Boson Sampling Scott Aaronson MIT Based on joint

Boson. Sampling Scott Aaronson (MIT) Based on joint work with Alex Arkhipov November 20, 2014 ar. Xiv: 1011. 3245

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

So the ECT is false … what more evidence could anyone want? Building a QC able to factor large numbers is damn hard! After 20 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?

Boson. Sampling (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

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 where is the matrix permanent n n submatrix of A corresponding to S

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 (Valiant 1979): #P-complete (meaning: as hard as any combinatorial counting problem) 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 Boson. Sampling 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 Boson. Sampling is #P=BPPNP solvable in classical polynomial time. Then P Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence Better Theorem: Suppose we can sample DA eventhat a weaker system can dopolynomial somethingtime. classically approximately in classical Thenhard 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 approximate Boson. Sampling would have the consequence that P#P=BPPNP

Related Work Valiant 2001, Terhal-Di. Vincenzo 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(X) for nonnegative X Gurvits 2002: O(n 2/ 2) classical randomized algorithm to approximate an n-photon amplitude to ± additive error (also, to compute k-mode marginal distribution in n. O(k) time)

OK, so why is it hard to sample the distribution over photon numbers classically? Given any matrix X Cn n, we can construct an m m unitary U (where m 2 n) as follows: Suppose we start with |I =|1, …, 1, 0, …, 0 (one photon in each of the first n modes), apply U, and measure. Then the probability of observing |I again is

Claim 1: p is #P-complete to estimate (up to a constant factor) This follows from Valiant’s famous result. Claim 2: Suppose we had a fast classical algorithm for boson sampling. Then we could estimate p in BPPNP— that is, using a randomized algorithm with an oracle for NP-complete problems This follows from a classical result of Goldwasser-Sipser Conclusion: Suppose we had a fast classical algorithm for boson sampling. Then P#P=BPPNP.

Unfortunately, this argument hinged on the hardness of estimating a single, exponentially-small probability p. As such, it’s not robust to realistic experimental error. Showing that a noisy Boson. Sampling device still samples a classically-intractable distribution is a much more complicated problem. As mentioned, we can do it, but only under an additional assumption (that estimating Gaussian permanents is #P-complete) A first step toward proving that conjecture, would simply be to understand the distribution of |Per(X)|2 for Gaussian X. Is it (as we conjecture) approximately lognormal?

Boson. Sampling Experiments In 2012, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3 -photon Boson. Sampling experiments, confirming that the amplitudes were given by 3 x 3 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 (1) you probably can’t without fault-tolerance, and (2) a classical computer probably couldn’t even verify the results!

Scattershot Boson. Sampling Exciting recent idea, proposed by Steve Kolthammer and others, 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 Boson. Sampling instance! Complexity analysis turns out to go through essentially without change

Polynomial-Time Verification of Boson. Sampling Devices? Idea 1: Let AS be the n n submatrix of A corresponding to output S. Let PS be the product of squared 2 -norms of AS’s rows. Check whether the observed distribution over PS is consistent with Boson. Sampling P under uniform distribution (a lognormal random variable) P under a Boson. Sampling distribution Idea 2: Let the scattering matrix U be a discrete Fourier transform. Then because of cancellations in the permanent, a ~1/n fraction of outcomes S should have probability 0. Check that these never occur.

Using Quantum Optics to Prove that the Permanent is #P-Complete [A. , Proc. Roy. Soc. 2011] Valiant showed that the permanent is #P-complete—but his proof required strange, custom-made gadgets We gave a new, arguably 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-complete quantities in their amplitudes

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