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The Computational Complexity of Linear Optics Scott Aaronson (MIT) Joint work with Alex Arkhipov vs

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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 more generic complexity assumptions than the hardness of F ACTORING Gives evidence that QCs have capabilities outside the polynomial hierarchy Only involves linear optics (With single-photon Fock state inputs, and nonadaptive multimode photon- detection measurements) Today, a different kind of hardness result for simulating quantum mechanics Disadvantages: Applies to relational problems (problems with many possible outputs) or sampling problems, not 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? 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 two plausible conjectures are true: the permanent of a Gaussian random matrix is (1) #P-hard to approximate, and (2) not too concentrated around 0. Then the output distribution of a linear-optics circuit cant even be approximately sampled efficiently classically, unless the polynomial hierarchy collapses. Our Results If our conjectures hold, then even a noisy linear-optics experiment can sample from a probability distribution that no classical computer can feasibly sample from

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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 seem 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 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 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 Extensions: - Even simulation of QM in PH would imply P #P = PH - Can design a single hard linear-optics circuit for each n - Can let the inputs be coherent rather than Fock states

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So why arent we done? Because real quantum experiments are subject to noise Would an efficient classical algorithm that sampled from a noisy distributionone that was only 1/poly(n)-close to the true distribution in variation distancestill collapse the polynomial hierarchy? Difficulty in showing this: The sampler might adversarially neglect to output the one submatrix whose permanent we care about! So well need to smuggle the P ERMANENT instance we care about into a random submatrix Main Result: Yes, assuming two plausible conjectures about random permanents (the PGC and the PCC)

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There exist ε,δ 1/poly(n) for which the following problem is #P-hard. Given a Gaussian random matrix X drawn from N(0,1) C n×n, output a complex number z such that with probability at least 1- over X. The Permanent-of-Gaussians Conjecture (PGC) We can prove the conjecture if =0 or =0! What makes it hard is the combination of average-case and approximation

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For all polynomials q, there exists a polynomial p such that for all n, The Permanent Concentration Conjecture (PCC) Empirically true! Also, we can prove it with determinant in place of permanent

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U 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 assuming the PGC and PCC, BPP NP =P #P and hence PH collapses Main Result

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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 lemma 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 Assuming the PCC, the above lets us estimate Per(A) itself in BPP NP And assuming the PGC, estimating Per(A) is #P-hard

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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 Photodetector arrays that can reliably distinguish 0 vs. 1 photon But crucially, no nonlinear optics or postselected measurements! Our Proposal: Concentrate on (say) n=20 photons and m=400 modes, so that classical simulation is nontrivial but not impossible

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Main Open Problems Prove the Permanent of Gaussians Conjecture! (That approximating the permanent of an N(0,1) Gaussian random matrix is #P-complete) Do our linear-optics experiment! Are there other (e.g., qubit-based) quantum systems for which approximate classical simulation would collapse PH? Can our linear-optics model solve classically-intractable decision problems? Prove the Permanent Concentration Conjecture! (That Pr[|Per(X)| 2 <n!/p(n)] < 1/q(n))

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