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

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!

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)

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

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

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

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

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

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

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

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

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.

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

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)

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

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

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

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

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

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

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))