1. Based on joint work with Alex Arkhipov
BosonSampling
Scott Aaronson (MIT)
Based on joint work with Alex Arkhipov
November 20, 2014
arXiv:

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 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?

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. nn submatrix of A corresponding to S
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

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 (Valiant 1979): #P-complete (meaning: as hard as any combinatorial counting problem)
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 approximate BosonSampling would have the consequence that P#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 measurements yield universal QC
Jerrum-Sinclair-Vigoda 2001: Fast classical randomized algorithm to approximate Per(X) for nonnegative X
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. 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2n) 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

11. 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.

12. 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 BosonSampling 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?

13. 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!

14. 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 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!

15. Scattershot BosonSampling
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 BosonSampling instance!
Complexity analysis turns out to go through essentially without change

16. Polynomial-Time Verification of BosonSampling 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 BosonSampling
P under uniform distribution (a lognormal random variable)
P under a BosonSampling 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.

17. Using Quantum Optics to Prove that the Permanent is #P-Complete [A
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:
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

18. 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 the polynomial hierarchy