The Computational Complexity of Linear Optics

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

2. In 1994, something big happened in the foundations of computer science, whose meaning is still debated today… Why exactly was Shor’s algorithm important? Boosters: Because it means we’ll build QCs! Skeptics: Because it means we won’t build QCs! Me: Even for reasons having nothing to do with building QCs!

3. Shor’s algorithm was a hardness resultfor one of the central computational problems of modern science: Quantum Simulation Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik) Shor’s Theorem: Quantum Simulation is not in probabilistic polynomial time, unless Factoring is also

4. Today, a different kind of hardness result for simulating quantum mechanics Advantages: Based on more “generic” complexity assumptions than the hardness of Factoring 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) 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

5. Bestiary of Complexity Classes P#P Counting Permanent BQP How complexity theorists say “such-and-such is damn unlikely”: “If such-and-such is true, then PH collapses to a finite level” PH xyz… NP BPP 3SAT Factoring P 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?” 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,” it’s damn good evidence that such-and-such is false

6. Our Results • 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=BPPNP). • 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 can’t even be approximately sampled efficiently classically, unless the polynomial hierarchy collapses. 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

7. 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

8. Particle Physics In One Slide There are two basic types of particle in the universe… All I can say is, the bosons got the harder job BOSONS FERMIONS Their transition amplitudes are given respectively by…

9. Linear Optics for Dummies (or computer scientists) Computational basis states have the form |S=|s1,…,sm, where s1,…,sm are nonnegative integers such that s1+…+sm=n n = # of identical photons m = # of modes For us, m>n Starting from a fixed initial state—say, |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 Here US,T is an nn matrix obtained by taking si copies of the ith row of U and tj copies of the jth column, for all i,j Then you get to measure (U)|I in the computational basis

10. Upper Bounds on the Power of Linear Optics Theorem (Feynman 1982, Abrams-Lloyd 1996):Linear-optics computation can be simulated inBQP Proof Idea: Decompose the mm unitary U into a product of O(m2) elementary “linear-optics gates” (beamsplitters and phaseshifters), then simulate each gate using polylog(n) standard qubit gates 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 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 nO(k) time

11. By contrast, exactly sampling the distribution over all n photons is extremely hard! Here’s why … Given any matrix ACn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

12. Claim 1: p is #P-complete to estimate (up to a constant factor) Idea: Valiant proved that the Permanent 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 BPPNP 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=BPPNP.

13. 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 BPPNP  PH If quantum mechanics could be efficiently simulated classically, then these two problems would become equivalent—thereby 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

14. So why aren’t we done? Because real quantum experiments are subject to noise Would an efficient classical algorithm that sampled from a noisy distribution—one that was only 1/poly(n)-close to the true distribution in variation distance—still collapse the polynomial hierarchy? Main Result: Yes, assuming two plausible conjectures about random permanents (the “PGC” and the “PCC”) Difficulty in showing this: The sampler might adversarially neglect to output the one submatrix whose permanent we care about! So we’ll need to “smuggle” the Permanent instance we care about into a randomsubmatrix

15. The Permanent-of-Gaussians Conjecture (PGC) 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)Cn×n, output a complex number z such that with probability at least 1- over X. We can prove the conjecture if =0 or =0! What makes it hard is the combination of average-case and approximation

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

17. Main Result Take a system of n identical photons with m=O(n2) modes. Put each photon in a known mode, then apply a Haar-random mm unitary transformation U: U Let D be the distribution that results from measuring the photons. Suppose there’s a fast classical algorithm that takes U as input, and samples any distribution even1/poly(n)-close to D in variation distance. Then assuming the PGC and PCC, BPPNP=P#P and hence PH collapses

18. Idea:Given a Gaussian random matrix A, we’ll “smuggle” A into the unitary transition matrix U for m=O(n2) photons—in 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 about—so 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 BPPNP Assuming the PCC, the above lets us estimate Per(A) itself in BPPNP And assuming the PGC, estimating Per(A) is #P-hard

19. Problem: Bosons like to pile on top of each other! Call a basis state S=(s1,…,sm) good if every si 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 Permanent 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kn2 modes, the probability of a collision is still at most (say) ½

20. 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

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