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quantum.phys.lsu

Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs. Jonathan P. Dowling. Louisiana State University Baton Rouge, Louisiana USA Computational Science Research Center Beijing, 100084, China. quantum.phys.lsu.edu.

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quantum.phys.lsu

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  1. Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs Jonathan P. Dowling Louisiana State University Baton Rouge, Louisiana USA Computational Science Research Center Beijing, 100084, China quantum.phys.lsu.edu QIM 19 JUN 13, Rochester BT Gard, et al., JOSA B Vol. 30, pp. 1538–1545 (2013). BT Gard, et al., arXiv:1304.4206.

  2. Buy This Book or The Cat Will (and Will Not)Die! 5 ★★★★★ REVIEWS! “I found myself LAUGHING OUT LOUD quite frequently.” “The book itself is fine and well-written … I can thoroughly recommend it.”

  3. Andrew White Classical Computers Can Very Likely Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary InputsBT Gard, RM Cross, MB Kim, H Lee, JPD, arXiv:1304.4206 Why We Thought Linear Optics Sucks at Quantum Computing Multiphoton Quantum Random Walks Generalized Hong-Ou-Mandel Effect Chasing Phases with Feynman Diagrams Two- and Three- Photon Coincidence What? The Fock! Slater Determinant vs. Slater Permanent This Does Not Compute! Experiments With Permanents!

  4. Why We Thought Linear Optics Sucks at Quantum Computing Blow Up In Energy!

  5. Why We Thought Linear Optics Sucks at Quantum Computing Blow Up In Time!

  6. Why We Thought Linear Optics Sucks at Quantum Computing Blow Up In Space!

  7. Why We Thought Linear Optics Sucks at Quantum Computing Nonlinear Optics — Including KLM— IS a Road to QC! Linear Optics Alone Can NOT Increase Entanglement— Even with Squeezed-State Inputs!

  8. Multi-Fock-Input Photonic Quantum Pachinko Detectors are Photon-Number Resolving

  9. Generalized Hong-Ou-Mandel B A No odds! (But we’ll get even.) N00N Components Dominate! (Bat State.)

  10. Schrödinger Picture: Feynman Paths “One photon only ever interferes with itself.” — P.A.M Dirac

  11. Schrödinger Picture: Feynman Paths HOM effect in two-photon coincidences Two photons interfere with each other! (Take that, and that, Dirac!)

  12. Schrödinger Picture: Feynman Paths GHOM effect Exploded Rubik’s Cube of Three-Photon Coincidences Three photons interfere with each other! (Take that, and that, and that, Dirac!)

  13. How Many Paths? Let Us Count the Ways. B A This requires 8 Feynman paths to compute. It rapidly goes to Helena Handbasket!

  14. How Many Paths? Let Us Count the Ways. L is total number of levels. N+M is the total number of photons.

  15. How Many Paths? Let Us Count the Ways. Total Number of Paths Choosing photon numbers N = M = 9 and level depth L = 16 , we have 2288 = 5×1086 total possible paths, which is about four orders of magnitude larger then the number of atoms in the observable universe. So Much For the Schrödinger Picture!

  16. Aaronson News From the Quantum Complexity Front? Borzhemoi! This I know from nothing! From the Quantum Blogosphere: http://quantumpundit.blogspot.com “… you have to talk about the complexity-theoretic difference between the n*n permanent and the n*n determinant.” — Scott “Shtetl-Optimized” Aaronson “What will happen to me if I don’t!?” — Jonathan “Quantum-Pundit” Dowling

  17. What ? The Fock ! — Heisenberg Picture M = 0 BS XFMRS Example: L=3. Powers of Operators in Expansion Generate Complete Orthonormal Set Of Basis Vectors for Hilbert Space.

  18. What ? The Fock ! — Heisenberg Picture The General Case: Multinomial Expansion! Dimension of Hilbert State Space for N Photons At Level L.

  19. What ? The Fock ! — Heisenberg Picture Computationally Complex Regime L = 69 and fix N = 2L – 1 = 137 The Heisenberg and Schrödinger Pictures are NOT Computationally Equivalent. (This Result is Implicit in the Gottesman-Knill Theorem.) This Blow Up Does NOT Occur for Coherent or Squeezed Input States.

  20. What ? The Fock ! — Heisenberg Picture Coherent-State No-Blow Theorem! Displacement Operator Input State Computationally Complex? Output is Product of Coherent States: Efficiently Computable

  21. What ? The Fock ! — Heisenberg Picture Squeezed-State No-Blow Theorem! Squeezed Vacuum Operator Input State Computationally Complex? Output Can Be Efficiently Transformed into 2L Single Mode Squeezers: Classically Computable.

  22. News From the Quantum Complexity Front!? Ref. A: “AA proved that classical computers cannot efficiently simulate linear optics interferometer … unless the polynomial hierarchy collapses…I cannot recommend publication of this work.” Ref: B: “… a much more physical and accessible approach to the result. If the authors … bolster their evidence … the manuscript might be suitable for publication in Physical Review A.

  23. News From the Quantum Complexity Front!? Response to Ref. A: “… very few physicists know what the polynomial hierarchy even is … Physical Review is physics journal and not a computer science journal. Response to Ref: B: “… the referee suggested publication in some form if we could strengthen the argument … we now hope the referee will endorse our paper for publication in PRA.”

  24. HOM Effect for Fermions! Spatial WF AntiSymmetric (Fermionic) Spatial WF Symmetric (Bosonic) Hilbert Space Dimension Not the Whole Story: Multi-Particle Wave Functions Must be Symmetrized! Fermions (Total WF AntiSymmetric) Bosons (Total WF Symmetric) Spatial WF Symmetric (Bosonic) Spatial WF AntiSymmetric (Fermionic) Effect Explains Bound State Of Neutral Hydrogen Molecule!

  25. Fermion Fock Dimension Blows Up Too!? Choosing Computationally Complex Regime: N = L. Hilbert Space Dimension Blow Up Necessary but NOT Sufficient for Computational Complexity — Gottesman & Knill Theorem

  26. A Shortcut Through Hilbert Space? Treat as Input-Output with Matrix Transfer! Efficient!!!O(L3)

  27. Must Properly Symmetrize Input State! Input/Output Problem BS XFRMs Insure Proper Symmetry All the Way Down Take coherence length >> L

  28. Determinant: (2L)! Steps + – + Permanent: (2L)! Steps + + + Laplace Decomposition

  29. Slater Determinant vs. ‘Slater’ Permanent Fermions: Dim(H) exponential Anti-Symmetric Wavefunction Slater Determinant: O(L2) Gaussian Elimination Does Compute! Bosons: Dim(H) exponential Symmetric Wavefunction Slater Permanent: O(22LL2)Ryser’s Algorithm (1963) Does NOT Compute! Hilbert Space Dimension Blow Up Necessary but NOT Sufficient!

  30. "Quantum Physics is NOT a Branch of Computer Science!" — D.F.V. P.D.Q. P.A.M. James Classical Computers Can Very Likely Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary InputsBT Gard, RM Cross, MB Kim, H Lee, JPD, arXiv:1304.4206 Why Linear Optics Should Suck at Quatum Computing Multiphoton Quantum Random Walks Generalized Hong-Ou-Mandel Effect Chasing Phases with Feynman Diagrams Two- and Three- Photon Coincidence What? The Fock! Slater Determinant vs. Slater Permanent This Does Not Compute!

  31. Lee Veronis Wilde Brown Kim Cooney Dowling Balouchi Gard Granier Jiang Olson Sheng Singh Xiao Bardhan Seshadreesan

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