Scott Aaronson (MIT) Talk at SITP, February 21, 2014 BosonSampling Scott Aaronson (MIT) Talk at SITP, February 21, 2014

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

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?

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

n identical photons enter, one per input mode 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Cmn, such that nn submatrix of A corresponding to S where is the matrix permanent For simplicity, I’m ignoring outputs with ≥2 photons per mode

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!)

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

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 approximate BosonSampling would already have the consequence that P#P=BPPNP

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 measurements yield universal QC Jerrum-Sinclair-Vigoda 2001: Fast classical randomized algorithm to approximate Per(A) for nonnegative A Gurvits 2002: Fast classical randomized algorithm to approximate n-photon amplitudes to ± additive error

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 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! Theoretical Challenge: Argue that, even with photon losses and messier initial states, you’re still solving a classically-intractable sampling problem

Scattershot BosonSampling Wonderful new idea, proposed by several experimental groups, for sampling a hard distribution even with highly unreliable (but heralded) photon sources, like Spontaneous Parametric Downconversion (SPDC) crystals The idea: Say you have 100 sources, of which only 10 (on average) generate a photon. Then just detect which sources succeeded, and use those to define your BosonSampling instance! Complexity analysis goes through essentially without change

Verifying BosonSampling Devices As mentioned before, 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”

Theorem (A. 2013): Let ACmn be Haar-random, where m>>n 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n submatrix 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 Recent realization: You can also use the number of multi-photon collisions to distinguish DA from DA’, the same distribution but with classical distinguishable particles A AS P under uniform distribution (a lognormal random variable) P under a BosonSampling distribution

Turning the Logic Around [A., Proc. Roy. Soc. 2011] Arkhipov and I used the #P-completeness of the permanent—a great discovery of CS theory from the 1970s—to argue that bosonic sampling is hard for a classical computer Later, I realized that one can also go in the reverse direction! Using the power of postselected linear-optical quantum computing—shown by [Knill-Laflamme-Milburn 2001]—and the connection between LOQC and the permanent, I gave a new, arguably-simpler proof that the permanent is #P-complete

Open Problems Prove that approximating the permanent of an i.i.d. Gaussian matrix is #P-hard! Can our linear-optics model solve a classically-intractable problem for which a classical computer can efficiently verify the answer? 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