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New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson Parts based on joint work with Alex Arkhipov

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In 1994, something big happened in the foundations of computer science, whose meaning is still debated today… Why exactly was Shors algorithm important? Boosters: Because it means well build QCs! Skeptics: Because it means we wont build QCs! Me: For reasons having nothing to do with building QCs!

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Shors algorithm was a hardness result for one of the central computational problems of modern science: Q UANTUM S IMULATION Shors Theorem: Q UANTUM S IMULATION is not in probabilistic polynomial time, unless F ACTORING is also Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

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Advantages of the new results: Based on generic complexity assumptions, rather than the classical hardness of F ACTORING Use only extremely weak kinds of quantum computing (e.g. nonadaptive linear optics) testable before Im dead? Give evidence that QCs have capabilities outside the entire polynomial hierarchy Today: A different kind of hardness result for simulating quantum mechanics Disadvantages: Apply to sampling problems (or to problems with many possible valid outputs), not decision problems Harder to convince a skeptic that your QC is solving the relevant hard problem Problems dont seem useful

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First Problem Given a random Boolean function f:{0,1} n {-1,1} Find subsets S 1,…,S k [n] of the input bits, most of whose parities are slightly better correlated than chance with f E.g., sample a subset S with probability where Distribution of these Fourier coefficients for a random S Distribution for the Ss that youre being asked to output

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This problem is trivial to solve using a quantum computer! H H H H H H f |0 Theorem 1: Any classical probabilistic algorithm to solve it (even approximately) must make exponentially many queries to f Theorem 2: This is true even if we imagine that P=NP, and that the classical algorithm can ask questions like Theorem 3: Even if we instantiate f by some explicit function (like 3SAT), any classical algorithm to solve the problem really accurately would imply P #P =BPP NP (meaning the polynomial hierarchy would collapse)

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Ideally, we want a simple, explicit quantum system Q, such that any classical algorithm that even approximately simulates Q would have dramatic consequences for classical complexity theory We argue that this possible, using non-interacting bosons BOSONSFERMIONS There are two basic types of particle in the universe… Their transition amplitudes are given respectively by… All I can say is, the bosons got the harder job…

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U Our Current Result Take a system of n photons with m=O(n 2 ) modes each. Put each photon in a known mode, then apply a random m m scattering matrix U: Let D be the distribution that results from measuring the photons. Suppose theres an efficient classical algorithm that samples any distribution even 1/n O(1) -close to D. Then in BPP NP, one can approximate the permanent of a matrix A of independent N(0,1) Gaussians, to additive error with high probability over A. Challenge: Prove the above problem is #P-complete

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Experimental Prospects What would it take to implement this experiment with photonics? Reliable phase-shifters Reliable beamsplitters Reliable single-photon sources Reliable photodetector arrays But crucially, no nonlinear optics or postselected measurements! Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible

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Summary Ive often said we have three choices: either (1)The Extended Church-Turing Thesis is false, (2)Textbook quantum mechanics is false, or (3)QCs can be efficiently simulated classically. For all intents and purposes?

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