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New Computational Insights from Quantum Optics Scott Aaronson Based on joint work with Alex Arkhipov

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The Extended Church- Turing Thesis (ECT) Everything feasibly computable in the physical world is feasibly computable by a (probabilistic) Turing machine But building a QC able to factor n>>143 is damn hard! Cant CS meet physics halfway on this one? I.e., show computational hardness in more easily-accessible quantum systems? Also, factoring is a special problem

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

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Can We Use Bosons to Calculate the Permanent? Explanation: Amplitudes arent 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 trueit would let us solve NP-complete problems and more using QC! So if n-boson amplitudes correspond to permanents…

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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) Shors algorithm, we get stronger evidence that a weaker system can do interesting quantum computations

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Crucial step we take: switching attention to sampling problems BQP P #P BPP BPP NP PH F ACTORING P ERMANENT 3SAT X Y … SampP SampBQP

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Valiant 2001, Terhal-DiVincenzo 2002, folklore: A QC built of noninteracting fermions can be efficiently simulated by a classical computer Related Work A. 2011: Generalization of Gurvitss algorithm that gives better approximation if A has repeated rows or columns 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 Gurvits 2005: O(n 2 / 2 ) classical randomized algorithm to approximate Per(A) to additive error |A| n Bremner, Jozsa, Shepherd 2011: Commuting Hamiltonians

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The Quantum Optics Model A rudimentary subset of quantum computing, involving only non-interacting bosons, and not based on qubits Classical counterpart: Galtons Board, on display at the Boston Museum of Science Using only pegs and non- interacting balls, you probably cant build a universal computer but you can do some interesting computations, like generating the binomial distribution!

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The Quantum Version Lets 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

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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 Well never create or destroy photons. So s 1 +…+s m =n is constant. Initial state: |I =|1,…,1,0,……,0 For us, m=poly(n) U

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You get to apply any m m unitary matrix U If n=1 (i.e., theres only one photon, in a superposition over the m modes), U acts on that photon in the obvious way 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

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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

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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.

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

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Estimating |Per(A)| 2, for most Gaussian matrices A, is a #P-hard problem Our Main Conjecture We can prove it if you replace estimating by calculating, or most by all If the conjecture holds, then even a noisy n-photon experiment could falsify the Extended Church Thesis, assuming P #P BPP NP ! Most of our paper is devoted to giving evidence for this conjecture

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There exist constants C,D and >0 such that for all n and >0, First step: understand the distribution of |Per(A)| 2 for random A Empirically true! Also, we can prove it with determinant in place of permanent Conjecture:

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The Equivalence of Sampling and Searching [A., CSR 2011] [A.-Arkhipov] gave a sampling problem solvable using quantum optics that seems hard classicallybut does that imply anything about more traditional problems? Recently, I found a way to convert any sampling problem into a search problem of equivalent difficulty Basic Idea: Given a distribution D, the search problem is to find a string x in the support of D with large Kolmogorov complexity

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

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Experimental Issues Basically, were asking for the n-photon generalization of the Hong-Ou-Mandel dip, where n = big as possible Our results suggest that such a generalized HOM dip would be evidence against the Extended Church-Turing Thesis Physicists: Sounds hard! But not as hard as full QC Remark: If n is too large, a classical computer couldnt even verify the answers! Not a problem yet though… Several experimental groups (Bristol, U. of Queensland) are currently working to do our experiment with 3 photons. Largest challenge they face: reliable single-photon sources

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Summary Thinking about quantum optics led to: - A new experimental quantum computing proposal - New evidence that QCs are hard to simulate classically - A new classical randomized algorithm for estimating permanents - A new proof of Valiants result that the permanent is #P-hard - (Indirectly) A new connection between sampling and searching

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Open Problems Find more ways for quantum complexity theory to meet the experimentalists halfway Bremner, Jozsa, Shepherd 2011: QC with commuting Hamiltonians can sample hard distributions Can our model solve classically-intractable decision problems? Prove our main conjecture: that Per(A) is #P-hard to approximate for a matrix A of i.i.d. Gaussians ($1,000)! Fault-tolerance in the quantum optics model?

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