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

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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: 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! Andrew White Experiments With Permanents!

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

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

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

Linear Optics Alone Can NOT Increase Entanglement— Even with Squeezed-State Inputs! Why We Thought Linear Optics Sucks at Quantum Computing

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

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

Schr ö dinger Picture: Feynman Paths “One photon only ever interferes with itself.” — P.A.M Dirac

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

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

How Many Paths? Let Us Count the Ways. A 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. So Much For the Schr ö dinger Picture! Total Number of Paths Choosing photon numbers N = M = 9 and level depth L = 16, we have = 5×10 86 total possible paths, which is about four orders of magnitude larger then the number of atoms in the observable universe.

News From the Quantum Complexity Front? From the Quantum Blogosphere: “… 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 Aaronson

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.

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

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. What ? The Fock ! — Heisenberg Picture

Coherent-State No-Blow Theorem! Displacement Operator Input State Computationally Complex? Output is Product of Coherent States: Efficiently Computable

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.

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.

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) Fermions (Total WF AntiSymmetric) Spatial WF Symmetric (Bosonic) Spatial WF AntiSymmetric (Fermionic) Effect Explains Bound State Of Neutral Hydrogen Molecule!

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

A Shortcut Through Hilbert Space? Treat as Input-Output with Matrix Transfer! Efficient!!! O(L 3 )

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

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

Slater Determinant vs. ‘Slater’ Permanent Fermions: Dim(H) exponential Anti-Symmetric Wavefunction Slater Determinant: O(L 2 ) Gaussian Elimination Does Compute! Hilbert Space Dimension Blow Up Necessary but NOT Sufficient! Bosons: Dim(H) exponential Symmetric Wavefunction Slater Permanent: O(2 2L L 2 ) Ryser’s Algorithm (1963) Does NOT Compute!

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: 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!

LeeVeronis WildeDowling Olson ShengSinghXiao Seshadreesan BalouchiGardGranierJiang Bardhan Brown Kim Cooney