Verification of BosonSampling Devices

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

7. 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 a fast classical algorithm for approximateBosonSampling would already have the consequence thatP#P=BPPNP

8. BosonSampling Experiments 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 # of experiments ≥ # of photons! 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: • Who cares? Take n=30 • 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) Histogram of (normalized) probabilities under a Haar-random BosonSampling distribution Under the uniform distribution

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

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

13. The Classical Mockup Challenge Given a matrix ACmn, is there some classically efficiently-samplable distribution CA, which is indistinguishable from the BosonSampling distribution DA by any polynomial-time algorithm? Observation: If we just wanted an efficiently-samplable distribution that’s indistinguishable from DA by any (say) n2-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!)