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“Computational Complexity and Physics”

“Computational Complexity and Physics”. Scott Aaronson (MIT) New Insights Into Computational Intractability Oxford University, October 3, 2013. So, I’ll just give two “tales from the trenches”: One about BosonSampling , one about black holes. What you should take away:. 1990s:.

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“Computational Complexity and Physics”

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  1. “Computational Complexity and Physics” Scott Aaronson (MIT) New Insights Into Computational Intractability Oxford University, October 3, 2013

  2. So, I’ll just give two “tales from the trenches”: One about BosonSampling, one about black holes What you should take away: 1990s: Quantum Physics Computational Complexity Shor & Grover Title is too broad! Today: Quantum Physics Computational Complexity

  3. BosonSampling (A.-Arkhipov 2011) Classical counterpart: Galton’s Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip A rudimentary type of quantum computing, involving only non-interacting photons

  4. n identical photons enter, one per input mode Assume for simplicity they all leave in different modes—there are possibilities The beamsplitter network defines a column-orthonormal matrix ACmn, such that In general, we consider a network of beamsplitters, with n input “modes” (locations) and m>>n output modes where nnsubmatrix of A corresponding to S is the matrix permanent For simplicity, I’m ignoring outputs with ≥2 photons per mode

  5. Example For Hong-Ou-Mandel experiment, In general, an nn complex permanent is a sum of n! terms, almost all of which cancel How hard is it to estimate the “tiny residue” left over? Answer:#P-complete, even for constant-factor approx (Contrast with nonnegative permanents!)

  6. So, Can We Use Quantum Optics to Solve a #P-Complete Problem? That sounds way too good to be true… Explanation: If X is sub-unitary, then |Per(X)|2 will usually be exponentially small. So to get a reasonable estimate of |Per(X)|2 for a given X, we’d generally need to repeat the optical experiment exponentially many times

  7. Better idea: Given ACmn as input, let BosonSampling be the problem of merely sampling from the same distribution DA that the beamsplitter network samples from—the one defined by Pr[S]=|Per(AS)|2 Theorem (A.-Arkhipov 2011): Suppose BosonSampling is solvable in classical polynomial time. Then P#P=BPPNP Better Theorem: Suppose we can sample DA even approximately in classical polynomial time. Then in BPPNP, it’s possible to estimate Per(X), with high probability over a Gaussian random matrix Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence that a weaker system can do something classically hard

  8. Experiments Last year, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3-photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 permanents # of experiments > # of photons!

  9. Obvious Challenges for Scaling Up: • Reliable single-photon sources (optical multiplexing?) • Minimizing losses • Getting high probability of n-photon coincidence • Goal (in our view): Scale to 10-30 photons • Don’t want to scale much beyond that—both because • you probably can’t without fault-tolerance, and • a classical computer probably couldn’t even verify the results! Theoretical Challenge: Argue that, even with photon losses and messier initial states, you’re still solving a classically-intractable sampling problem

  10. Recent Criticisms of Gogolin et al. (arXiv:1306.3995) Suppose you ignore which actual photodetectors light up, and count only the number of timeseach output configuration occurs. In that case, the BosonSampling distribution DA is exponentially-close to the uniform distribution U Response:Why would you ignore which detectors light up?? The output of almost any algorithm is also gobbledygook if you ignore the order of the output bits…

  11. Recent Criticisms of Gogolin et al. (arXiv:1306.3995) OK, so maybe DA isn’t close to uniform. Still, the very same arguments we gave for why polynomial-time classical algorithms can’t sample DA, suggest that they can’t even distinguish DA from U! Response:That’s why we said to focus on 10-30 photons—a range where a classical computer can verify a BosonSampling device’s output, but the BosonSampling device might be “faster”!(And 10-30 photons is probably the best you can do anyway, without quantum fault-tolerance)

  12. More Decisive Responses(A.-Arkhipov, arXiv:1309.7460) Theorem: Let ACmn be a Haar-random BosonSampling matrix, where m≥n5.1/. Then with 1-O() probability over A, the BosonSampling distribution DA has Ω(1) variation distance from the uniform distribution U Histogram of (normalized) probabilities under DA Under U Necessary, though not sufficient, for approximately sampling DA to be hard

  13. Theorem (A. 2013): Let ACmn be Haar-random, where m>>n. Then there is a classical polynomial-time algorithm C(A) that distinguishes DA from U (with high probability over A and constant bias, and using only O(1) samples) Strategy: Let AS be the nnsubmatrix of A corresponding to output S. Let P be the product of squared 2-norms of AS’s rows. If P>E[P], then guess S was drawn from DA; otherwise guess S was drawn from U P under uniform distribution (a lognormal random variable) AS P under a BosonSampling distribution A

  14. Using Quantum Optics to Prove that the Permanent is #P-Complete[A., Proc. Roy. Soc. 2011] Valiant showed that the permanent is #P-complete—but his proof required strange, custom-made gadgets • We gave a new, arguably more transparent proof by combining three facts: • n-photon amplitudes correspond to nn permanents • (2) Postselected quantum optics can simulate universal quantum computation [Knill-Laflamme-Milburn 2001] • (3) Quantum computations can encode #P-complete quantities in their amplitudes

  15. Black Holes and Computational Complexity?? QAM AM QSZK SZK BQP BPP YES!! Amazing connection made this year by Harlow & Hayden But first, we need to review 40 years of black hole history

  16. Bekenstein1972, Hawking 1975: Black holes have entropy and temperature! They emit radiation The Information Loss Problem: Calculations suggest that Hawking radiation is thermal—uncorrelated with whatever fell in. So, is infalling information lost forever? Would violate the unitarity / reversibility of QM OK then, assume the information somehow gets out! The Xeroxing Problem: How could the same qubit | fall inexorably toward the singularity, and emerge in Hawking radiation? Would violate the No-Cloning Theorem Black Hole Complementarity (Susskind, ‘t Hooft): An external observer can describe everything unitarily without including the interior at all! Interior should be seen as “just a scrambled re-encoding” of the exterior degrees of freedom

  17. The Firewall Paradox (AMPS 2012) R = Faraway Hawking Radiation H = Just-Emitted Hawking Radiation B = Interior of “Old” Black Hole Near-maximal entanglement Also near-maximal entanglement Violates “monogamy of entanglement”! The same qubit can’t be maximally entangled with 2 things

  18. Harlow-Hayden 2013 (arXiv:1301.4504): Striking argument that Alice’s decoding task would require exponential timeComplexity theory to the rescue of quantum field theory?? The HH decoding problem: Given an n-qubit pure state |BHR produced by a known, poly-size quantum circuit. Promised that, by acting only on R (the “Hawking radiation”), it’s possible to distill an EPR pair (|00+|11)/√2 between R and B (the “black hole interior”). Distill such a pair, by applying a unitary transformation UR to the qubits in R. Theorem (HH): This decoding problem is QSZK-hard. So presumably intractable, unless QSZK=BQP (which we have oracle evidence against)!

  19. Improvement (A. 2013): Suppose there are poly-size quantum circuits for the HH decoding problem. Then any OWF f:{0,1}n{0,1}p(n) can be inverted in quantum polynomial time. Proof:Assume for simplicity that f is injective. Consider Suppose applying UR to R decodes an EPR pair between R and B. Then for some {|x}x, {|x}x, we must have Generalizing to arbitrary OWFs requires techniques similar to those used in HILL Theorem! Furthermore, to get perfect entanglement, we need |x=|x for all x! So from UR, we can get unitaries V,W such that

  20. Other Exciting Recent Complexity/Physics Connections Quantum PCP Conjecture. “Is approximate quantum MAX-k-SAT hard for quantum NP?” If so, will require the construction exotic condensed matter systems, exhibiting room-temperature entanglement! Beautiful recent survey by Aharonov et al. (arXiv:1309.7495) Using Entanglement to Steer Quantum Systems. Can use Bell inequality violation for guaranteed randomness expansion (Vazirani-Vidick), blind and authenticated QC with a classical polynomial-time verifier, QMIP=MIP* (Reichardt-Unger-Vazirani)… I hope for, and expect, many more such striking connections! Indeed, making these connections possible might be the most important result of quantum computing research—even if useful QCs eventually get built!

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