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New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov

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Computer Scientist / Physicist Nonaggression Pact You tolerate these complexity classes: P NP BPP BQP #P PH And I dont inflict these on you: AM AWPP BQP/qpoly MA P/poly PSPACE QCMA QIP QMA SZK YQP

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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: Even 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: Based on a more generic complexity assumption than the hardness of F ACTORING Gives evidence that QCs have capabilities outside the entire polynomial hierarchy Only involves linear optics! (With single-photon Fock state inputs, and nonadaptive multimode photon- detection measurements) Today: A new kind of hardness result for simulating quantum mechanics Disadvantages: Applies to relational problems (problems with many possible valid outputs) or sampling problems, not to decision problems Harder to convince a skeptic that your QC is indeed solving the relevant hard problem Less relevant for the NSA

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Example of a PH problem: For all n-bit strings x, does there exist an n-bit string y such that for all n-bit strings z, (x,y,z) holds? Before We Go Further, A Bestiary of Complexity Classes… Just as they believe P NP, complexity theorists believe that PH is infinite So if you can show such-and-such is true PH collapses to a finite level, its damn good evidence that such-and-such is false BQP P #P BPP P NP PH F ACTORING P ERMANENT C OUNTING 3SAT X Y Z … How complexity theorists say such-and-such is damn unlikely: If such-and-such is true, then PH collapses to a finite level

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Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then the polynomial hierarchy collapses (indeed P #P =BPP NP ). Indeed, even if such a distribution can be sampled by a classical computer with an oracle for the polynomial hierarchy, still the polynomial hierarchy collapses. Suppose the output distribution of any linear-optics circuit can even be approximately sampled efficiently classically. Then in BPP NP, one can nontrivially approximate the permanent of a matrix of independent N(0,1) Gaussian entries (with high probability over the choice of matrix). Permanent-of-Gaussians Conjecture (PGC): The above problem is #P-complete (i.e., as hard as worst-case P ERMANENT ) Our Results If the PGC is true, then even a noisy linear-optics experiment can sample from a probability distribution that no classical computer can feasibly sample from, unless the polynomial hierarchy collapses

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Related Work Knill, Laflamme, Milburn 2001: Linear optics with adaptive measurements yields universal QC Valiant 2002, Terhal-DiVincenzo 2002: Noninteracting fermions can be simulated in P A. 2004: Quantum computers with postselection on unlikely measurement outcomes can solve hard counting problems (PostBQP=PP) Shepherd, Bremner 2009: Instantaneous quantum computing can solve sampling problems that might be hard classically Bremner, Jozsa, Shepherd 2010: Efficient simulation of instantaneous quantum computing would collapse PH

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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 Particle Physics In One Slide

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Starting from a fixed initial statesay, |I =|1,…,1,0,…0 you get to choose any m m mode-mixing unitary U U induces an unitary (U) on n-photon states, defined by Linear Optics for Dummies (or computer scientists) Computational basis states have the form |S =|s 1,…,s m, where s 1,…,s m are nonnegative integers such that s 1 +…+s m =n n = # of identical photons m = # of modes For us, m>n Then you get to measure (U)|I in the computational basis Here U S,T is an n n matrix 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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Theorem (Feynman 1982, Abrams-Lloyd 1996): Linear-optics computation can be simulated in BQP Proof Idea: Decompose the m m unitary U into a product of O(m 2 ) elementary linear-optics gates (beamsplitters and phaseshifters), then simulate each gate using polylog(n) standard qubit gates Theorem (Gurvits): There exist classical algorithms to approximate S| (U)|T to additive error in randomized poly(n,1/ ) time, and to compute the marginal distribution on photon numbers in k modes in n O(k) time Theorem (Bartlett-Sanders et al.): If the inputs are Gaussian states and the measurements are homodyne, then linear- optics computation can be simulated in P Upper Bounds on the Power of Linear Optics

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By contrast, exactly sampling the distribution over all n photons is extremely hard! Heres why … Given any matrix A C n n, we can construct an m m mode- mixing 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 known (classical) reductions to go from a multiplicative approximation of |Per(A)| 2 to Per(A) itself. Claim 2: Suppose we had a fast classical algorithm for linear-optics 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 linear-optics sampling. Then P #P =BPP NP.

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High-Level Idea Estimating a sum of exponentially many positive or negative numbers: #P-complete Estimating a sum of exponentially many nonnegative numbers: Still hard, but known to be in BPP NP PH If quantum mechanics could be efficiently simulated classically, then these two problems would become equivalentthereby placing #P in PH, and collapsing PH So why arent we done? Because real quantum experiments are subject to noise Would an efficient classical algorithm that sampled from a noisy distribution still collapse the polynomial hierarchy?

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U Main Result: Take a system of n identical photons with m=O(n 2 ) modes. Put each photon in a known mode, then apply a Haar-random m m unitary transformation U: Let D be the distribution that results from measuring the photons. Suppose theres a fast classical algorithm that takes U as input, and samples any distribution even 1/poly(n)-close to D in variation distance. Then in BPP NP, one can estimate the permanent of a matrix A of i.i.d. N(0,1) complex Gaussians, to additive error with high probability over A. Permanent-of-Gaussians Conjecture (PGC): This problem is #P-complete

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PGC Hardness of Linear-Optics Sampling Idea: Given a Gaussian random matrix A, well smuggle A into the unitary transition matrix U for m=O(n 2 ) photonsin such a way that S| (U)|I =Per( A), for some basis state |S Useful fact we rely on: given a Haar-random m m unitary matrix, an n n submatrix looks approximately Gaussian Then the classical sampler has no way of knowing which submatrix of U we care aboutso even if it has 1/poly(n) error, with high probability it will return |S with probability |Per( A)| 2 Then, just like before, we can use approximate counting to estimate Pr[|S ] |Per( A)| 2 in BPP NP, and thereby solve a #P-complete problem

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Problem: Bosons like to pile on top of each other! Call a basis state S=(s 1,…,s m ) good if every s i is 0 or 1 (i.e., there are no collisions between photons), and bad otherwise If bad basis states dominated, then our sampling algorithm might work, without ever having to solve a hard P ERMANENT instance Furthermore, the bosonic birthday paradox is even worse than the classical one! rather than ½ as with classical particles Fortunately, we show that with n bosons and m kn 2 modes, the probability of a collision is still at most (say) ½

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Experimental Prospects What would it take to implement the requisite experiment? Reliable phase-shifters and beamsplitters, to implement an arbitrary unitary on m photon modes Reliable single-photon sources Fock states, not coherent states Photodetector arrays that can reliably distinguish 0 vs. 1 photon But crucially, no nonlinear optics or postselected measurements! Our Proposal: Concentrate on (say) n 30 photons and m 1000 modes, so that classical simulation is difficult but not impossible

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Open Problems Prove the Permanent of Gaussians Conjecture! Would imply that even approximate classical simulation of linear-optics circuits would collapse PH 140Fr Do a linear-optics experiment that solves a classically-intractable sampling problem! ? What are the exact resource requirements? E.g., can our experiment be done using a log(n)-depth linear-optics circuit? Are there other quantum systems for which approximate classical simulation would collapse PH?

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