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BosonSampling

Delve into the world of BosonSampling, Scott Aaronson's Extended Church-Turing Thesis, and quantum computing implications. Discover the complexity, challenges, experiments, and future possibilities in quantum optics.

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BosonSampling

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  1. BosonSampling Scott Aaronson (MIT) April 18, 2014

  2. The Extended Church-Turing Thesis (ECT) Everything feasibly computable in the physical world is feasibly computable by a (probabilistic) Turing machine Shor’s Theorem:Quantum Simulation has no efficient classical algorithm, unless Factoring does also

  3. So the ECT is false … what more evidence could anyone want? • Building a QC able to factor large numbers is damn hard! After 16 years, no fundamental obstacle has been found, but who knows? • Can’t we “meet the physicists halfway,” and show computational hardness for quantum systems closer to what they actually work with now? • Factoring might be have a fast classical algorithm! At any rate, it’s an extremely “special” problem • Wouldn’t it be great to show that if, quantum computers can be simulated classically, then (say) P=NP?

  4. BosonSampling (A.-Arkhipov 2011) Classical counterpart: Galton’s Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip A rudimentary type of quantum computing, involving only non-interacting photons

  5. n identical photons enter, one per input mode Assume for simplicity they all leave in different modes—there are possibilities The beamsplitter network defines a column-orthonormal matrix ACmn, such that In general, we consider a network of beamsplitters, with n input “modes” (locations) and m>>n output modes where nnsubmatrix of A corresponding to S is the matrix permanent

  6. Example For Hong-Ou-Mandel experiment, In general, an nn complex permanent is a sum of n! terms, almost all of which cancel How hard is it to estimate the “tiny residue” left over? Answer:#P-complete, even for constant-factor approx (Contrast with nonnegative permanents!)

  7. So, Can We Use Quantum Optics to Solve a #P-Complete Problem? That sounds way too good to be true… Explanation: If X is sub-unitary, then |Per(X)|2 will usually be exponentially small. So to get a reasonable estimate of |Per(X)|2 for a given X, we’d generally need to repeat the optical experiment exponentially many times

  8. Better idea: Given ACmn as input, let BosonSampling be the problem of merely sampling from the same distribution DA that the beamsplitter network samples from—the one defined by Pr[S]=|Per(AS)|2 Theorem (A.-Arkhipov 2011): Suppose BosonSampling is solvable in classical polynomial time. Then P#P=BPPNP Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence that a weaker system can do something classically hard Better Theorem: Suppose we can sample DA even approximately in classical polynomial time. Then in BPPNP, it’s possible to estimate Per(X), with high probability over a Gaussian random matrix We conjecture that the above problem is already #P-complete. If it is, then even a fast classical algorithm for approximateBosonSampling would have the consequence thatP#P=BPPNP

  9. Related Work Valiant 2001, Terhal-DiVincenzo 2002, “folklore”:A QC built of noninteracting fermions can be efficiently simulated by a classical computer Knill, Laflamme, Milburn 2001: Noninteracting bosons plus adaptive measurementsyield universal QC Jerrum-Sinclair-Vigoda 2001: Fast classical randomized algorithm to approximate Per(A) for nonnegative A Gurvits 2002: O(n2/2) classical randomized algorithm to approximate an n-photon amplitude to ± additive error (also, to compute k-mode marginal distribution in nO(k) time)

  10. BosonSampling Experiments In 2012, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3-photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 permanents # of experiments > # of photons!

  11. Obvious Challenges for Scaling Up: • Reliable single-photon sources (optical multiplexing?) • Minimizing losses • Getting high probability of n-photon coincidence • Goal (in our view): Scale to 10-30 photons • Don’t want to scale much beyond that—both because • you probably can’t without fault-tolerance, and • a classical computer probably couldn’t even verify the results!

  12. Scattershot BosonSampling Exciting new idea, proposed by Steve Kolthammer, for sampling a hard distribution even with highly unreliable (but heralded) photon sources, like SPDCs The idea: Say you have 100 sources, of which only 10 (on average) generate a photon. Then just detect which sources succeed, and use those to define your BosonSampling instance! Complexity analysis goes through essentially without change Issues:Increases depth of optical network needed. Also, if some sources generate ≥2 photons, need a new hardness assumption

  13. Open Problems Prove that Gaussian permanent approximation is #P-hard (first step: understand distribution of Gaussian permanents) Can the BosonSampling model solve classically-hard decision problems? With verifiable answers? Can one efficiently sample a distribution that can’t be efficiently distinguished from BosonSampling? Similar hardness results for other natural quantum systems (besides linear optics)?Bremner, Jozsa, Shepherd 2010: Another system for which exact classical simulation would collapse PH

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