1. Verification of BosonSampling Devices Scott Aaronson (MIT) Talk at Simons Institute, February 28, 2014

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

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? F ACTORING 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) A rudimentary type of quantum computing, involving only non-interacting photons Classical counterpart: Galton’s Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip

5. In general, we consider a network of beamsplitters, with n input “modes” (locations) and m>>n output modes 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  C m  n, such that where is the matrix permanent n  n submatrix of A corresponding to S

6. So, Can We Use Quantum Optics to Solve a #P-Complete Problem? 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 That sounds way too good to be true…

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

8. BosonSampling Experiments # of experiments ≥ # of photons! Last year, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3- and 4- photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 and 4x4 permanents Obvious challenge for scaling up: Need n-photon coincidences (requires either postselection or deterministic single-photon sources) Recent idea: Scattershot BosonSampling

9. Verifying BosonSampling Devices Crucial difference from factoring: Even verifying the output of a claimed BosonSampling device would presumably take exp(n) time, in general! Recently underscored by [Gogolin et al. 2013] (alongside specious claims…) Our responses: (1)Who cares? Take n=30 (2) If you do care, we can show how to distinguish the output of a BosonSampling device from all sorts of specific “null hypotheses”

10. Is a BosonSampling device’s output just uniform noise? No way, not even close (A.-Arkhipov, arXiv:1309.7460) Under the uniform distribution Histogram of (normalized) probabilities under a Haar- random BosonSampling distribution

11. Theorem (A. 2013): Let A  C m  n be Haar-random, where m>>n. Then there’s a classical polytime algorithm C(A) that distinguishes the BosonSampling distribution D A from the uniform distribution U (whp over A, and using only O(1) samples) Strategy: Let A S be the n  n submatrix of A corresponding to output S. Let P be the product of squared 2-norms of A S ’s rows. If P>E[P], then guess S was drawn from D A ; otherwise guess S was drawn from U P under uniform distribution (a lognormal random variable) P under a BosonSampling distribution A ASAS

12. Recent realization: You can use the number of multi- photon collisions to efficiently distinguish D A from E A Given a matrix, A, let E A be like the BosonSampling distribution D A, but with distinguishable particles: Observe that the row-norm estimator, P, fails completely to distinguish D A from E A ! (Why?) Conjecture: Could also distinguish without looking at collisions

13. The Classical Mockup Challenge Given a matrix A  C m  n, is there some classically efficiently-samplable distribution C A, which is indistinguishable from the BosonSampling distribution D A by any polynomial-time algorithm? Observation: If we just wanted an efficiently-samplable distribution that’s indistinguishable from D A by any (say) n 2 -time algorithm, that’s trivial to get! Brandao: We can even get such a mockup distribution with a large min-entropy, using Trevisan-Tulsiani-Vadhan

14. The NP Challenge Can our linear-optics model solve a classically-intractable problem (say, a search or decision problem) for which a classical computer can efficiently verify the answer? Given an n  n matrix with large (1/poly(n)) permanent, can one “smuggle” it as a submatrix of a unitary matrix? What kinds of (sub)unitary matrices can have ≥1/poly(n) permanents? Must every such matrix be “close to the identity,” in some sense? Arkhipov: Every unitary with permanent ≥1-1/e has a “large” diagonal

15. The Interactive Protocol Challenge Can a BosonSampling device convince a classical skeptic of its post-classical powers via an interactive protocol? Arora et al. 2012: An oracle for Gaussian permanent estimation would be self-checkable. (But alas, a BosonSampling device is not such an oracle!)