1 / 27

) New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers (

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.

cynthiadalton
Download Presentation

) New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers (

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  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: 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: New kinds of hardness results for simulating quantum mechanics Advantages of the new results: 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? Disadvantages: 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” (?)

  5. 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

  6. 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.

  7. BQP vs. PH: A Timeline 1990 1995 2000 2005 2010 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 RFSMA Natural conjecture: RFSPH Alas, we can’t even prove RFSAM!

  8. Fourier Sampling Problem Given oracle access to a random Boolean function The Task: Output strings z1,…,zn, at least 75% of which satisfy and at least 25% of which satisfy where

  9. Fourier Sampling Is In BQP |0 H H Repeat n times; output whatever you see Algorithm: |0 H f H |0 H H Distribution over Fourier coefficients Distribution over Fourier coefficients output by quantum algorithm

  10. 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 Obvious problem: How do we know C will output the particular s we’re interested in, thereby revealing anything about ? 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

  11. Decision Version: Fourier Checking Given oracle access to two Boolean functions • Decide whether • f,g are drawn from the uniform distribution U, or • f,g are drawn from the following “forrelated” distribution F: pick a random unit vector then let

  12. Fourier Checking Is In BQP |0 H H H |0 H f H g H |0 H H H Probability of observing |0n:

  13. Intuition: Fourier Checking Shouldn’t Be In PH • Why? • For any individual s, computing the Fourier coefficient 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?

  14. 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, Key Definition: A distribution D is -almost k-wise independent if for all k-terms 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

  15. Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us: Let f:{0,1}n{0,1} be computed by a circuit of size and depth O(1). Then for all n(1)-wise independent distributions D, Razborov’08 dramatically simplified Bazzi’s proof Finally, Braverman’09 proved the whole thing Bazzi’07 proved the depth-2 case Alas, we need the… “Generalized Linial-Nisan Conjecture”: Let f be computed by a circuit of size and depth O(1). Then for all 1/n(1)-almost n(1)-wise independent distributions D,

  16. 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? Result/Observation: Suppose QSamplingBPP. Then P#P=BPPNP (so in particular, PH collapses to the third level) 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.

  17. WhyQSampling Is Hard Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit: |0 H H |0 H f H |0 H H Then the probability of observing the all-0 string is

  18. Claim 1: p is #P-hard to estimate (up to a constant factor) • Related to my result that PostBQP=PP • Proof: If we can estimate p, then we can also compute xf(x) using binary search and padding Claim 2: Suppose QSamplingBPP. Then we could estimate p in BPPNP Proof: Let M be a classical algorithm for QSampling, and let r be its randomness. Use approximate counting to estimate Conclusion: Suppose QSamplingBPP. Then P#P=BPPNP

  19. Ideally, we want asimple, 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 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…

  20. 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 mm 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 with high probability over X.

  21. 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 with 1-1/poly(n) probability over X • “But isn’t the permanent easy to approximate, by Jerrum-Sinclair-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

  22. PGCHardness 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 mm unitary matrix, an nn 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 where US is an nn matrix containing si copies of the ith row of U (first n columns only) Neat Fact:The pS’s sum to 1

  23. 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! rather than ½ as with classical particles Fortunately, we show that with n bosons and mkn2 boxes, the probability of a collision is still at most (say) ½

  24. Experimental Prospects • What would it take to implement the requisite experiment with photonics? • Reliable phase-shifters and beamsplitters, to implement a Haar-random 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

  25. Prize Problems Prove the Generalized Linial-Nisan Conjecture!Yields an oracle A such that BQPAPHA Prove Generalized L-N even for the special case of DNFs.Yields an oracle A such that BQPAAMA Prove the Permanent of Gaussians Conjecture!Would imply that even approximate classical simulation of linear-optics circuits would collapsePH NIS500 $100 $200

  26. 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?

  27. Summary • I like to say that we have three choices: either • The Extended Church-Turing Thesis is false, • Textbook quantum mechanics is false, or • QCs can be efficiently simulated classically. For all intents and purposes?

More Related