1. BosonSampling Scott Aaronson (MIT) ICMP 2015, Santiago, Chile Based mostly on joint work with Alex Arkhipov

2. What This Talk Won’t Have What It Will Have P #P Oracle for Counting Problems NP Efficiently Checkable Problems P Efficiently Solvable Problems PH Constant Number of NP Quantifiers

3. Shor’s Theorem: Q UANTUM S IMULATION has no efficient classical algorithm, unless F ACTORING does also The Extended Church- Turing Thesis (ECT) Everything feasibly computable in the physical world is feasibly computable by a (probabilistic) Turing machine

4. So the ECT is false … what more evidence could anyone want? Building a QC able to factor large numbers is 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? F ACTORING 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?

5. BOSONS FERMIONS Our Starting Point In P#P-complete [Valiant] All I can say is, the bosons got the harder job

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

7. 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 =BPP NP 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

8. Valiant 2001, Terhal-DiVincenzo 2002, “folklore”: A QC built of noninteracting fermions can be efficiently simulated by a classical computer Related Work 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

9. 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!

10. 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

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

12. 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 U S,T is an n  n submatrix of U (possibly with repeated rows and columns), obtained by taking s i copies of the i th row of U and t j copies of the j th column for all i,j

13. Beautiful Alternate Perspective The “state” of our computer, at any time, is a degree-n polynomial over the variables x=(x 1,…,x m ) (n<<m) Initial state: p(x) := x 1  x n 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

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

15. Claim 1: p is #P-complete to estimate (up to a constant factor) Idea: Valiant proved that the P ERMANENT 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 BPP NP 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 =BPP NP.

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

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

18. “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

19. BosonSampling Experiments # of experiments > # of photons! 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

20. 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! Challenges for Scaling Up: -Reliable single-photon sources (optical multiplexing?) -Minimizing losses -Getting high probability of n-photon coincidence

21. 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

22. 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

23. 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/n O(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

24. 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 What can we say about these quantities? Are they also (plausibly) #P-hard to approximate? Work in progress with Daniel Brod

25. 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

26. Bonus: Rise and Fall of “Complexity” But how to quantify? One simpleminded measure: apparent complexity. The Kolmogorov complexity (estimated, say, by GZIP file size) of a coarse-grained (de-noised) description of our thermodynamic mixing process. Does it rise and then fall? Sean Carroll’s example:

27. The Coffee Automaton A., Carroll, Mohan, Ouellette, Werness 2015: A probabilistic n  n reversible system that starts half “coffee” and half “cream.” At each time step, we randomly “shear” half the coffee cup horizontally or vertically (assuming a toroidal cup) We prove that the apparent complexity of this image has a rising-falling pattern, with a maximum of at least ~n 1/6