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

2. 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

3. 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!

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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?

15. 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

16. 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

17. 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

18. 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

19. 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?