Boson Sampling Scott Aaronson MIT ICMP 2015 Santiago

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

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

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 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?

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

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

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

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

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 noninteracting balls, you probably can’t build a universal computer— but you can do some interesting computations, like generating the binomial distribution!

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

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

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

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

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

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/n. O(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

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 is 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.

“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

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!

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

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

Can Boson. Sampling 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

Boson. Sampling 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

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 Boson. Sampling 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

Bonus: Rise and Fall of “Complexity” Sean Carroll’s example: 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?

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) Compressed File Size We prove that the apparent complexity of this image has a rising-falling pattern, with a maximum of at least ~n 1/6 500 450 400 350 300 250 200 150 100 50 0 -100 300 500 Time Steps 700 900