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עדויות חדשות שקשה לדמות את מכניקת הקוונטים עם מחשבים קלאסיים )New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers( סקוט אהרונסון )Scott Aaronson( MIT 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: For reasons having nothing to do with building QCs! Shor’s algorithm was a hardness result for 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 Today: New kinds of hardness results for simulating quantum mechanics Advantages of the new results: Disadvantages: Based on “generic” complexity assumptions, rather than the classical hardness of FACTORING Give evidence that QCs have capabilities outside the entire polynomial hierarchy Use only extremely weak kinds of QC (e.g. nonadaptive linear optics)—testable before I’m dead? Most apply to sampling problems (or problems with many possible valid outputs), rather than decision problems Harder to convince a skeptic that your QC is indeed solving the relevant hard problem Problems not “useful” (?) Results (from arXiv:0910.4698) There exist black-box sampling and relational problems in BQP that are not in BPPPH Assuming the “Generalized Linial-Nisan Conjecture,” there exists a black-box decision problem in BQP but not in PH Original Linial-Nisan Conjecture was recently proved by Braverman, after being open for 20 years Unconditionally, there exists a black-box decision problem that requires (N) queries classically ((N1/4) even using postselection), but only O(1) queries quantumly Results (from recent joint work with Alex Arkhipov) Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then P#P=BPPNP, and hence PH collapses. Indeed, even if such a distribution can be sampled in BPPPH, still PH collapses. Suppose the output distribution of any linear-optics circuit can even be approximately sampled in BPP. Then a BPPNP machine can approximate the permanent of a matrix of independent N(0,1) Gaussians. Conjecture: The above problem is #P-complete. BQP vs. PH: A Timeline Bernstein and Vazirani define BQP They construct an oracle problem, RECURSIVE FOURIER SAMPLING, that has quantum query complexity n but classical query complexity n(log n) First example where quantum is superpolynomially better! A simple extension yields RFSMA Natural conjecture: RFSPH Alas, we can’t even prove RFSAM! Fourier Sampling Problem Given oracle access to a random Boolean function f : 0,1 1,1 n The Task: Output strings z1,…,zn, at least 75% of which satisfy and at least 25% of which satisfy where ˆf z : 1 n/2 2 fˆ zi 2 1 z x x0,1n f x fˆ zi 1 FOURIER SAMPLING Is In BQP |0 Algorithm: H |0 H |0 H H f H H Repeat n times; output whatever you see Distribution over Fourier coefficients Distribution over Fourier coefficients output by quantum algorithm FOURIER SAMPLING Is Not In PH Key Idea: Show that, if we had a constant-depth 2poly(n)-size circuit C for FOURIER SAMPLING, then we could violate a known AC0 lower bound, by “sneaking a MAJORITY problem” into the estimation of some random Fourier coefficient fˆ s Obvious problem: How do we know C will output the particular s we’re interested in, thereby revealing anything about fˆ s ? We don’t! (Indeed, there’s only a ~1/2n chance it will) But we have a long time to wait, since our reduction can be nondeterministic! That just adds more layers to the AC0 circuit Decision Version: FOURIER CHECKING Given oracle access to two Boolean functions f , g : 0,1 1,1 n Decide whether (i) f,g are drawn from the uniform distribution U, or (ii) f,g are drawn from the following “forrelated” n distribution F: pick a random unit vector v 2 , then let f x : sgn vx , g x : sgn vˆx FOURIER CHECKING Is In BQP |0 H |0 H |0 H H f H H H g H H Probability of observing |0n: 2 n 2 if f,g are random 1 x y f x 1 g y 3n 2 x , y0,1n 1 if f,g are forrelated Intuition: FOURIER CHECKING Shouldn’t Be In PH Why? • For any individual s, computing the Fourier coefficient fˆ s is a #P-complete problem • f and g being forrelated is an extremely “global” property: no polynomial number of f(x) and g(y) values should reveal much of anything about it But how to formalize and prove that? A k-term is a product of k literals of the form xi or 1-xi A distribution D over {0,1}N is k-wise independent if for all k-terms C, 1 PrD C PrU C 2 k Key Definition: A distribution D is -almost k-wise independent if for all k-terms C, PrD C 1 1 PrU C Approximation is multiplicative, not additive … that’s important! Theorem: For all k, the forrelated distribution F is O(k2/2n/2)-almost k-wise independent Proof: A few pages of Gaussian integrals, then a discretization step Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us: n o 1 Let f:{0,1}n{0,1} be computed by a circuit of size 2 and depth O(1). Then for all n(1)-wise independent distributions D, Pr f x Pr n f x o1. x~ D x0,1 Razborov’08 Finally,Bazzi’07 Braverman’09 dramatically Alas, proved we need proved the simplified depth-2 the… the whole Bazzi’s case thing proof “Generalized Linial-Nisan Conjecture”: Let f be computed n o 1 by a circuit of size 2 and depth O(1). Then for all 1/n(1)-almost n(1)-wise independent distributions D, Pr f x Pr n f x o1. x~ D x0,1 Coming back to our result for relational problems: what was surprising was that we showed hardness of a BQP sampling problem, using a nondeterministic reduction from MAJORITY—a “#P” problem! This raises a question: is something similar possible in the unrelativized (non-black-box) world? Indeed it is. Consider the following problem: QSAMPLING: Given a quantum circuit C, which acts on n qubits initialized to the all-0 state. Sample from C’s output distribution. Result/Observation: Suppose QSAMPLINGBPP. Then P#P=BPPNP (so in particular, PH collapses to the third level) Why QSAMPLING Is Hard Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit: |0 H |0 H |0 H H f H H Then the probability of observing the all-0 string is 1 p : 2 n 2 f x n x0,1 2 Claim 1: p is #P-hard to estimate (up to a constant factor) Related to my result that PostBQP=PP Claim 2: Suppose QSAMPLINGBPP. Then we could estimate p in BPPNP Proof: Let M be a classical algorithm for QSAMPLING, and Proof: If we can estimate p, let r be its randomness. Use then we can also compute approximate counting to xf(x) using binary search n 2 estimate Pr M r outputs 0 and padding r 1 f x 2n n 2 Conclusion: Suppose QSAMPLING x0,1BPP. Then P#P=BPPNP p : Ideally, we want a simple, explicit quantum system Q, such that any classical algorithm that even approximately simulates Q would have dramatic consequences for classical complexity theory We believe this is possible, using non-interacting bosons There twosay basic of particle in the universe… Allare I can is,types the bosons got the harder job… BOSONS FERMIONS Their transition amplitudes are given respectively by… Per A n a S n i 1 i, i Det A 1 sgn S n n a i, i 1 i Our Result: Take a system of n identical photons, with m=O(n2) modes (basis states) each. Put each photon in a known mode, then apply a random mm scattering matrix U: U Conjecture: This problem is #P-complete Let D be the distribution that results from measuring the photons. Suppose there’s an efficient classical algorithm that samples any distribution even 1/poly(n)-close to D in variation distance. Then in BPPNP, one can estimate the permanent of a matrix X of i.i.d. N(0,1) complex Gaussians, to additive error n! with high probability over X. n O 1 , The Permanent of Gaussians Conjecture (PGC) Given a matrix X of i.i.d, N(0,1) complex Gaussians, it is #P-complete to approximate Per(X) to within n! , poly n with 1-1/poly(n) probability over X “But isn’t the permanent easy to approximate, by JerrumSinclair-Vigoda?” Yes—for nonnegative matrices. For general matrices, can get huge cancellations between positive and negative terms, and indeed even approximating the permanent is #P-complete in the worst case Intuition for PGC: We know computing the permanent of a random matrix is #P-complete—over finite fields. “Merely” need to extend that result to the reals or complex numbers! Basic difficulty: When doing LFKN-style interpolation, errors in the permanent estimates can blow up exponentially PGCHardness of BOSONSAMPLING Idea: Given a Gaussian random matrix X, we’ll “smuggle” X into the unitary transition matrix U for m=O(n2) bosons Useful fact we rely on: given a Haar-random mm unitary matrix, an nn submatrix looks approximately Gaussian Suppose that initially, modes 1,…,n contain one boson each while modes n+1,…,m are unoccupied. Then after applying U, we observe a configuration (list of occupation numbers) s1,…,sm, with probability pS : Per U S 2 s1! sm ! Neat Fact: The pS’s sum to 1 where US is an nn matrix containing si copies of the ith row of U (first n columns only) Problem: Bosons like to pile on top of each other! Call a configuration S=(s1,…,sm) good if every si is 0 or 1 (i.e., there are no collisions between bosons), and bad otherwise If bad configurations 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! 2 Prboth particles land in the same box , 3 rather than ½ as with classical particles Fortunately, we show that with n bosons and mkn2 boxes, the probability of a collision is still at most (say) ½ Experimental Prospects What would it take to implement the requisite experiment with photonics? • Reliable phase-shifters and beamsplitters, to implement a Haarrandom unitary on m photon modes • Reliable single-photon sources • Reliable photodetector arrays But crucially, no nonlinear optics or postselected measurements! Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible Prize Problems Prove the Generalized Linial-Nisan Conjecture! $200 Yields an oracle A such that BQPAPHA Prove Generalized L-N even for the special case of DNFs. $100 Yields an oracle A such that BQPAAMA Prove the Permanent of Gaussians Conjecture! Would imply that even approximate classical simulation of NIS500 linear-optics circuits would collapse PH More Open Problems (no prizes) Can we “instantiate” FOURIER CHECKING by an explicit (unrelativized) problem? Can we use BOSONSAMPLING to solve any decision problem outside BPP? Can you convince a skeptic (who isn’t a BPPNP machine) that your QC is indeed doing BOSONSAMPLING? Can we get unlikely classical complexity consequences from P=BQP or PromiseP=PromiseBQP? Summary I like to say that we have three choices: either (1) The Extended Church-Turing Thesis is false, (2) Textbook quantum mechanics is false, or (3) QCs can be efficiently simulated classically. For all intents and purposes?