* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Why We Thought Linear Optics Sucks at Quantum Computing
Double-slit experiment wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum fiction wikipedia , lookup
Identical particles wikipedia , lookup
Boson sampling wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum field theory wikipedia , lookup
Bell's theorem wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum entanglement wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Erwin Schrödinger wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum computing wikipedia , lookup
Hartree–Fock method wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Quantum machine learning wikipedia , lookup
Scalar field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Path integral formulation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Bra–ket notation wikipedia , lookup
Density matrix wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum key distribution wikipedia , lookup
Canonical quantization wikipedia , lookup
Coherent states wikipedia , lookup
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. 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 wellwritten … I can thoroughly recommend it.” Classical Computers Can Very Likely Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs BT Gard, RM Cross, MB Kim, H Lee, JPD, arXiv:1304.4206 Why We Thought Linear Optics Sucks at Quantum Computing Multiphoton Quantum Random Walks Experiments With Permanents! Generalized Hong-Ou-Mandel Effect Chasing Phases with Feynman Diagrams Two- and Three- Photon Coincidence QuickTime™ and a decompressor are needed to see this picture. What? The Fock! Slater Determinant vs. Slater Permanent Andrew White Why We Thought Linear Optics Sucks at Quantum Computing QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Blow Up In Energy! Why We Thought Linear Optics Sucks at Quantum Computing QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Blow Up In Time! Why We Thought Linear Optics Sucks at Quantum Computing QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Blow Up In Space! Why We Thought Linear Optics Sucks at Quantum Computing QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Linear Optics Alone Can NOT Increase Entanglement— Even with Squeezed-State Inputs! Multi-Fock-Input Photonic Quantum Pachinko Detectors are Photon-Number Resolving Generalized Hong-Ou-Mandel 2 2 A out 3 4 8 A 1 0 B 2 2 B A 3 2 B 0 8 A 4 B No odds! (But we’ll get even.) N00N Components Dominate! (Bat State.) Schrödinger Picture: Feynman Paths “One photon only ever interferes with itself.” — P.A.M Dirac Schrödinger Picture: Feynman Paths HOM effect in two-photon coincidences Two photons interfere with each other! (Take that, and that, Dirac!) Schrödinger Picture: Feynman Paths Exploded Rubik’s Cube of Three-Photon Coincidences Three photons interfere with each other! (Take that, and that, and that, Dirac!) GHOM effect How Many Paths? Let Us Count the Ways. 2 2 A out 3 4 8 A 1 0 B 2 2 B A 3 2 B 0 8 A 4 B This requires 8 Feynman paths to compute. It rapidly goes to Helena Handbasket! How Many Paths? Let Us Count the Ways. L is total number of levels. N+M is the total number of photons. How Many Paths? Let Us Count the Ways. Total Number of Paths P L, N, M 2 L N M 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! News From the Quantum Complexity Front? Aaronson 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 What ? The Fock ! — Heisenberg Picture N M=0 †0 1 â BS XFMRS 6 l 1 3 l 0 1 0† N 1 0 2 â 0 irâ tâ †1 1 †1 2 0 0 0 0 2 / N! 1 †0 2 â tâ irâ †1 1 1 irt 2 â1† 3 r 2tâ2† 3 ir t 2 r 2 â3† 3 2r 2tâ4† 3 irt 2 â5† 3 t 3â6† 3 N! †1 2 N 6 0 l 1 Example: L=3. Powers of Operators in Expansion Generate Complete Orthonormal Set Of Basis Vectors for Hilbert Space. 3 l What ? The Fock ! — Heisenberg Picture The General Case: Multinomial Expansion! L N 1 L †L n , n , ... n l âk N! N 2 L n 1 2 2 L 1 k 2 L l 1 l nk 0 L Dimension of N 2L 1 Hilbert State dim H N, L N Space for N Photons At Level L. What ? The Fock ! — Heisenberg Picture N 2L 1 Computationally Complex Regime dim H N : 2 2N N 2 N 2 /2 : P[N] L = 69 and fix N = 2L – 1 = 137 dim H 137 : 1081 4 102845 : P N 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. What ? The Fock ! — Heisenberg Picture Coherent-State No-Blow Theorem! D̂10 exp â1†0 *â10 D̂ 0 0 1 0 1 0 0 Displacement Operator Input State 2 n N 2L 1 2 L 2L L †L exp l âl H.c. 0 l 1 exp 2L l 1 2L D̂ l 1 L l L l â H.c. 0 L L †L l l 0 2L L lL l 1 L l Computationally Complex? L Output is Product of Coherent States: Efficiently Computable What ? The Fock ! — Heisenberg Picture Squeezed-State No-Blow Theorem! Ŝ10 exp * â10 0 1 0 0 2 2 / 2 â1†0 Ŝ10 0 0 1 0 2 Squeezed Vacuum Operator Input State 0 2 n N 2L 1 L 2 2 2L 2 L exp * l*âlL l âl† L / 2 0 l 1 l 1 Computationally Complex? L Output Can Be Efficiently Transformed into 2L Single Mode Squeezers: Classically Computable. 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.” QuickTime™ and a decompressor are needed to see this picture. 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. 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.” Hilbert Space Dimension Not the Whole Story: Multi-Particle Wave Functions Must be Symmetrized! Bosons (Total WF Symmetric) Aout 0 Ain Bout 0 Bin Aout Ain Bin Ain Bout Spatial WF Symmetric (Bosonic) Fermions (Total WF AntiSymmetric) Ain Aout Bin Bout Spatial WF AntiSymmetric (Fermionic) Bin Ain Bin Ain Bin Effect Explains Bound State Of Neutral Hydrogen Molecule! Aout Bout Aout Bout Spatial WF AntiSymmetric (Fermionic) Aout 0 Bout 0 Aout Spatial WF Symmetric (Bosonic) Bout Fermion Fock Dimension Blows Up Too!? 0 11 0 0 0 2 B11 d 1 1 0 11 1 12 1 B12 1 2 0 3 1 13 B13 12 2 2 23 0 2 1 1 3 B22 22 1 3 1 33 3 3 2 2 B23 32 2 42 3 3 4 34 53 3 5 2 0 4 B33 2 l 2 5 3 6 l 3 L 63 2L 2N dim H N, L : 2 / N N Choosing Computationally Complex Regime: N = L. Hilbert Space Dimension Blow Up Necessary but NOT Sufficient for Computational Complexity — Gottesman & Knill Theorem A Shortcut Through Hilbert Space? Treat as Input-Output with Matrix Transfer! f /b in 0 11 0 0 0 2 B11 d 1 1 0 11 1 12 1 B12 1 2 0 3 1 13 B13 12 2 1 2 23 33 f /b out 0 2 1 1 3 B22 22 3 1 3 3 2 2 B23 32 2 42 3 3 4 34 M1M 2 M3 3 5 53 f /b in 2 0 4 B33 63 2 l 2 5 3 6 l 3 L M1 M2 M3 Efficient!!! O(L3) Must Properly Symmetrize Input State! f in b in TotallyΚAnti-Symmetric Slater Determinant of Matrix TotallyΚSymmetricΚ 'Slater' Permanent of Matrix 0 11 0 0 0 Input/Output Problem 2 B11 d 1 1 0 11 1 12 1 B12 1 2 0 3 1 13 f out b out B13 12 2 2 23 0 2 1 1 3 B22 22 1 3 1 33 3 3 2 2 B23 32 2 42 3 3 4 34 53 3 5 2 0 4 B33 2 l 2 5 3 6 l 3 L BS XFRMs Insure Proper Symmetry All the Way Down 63 TotallyΚAnti-Symmetric TotallyΚSymmetric Take coherence length >> L Laplace Decomposition + + – + + Determinant: (2L)! Steps + Permanent: (2L)! Steps Slater Determinant vs. ‘Slater’ Permanent 0 11 0 0 0 2 B11 d 1 1 0 11 1 12 1 B12 1 2 0 3 1 3 1 B13 12 2 22 2 3 2 3 0 2 1 1 3 B22 1 1 3 3 3 3 2 2 B23 32 2 42 3 3 4 3 4 3 5 3 5 2 0 4 B33 2 l 2 5 3 6 3 6 l 3 L 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! Classical Computers Can Very Likely Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs BT 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! a dna ™emiTkciuQ rosserpmoced .erutcip siht ees ot dedeen era QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Lee Dowling Veronis QuickTime™ and a decompressor are needed to see this picture. Brown QuickTime™ and a decompressor are needed to see this picture. Balouchi QuickTime™ and a decompressor are needed to see this picture. Olson QuickTime™ and a decompressor are needed to see this picture. Bardhan QuickTime™ and a decompressor are needed to see this picture. Seshadreesan Wilde QuickTime™ and a decompressor are needed to see this picture. Cooney QuickTime™ and a decompressor are needed to see this picture. Kim QuickTime™ and a decompressor are needed to see this picture. Gard Granier QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Sheng Singh QuickTime™ and a decompressor are needed to see this picture. Jiang QuickTime™ and a decompressor are needed to see this picture. Xiao