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New Computational Insights from Quantum Optics Scott Aaronson

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What Is Quantum Optics? A rudimentary type of quantum computing, involving only non-interacting photons Classical counterpart: Galtons Board, on display at (e.g.) the Boston Museum of Science Using only pegs and non- interacting balls, you probably cant build a universal computer but you can do some interesting computations, like generating the binomial distribution!

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The Quantum Counterpart Lets replace the balls by identical single photons, and the pegs by beamsplitters Then the fact that photons obey Bose statistics leads to strange phenomena, like the Hong-Ou-Mandel dip The two photons are now correlated, even though they never interacted!

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Whats Going On? The amplitude for an n-photon final state in an optical experiment is a permanent: For example, the amplitude of the final state |1,1 in the Hong-Ou-Mandel experiment is where A=(a ij ) is an n n matrix of transition amplitudes for the individual photons The two contributions to the amplitude interfere destructively, cancelling each other out!

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So, Can We Use Quantum Optics to Calculate the Permanent? Explanation: To get a reasonable estimate of Per(A), you might need to repeat the optical experiment exponentially many times Theorem (Gurvits 2005): Theres an O(n 2 / 2 ) classical randomized algorithm to estimate the probability that there will be one photon in each of n slots, to accuracy A. 2011: Gurvitss algorithm can be generalized to estimate probabilities of arbitrary final states That sounds way too good to be trueit would let us solve NP-complete problems and more using QC!

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Even so, the fact that amplitudes are permanents does let us Use Quantum Optics to Solve Hard Sampling Problems [A.-Arkhipov, STOC 2011] Our Basic Result: Suppose there were a polynomial- time classical randomized algorithm that took as input a description of a quantum optics experiment, and output a sample from the correct final distribution over n-photon states. Then the polynomial hierarchy would collapse. Motivation: Compared to (say) Shors factoring algorithm, we get stronger evidence that a weaker system can do interesting quantum computations

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The Equivalence of Sampling and Searching [A., CSR 2011] [A.-Arkhipov] gave a sampling problem solvable using quantum optics that seems hard classicallybut does that imply anything about more traditional problems? Recently, I found a way to convert any sampling problem into a search problem of equivalent difficulty Basic Idea: Given a distribution D, the search problem is to find a string x in the support of D with large Kolmogorov complexity

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Using Quantum Optics to Prove that the Permanent is #P-Hard [A., Proc. Roy. Soc. 2011] Valiant famously showed that the permanent is #P-hard but his proof required strange, custom-made gadgets We gave a new, more transparent proof by combining three facts: (1)n-photon amplitudes correspond to n n permanents (2) Postselected quantum optics can simulate universal quantum computation [Knill-Laflamme-Milburn 2001] (3) Quantum computations can encode #P-hard quantities in their amplitudes

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Summary Thinking about quantum optics led to: - A new experimental quantum computing proposal - New evidence that QCs are hard to simulate classically - A new classical randomized algorithm for estimating permanents - A new proof of Valiants result that the permanent is #P-hard - (Indirectly) A new connection between sampling and searching

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Future Directions Do our optics experiment! Were in contact with two groups working to do so: Terry Rudolphs at Imperial College London and Andrew Whites in Brisbane, Australia Current focus: 3-4 photons Prove that even approximate classical simulation of our experiment is infeasible assuming PH is infinite Most of [A.-Arkhipov 2011] is devoted to a program for proving this, but big pieces remain Find more ways for quantum complexity theory to meet the experimentalists halfway

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