factoring? DIVIDING LINE AmpP I hereby Circuit propose a complexity Tree P theory of pure TSH quantum OTree states n H2 Vidal MOTree 2 2 one of whose goals is to 1 1 study possible Sure/Shor Strict containment Containment separators. Non-containment Classical The tree size of an n-qubit state | is the minimum size of a tree of linear combinations and tensor products that represents |. (Size = # of leaf vertices) 00 01 10 11 Example: 1 2 + 1 1 1 + 1 1 + 1 |11 |12 2 2 2 2 |01 |11 |02 |12 | has tree size 6 Actual Technical Result A., quant-ph/0311039 Codewords of random stabilizer codes have superpolynomial tree size If C = {x | Axb(mod 2)}, where A is chosen uniformly at random from n1/ 3 n then with high probability 2 , 1 C C x xC requires trees of size nclog n even to approximate well Proof uses recent breakthrough of Ran Raz Conjecture: Same lower bound holds for states arising in Shor‘s algorithm Recent (as in last week) Developments 2-D cluster states (as proposed by Briegel and Raussendorf) have tree size nclog n. Not true for 1-D 0 0 1 0 1 0 1 1 1 0 NOTE1 FOR 0 0 1 1 PHYSICISTS: 1 0 1 0 0 1 0 0 1 1 I only care about Explicit (non-random) coset states, obtained by concatenating Reed-Solomon and Hadamard codes, have tree size qubit states nclog n Exponential lower bounds on ―manifestly orthogonal‖ tree size. So… Unless there‘s a clear, consistent dividing line between what we‘ve seen and what QM predicts we‘ll see, we ought to worry now about the ―quantum computing picture of reality‖ Could all paths of a maze be traversed simultaneously? In order to win the lottery, prove PNP, date a supermodel, etc., is it enough for it to be possible that you achieve these things? BBBV’97 Hybrid Argument Can a quantum algorithm that makes fewer than N queries find 1 marked item out of N? In the case that no items are marked, some item must have aDUDE!!! small total probability of being queried Everyone! The Mark that item and rerun the algorithm marked item! Someone must be Over here!!! screaming about a marked item… too bad quantum mechanics is linear 1|1 2|2 3|3 4|4 5|5 6|6 7|7 8|8 9|9 Actual Technical Result II A., quant-ph/0402095 Is there some initial state—even a highly entangled, not efficiently preparable one—that would let a quantum computer solve NP-complete problems in polynomial time? After all, such a state might encode information about every MAX CLIQUE problem of size n! We would therefore evade the BBBV conclusion Theorem: Relative to some oracle, NP BQP/qpoly Proof Idea Can be reduced to showing a direct product theorem for quantum search: Given N items, K of which are marked, if we don‘t have enough queries to find even one marked item, then the probability of finding all K of them decreases exponentially in K. Klauck gave an incorrect proof of this. I give the first correct proof, using the polynomial method of Beals et al. Recently improved by Klauck, Špalek, and de Wolf. So How Should You Solve NP- Complete Problems? Measure electron spins to guess a random solution. If the solution is wrong, kill yourself. If the solution is right, destroy the human race. If wrong, cause it to exist for billions of years. Actual Technical Result III (A., quant-ph/0401062): Let PostBQP be the class of problems solvable in quantum polynomial time using ―postselection.‖ Then PostBQP = PP Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows: 1 0 1 2 0 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows: 1 0 1 2 0 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows: 1 0 1 2 0 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows: 1 0 1 2 0 Observation: Given custom-designed 1-qubit nonlinear gates, we could even solve PSPACE- complete problems efficiently (but not more) But what about ―realistic‖ nonlinear gates (e.g. Weinberg‘s) subject to small environmental error? Abrams and Lloyd‘s claim to solve NP-complete problems in this setting seems incorrect Open Problem: The Two-Edged Sword Can we amplify an exponentially small success probability without also amplifying exponentially small errors? (Maybe Gwyneth would be better off without trans-universe communication!) Conclusion: The Garden of Forking Paths P Determinism BPP Randomness / 2 NP / PP Postselection BQP Quantumness ―Computers are useless. They only give you answers.‖ –Picasso

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polynomial time, quantum computer, Quantum computation, Quantum Computing, error probability, complexity classes, quantum state, quantum polynomial time, quantum algorithm, quantum computers

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posted: | 4/28/2010 |

language: | English |

pages: | 28 |

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