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Quantum Polynomial Time and the Human Condition Scott Aaronson (UC Berkeley)

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1 Quantum Polynomial Time and the Human Condition Scott Aaronson (UC Berkeley)

2 Computers are useless. They only give you answers. –Pablo Picasso DUNCE Not merely a false statement, but a pompous and asinine one

3 Computers CAN ask questions

4 Answers are not always useless

5 (Picasso would say Im not addressing the meat of his objection)

6 Computers led to some of the deepest questions ever asked Could a machine be conscious? If you can recognize good ideas, can you also have them? (Does P=NP?) Can quantum parallelism be harnessed to solve astronomically hard problems? Goal of Talk: Show that this question is useful in Picassos sense

7 Wonderful Life

8 These movies dont take their premise to its logical conclusion. Why cant you learn from 2 n -1 alternate realities instead of just one?

9 Quarantine Let a computer smearwith 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

10 Quarantine See, quantum computers, by taking advantage of weird quantum phenomena which make no sense and no one understands but the numbers work out so shut the fuck up and take itquantum computers are able to compute all possible computations at the same time, by existing simultaneously in an infinite number of parallel universes. Popular Eschatology Unlike a laboratory rat or an ordinary computer, which must probe the pathways one at a time, the quantum computer can simultaneously traverse every twist and turn and immediately converge upon the prize. –George Johnson, Slate The Popularizers Have Spoken

11 Quarantine [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 couldnt 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

12 Quarantine 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 Ridiculous! Nature couldnt 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

13 Exactly what property separates the Sure States we know we can prepare, from the Shor States that suffice for factoring? DIVIDING LINE Crucial Question for Me

14 I hereby propose a complexity theory of pure quantum states one of whose goals is to study possible Sure/Shor separators. Classical Vidal Circuit AmpP MOTree OTree TSH Tree P Strict containment Containment Non-containment

15 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) Example: + |1 1 | |0 1 |1 1 |0 2 |1 2 | has tree size 6

16 Actual Technical Result A., quant-ph/ If C = {x | Ax b(mod 2)}, where A is chosen uniformly at random from then with high probability requires trees of size n c log n even to approximate well Proof uses recent breakthrough of Ran Raz Conjecture: Same lower bound holds for states arising in Shors algorithm Codewords of random stabilizer codes have superpolynomial tree size

17 Recent (as in last week) Developments 2-D cluster states (as proposed by Briegel and Raussendorf) have tree size n c log n. Not true for 1-D Explicit (non-random) coset states, obtained by concatenating Reed-Solomon and Hadamard codes, have tree size n c log n Exponential lower bounds on manifestly orthogonal tree size NOTE FOR PHYSICISTS: I only care about qubit states

18 So… Unless theres a clear, consistent dividing line between what weve seen and what QM predicts well 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 P NP, date a supermodel, etc., is it enough for it to be possible that you achieve these things?

19 BBBV97 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 a small total probability of being queried Mark that item and rerun the algorithm 1 |1 2 |2 3 |3 4 |4 5 |5 6 |6 7 |7 8 |8 9 |9 DUDE!!! Everyone! The marked item! Over here!!! Someone must be screaming about a marked item… too bad quantum mechanics is linear

20 Actual Technical Result II A., quant-ph/ Is there some initial stateeven a highly entangled, not efficiently preparable onethat 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

21 Proof Idea Can be reduced to showing a direct product theorem for quantum search: Given N items, K of which are marked, if we dont 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.

22 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/ ): Let PostBQP be the class of problems solvable in quantum polynomial time using postselection. Then PostBQP = PP THATS IT?

23 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows:

24 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows:

25 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows:

26 Nonlinear Quantum Computing Abrams & Lloyd 1998: We could solve NP- complete problems efficiently given a 1-qubit nonlinear gate that acts as follows:

27 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. Weinbergs) subject to small environmental error? Abrams and Lloyds 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!)

28 Conclusion: The Garden of Forking Paths DeterminismRandomness PostselectionQuantumness Computers are useless. They only give you answers. –Picasso PBPP NP / PPBQP


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