1. BosonSampling Scott Aaronson (University of Texas, Austin)
Conference on Integrated Quantum Photonics Rome, September 26, 2017
Based mostly on joint work with Alex Arkhipov

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 hard! After 23 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. Our Starting Point In P #P-complete [Valiant] BOSONS FERMIONS
Seems unfair that the bosons got the harder job
In P
#P-complete [Valiant]
BOSONS
FERMIONS

5. Can We Use Bosons to Calculate the Permanent?
So if n-boson amplitudes correspond to permanents…
Can We Use Bosons to Calculate the Permanent?
That sounds way too good to be true—it would let us solve NP-complete problems and more using QC!
Explanation: Amplitudes aren’t directly observable.
To get a reasonable estimate of Per(A), you might need to repeat the experiment exponentially many times

6. Then P#P=BPPNP and the polynomial hierarchy collapses.
Basic Result: Suppose there were a polynomial-time classical randomized algorithm that took as input a description of a noninteracting-boson experiment, and that output a sample from the correct final distribution over n-boson states.
Then P#P=BPPNP and the polynomial hierarchy collapses.
Motivation: Compared to (say) Shor’s algorithm, we get “stronger” evidence that a “weaker” system can do interesting quantum computations

7. 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
Bremner-Jozsa-Shepherd 2011 (independent of us): Analogous hardness results for simulating “commuting Hamiltonian” quantum computers

8. The Quantum Optics Model
A rudimentary subset of quantum computing, involving only non-interacting bosons, and not based on qubits
Classical counterpart: Galton’s Board, on display at many science museums
Using only pegs and non-interacting balls, you probably can’t build a universal computer—but you can do some interesting computations, like generating the binomial distribution!

9. The Quantum Version
Let’s replace the balls by identical single photons, and the pegs by beamsplitters
Then we see strange things like the Hong-Ou-Mandel dip
The two photons are now correlated, even though they never interacted!
Explanation involves destructive interference of amplitudes: Final amplitude of non-collision is

10. Getting Formal
The basis states have the form |S=|s1,…,sm, where si is the number of photons in the ith “mode”
We’ll never create or destroy photons. So s1+…+sm=n is constant.
For us, m=nO(1) U
Initial state: |I=|1,…,1,0,……,0 

11. You get to apply any mm unitary matrix U—say, using a collection of 2-mode beamsplitters
In general, there are ways to distribute n identical photons into m modes
U induces an MM unitary (U) on the n-photon states as follows:
Here US,T is an nn submatrix of U (possibly with repeated rows and columns), obtained by taking si copies of the ith row of U and tj copies of the jth column for all i,j

12. Beautiful Alternate Perspective
The “state” of our computer, at any time, is a degree-n polynomial over the variables x=(x1,…,xm) (n<<m)
Initial state: p(x) := x1xn
We can apply any mm unitary transformation U to x, to obtain a new degree-n polynomial
Then on “measuring,” we see the monomial with probability

13. OK, so why is it hard to sample the distribution over photon numbers classically?
Given any matrix A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

14. Claim 1: p is #P-complete to estimate (up to a constant factor)
Idea: Valiant proved that the Permanent is #P-complete.
Can use a classical reduction to go from a multiplicative approximation of |Per(A)|2 to Per(A) itself.
Claim 2: Suppose we had a fast classical algorithm for boson sampling. Then we could estimate p in BPPNP
Idea: Let M be our classical sampling algorithm, and let r be its randomness. Use approximate counting to estimate
Conclusion: Suppose we had a fast classical algorithm for boson sampling. Then P#P=BPPNP.

15. The Elephant in the Room
The previous result hinged on the difficulty of estimating a single, exponentially-small probability p—but what about noise and error?
The “right” question: can a classical computer efficiently sample a distribution with 1/nO(1) variation distance from the boson distribution?
Our Main Result: Suppose it can. Then there’s a BPPNP algorithm to estimate |Per(A)|2, with high probability over a Gaussian matrix

16. Our Main Conjecture
Estimating |Per(A)|2, with high probability over i.i.d. Gaussian A, is a #P-hard problem
If this conjecture holds, then even a noisy n-photon experiment could falsify the Extended Church Thesis, assuming P#PBPPNP!
Much of our work was devoted to giving evidence for this conjecture
What makes the Gaussian ensemble special? Theorem: It arises by considering sufficiently small submatrices of Haar-random unitary matrices.

17. “Easier” problem: Just show that, if A is an i. i. d
“Easier” problem: Just show that, if A is an i.i.d. Gaussian matrix, then |Per(A)|2 is approximately a lognormal random variable (as numerics suggest), and not so concentrated around 0 as to preclude its being hard to estimate
Can prove for determinant in place of permanent. For permanent, best known anti-concentration results [Tao-Vu] are not yet strong enough for us
Can calculate E[|Per(A)|2]=n! and E[|Per(A)|4]=(n+1)(n!)2, but not strong enough to imply anti-concentration result

18. BosonSampling Experiments
Initial experiments with 3 photons (groups in Rome, Oxford, Vienna, and Brisbane)
Carolan et al. 2015: With 6 photons, but initial states of the form |3,3
Wang et al. 2017: With 5 photons, initial states of the form |1,1,1,1,1

19. 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!

20. Scattershot BosonSampling
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

21. 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

22. Can BosonSampling Solve Non-Sampling Problems
Can BosonSampling Solve Non-Sampling Problems? (Could it even have cryptographic applications?)
Idea: What if we could “smuggle” a matrix A with huge permanent, as a submatrix of a larger unitary matrix U? Finding A could be hard classically, but shooting photons into an interferometer network would easily reveal it
Pessimistic Conjecture: If U is unitary and |Per(U)|1/nO(1), then U is “close” to a permuted diagonal matrix—so it “sticks out like a sore thumb”
A.-Nguyen, Israel J. Math 2014: Proof of a weaker version of the pessimistic conjecture, using inverse Littlewood-Offord theory

23. BosonSampling with Lost Photons
Suppose we have n+k photons in the initial state, but k are randomly lost. Then the probability of each output has the form
A.-Brod 2016: For any constant number of losses, k=O(1), the above quantities are #P-hard to approximate, assuming |Per(A)|2 itself is.
We don’t know what happens for larger k...

24. Summary
Intuition suggests that not merely quantum computers, but many natural quantum systems, should be intractable to simulate on classical computers, because of the exponentiality of the wavefunction
BosonSampling provides a clear example of how we can formalize this intuition—or at least, base it on “standard” conjectures in theoretical computer science.
It’s also brought QC theory into closer contact with experiment. And it’s highlighted the remarkable connection between bosons and the matrix permanent. Future progress may depend on solving hard open problems about the permanent