Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs quantum.phys.lsu.edu Louisiana.

Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with

Arbitrary Inputs

quantum.phys.lsu.edu

Louisiana State UniversityBaton Rouge, Louisiana USA

Computational Science Research CenterBeijing, 100084, China

QIM 19 JUN 13, Rochester

BT Gard, et al., JOSA B Vol. 30, pp. 1538–1545 (2013).BT Gard, et al., arXiv:1304.4206.

Jonathan P. Dowling

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.”

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:1304.4206Why We Thought Linear Optics Sucks at Why We Thought Linear Optics Sucks at

Quantum ComputingQuantum Computing

Multiphoton Quantum Random WalksMultiphoton Quantum Random Walks

Generalized Hong-Ou-Mandel EffectGeneralized Hong-Ou-Mandel Effect

Chasing Phases with Feynman DiagramsChasing Phases with Feynman Diagrams

Two- and Three- Photon Coincidence Two- and Three- Photon Coincidence

What? The Fock!What? The Fock!

Slater Determinant vs. Slater PermanentSlater Determinant vs. Slater Permanent

This Does Not Compute!This Does Not Compute!

QuickTime™ and a decompressor

are needed to see this picture.

Andrew White

Experiments

With Permanents!



Why We Thought Linear Optics Sucks at Quantum ComputingWhy We Thought Linear Optics Sucks at Quantum Computing



BlowUpIn

Energy!

BlowUpIn

Time!






BlowUpIn

Space!






Linear Optics Alone Can NOTIncrease Entanglement—

Even withSqueezed-State Inputs!






Multi-Fock-Input Photonic Quantum Pachinko

Detectors are Photon-Number Resolving

Generalized Hong-Ou-Mandel

2

2

ψ

out=384

A0

B−122

A2

B+380

A4

B

No odds! (But we’ll get even.)N00N Components Dominate! (Bat

State.)

A B

Schrödinger 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


Three photons interfere with each other!

(Take that, and that, and that, Dirac!)

GHOM effect Exploded

Rubik’s Cube of Three-Photon Coincidences


How Many Paths? Let Us Count the Ways.

2

2

ψ

out=384

A0

B−122

A2

B+380

A4

B

A B

This requires 8 Feynman paths to compute.

It rapidly goes to Helena Handbasket!


L is total number of levels.N+M is the total number of photons.


So Much For the Schrödinger Picture!

P L, N , M[ ] =2L N+M( )

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.

News From the Quantum Complexity Front?

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

Aaronson

What ? The Fock ! — Heisenberg Picture

a1†0 =ira1

†1 −ta2†1 a2

†0 =ta1†1 −ira2

†1

N1

00

2

0 = a10 †( )

N0 1

0 0 2

0 / N!M = 0

BS XFMRS

ψ

l

3

l =1

6

∏ =1

N !−irt 2a1

† 3 + r2ta2† 3 + ir t 2 − r2

( ) a3† 3 − 2r2ta4

† 3 − irt 2a5† 3 − t 3a6

† 3( )

N0

l

3

l =1

6

∏

Example: L=3. Powers ofOperators in Expansion

Generate Complete Orthonormal Set

Of Basis Vectors for Hilbert Space.


ψ L=

1

N !

Nn1, n2 , ... n2 L

⎛

⎝⎜

⎞

⎠⎟

N = nll =1

2 L

∑∑ α l

Lak† L( )

nk0

L

1≤k ≤2 L∏

dim H N , L( )⎡⎣ ⎤⎦=N + 2L−1

N⎛

⎝⎜⎞

⎠⎟

Dimension ofHilbert State

Space for N Photons At Level L.

The General Case: Multinomial Expansion!

N =2L−1 Computationally Complex Regime

dim H N( )⎡⎣ ⎤⎦ :

22N

πN<< 2N2 /2 : P[N]

L = 69 and fix N = 2L – 1 = 137

dim H 137( )⎡⎣ ⎤⎦ : 10

81 << 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.



Coherent-State No-Blow Theorem!

β 2= n = N = 2L +1

ψ L= exp β α l

L

l =1

2 L

∑ al† L − H.c.

⎛⎝⎜

⎞⎠⎟

0L =

exp βα lLal

† L − H.c.( ) 0L

l =1

2 L

∏ =

DlL βα l

L( ) 0L

l =1

2 L

∏ = βα lL

l

L

l =1

2 L

∏

Displacement Operator

Input State

Computationally Complex?

Output is Product of Coherent States: Efficiently Computable


Squeezed-State No-Blow Theorem!

n =N =2L +1

Squeezed Vacuum Operator

Input State

Computationally Complex?

Output Can Be Efficiently Transformed into 2L Single Mode Squeezers: Classically Computable.

ξ1

00

2

0= S1

0 ξ( ) 01

00

2

0

S10 ξ( ) =exp ξ* a1

0( )2−ξ a1

† 0( )2

( ) / 2⎡⎣⎢

⎤⎦⎥

ψ L= exp ξ * α l

*

l =1

2 L

∑ alL⎛

⎝⎜⎞⎠⎟

2

− ξ α ll =1

2 L

∑ al† L⎛

⎝⎜⎞⎠⎟

2⎛

⎝⎜

⎞

⎠⎟ / 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0L

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)

↑

Ain

↑Bin

↑↑

Aout

0Bout

+ 0Aout

↑↑Bout

↑

Ain

↑Bin

↑

Aout

↑Bout

↑

Ain

↓Bin

−↓Ain

↑Bin

↑

Aout

↓Bout

−↓Aout

↑Bout

↑↓Aout

0Bout

+ 0Aout

↑↓Bout

Spatial WF Symmetric (Bosonic)

Spatial WF Symmetric (Bosonic)Spatial WF AntiSymmetric (Fermionic)

Spatial WF AntiSymmetric (Fermionic)

↑

Ain

↓Bin

−↓Ain

↑Bin

↑Ain

↓Bin

−↓Ain

↑Bin

Effect Explains Bound State Of Neutral Hydrogen Molecule!

Fermion Fock Dimension Blows Up Too!?110 020

101 ψ11 ψ21ϕ21ϕ11

102 052

B11

B12 B22

B13 B23 B33

l=0

l=1

l=2

l=3=L∇13 ∇23 ∇33 ∇43 ∇53 ∇63

ϕ22ϕ12 ϕ42ϕ32ψ12 ψ22ψ32 ψ42ψ13 ψ23 ψ33 ψ43 ψ53 ψ63

d 031

Hilbert Space Dimension Blow Up Necessary but NOT Sufficient for Computational Complexity — Gottesman & Knill Theorem

dim H N , L( )⎡⎣ ⎤⎦=

2LN

⎛

⎝⎜⎞

⎠⎟: 22N / πN

Choosing Computationally Complex Regime: N = L.

A Shortcut Through Hilbert Space?Treat as Input-Output with Matrix Transfer!

110 020

101 ψ11 ψ21ϕ21ϕ11

102 052

B11

B12 B22

B13 B23 B33

l=0

l=1

l=2

l=3=L∇13 ∇23 ∇33 ∇43 ∇53 ∇63

ϕ22ϕ12 ϕ42ϕ32ψ12 ψ22ψ32 ψ42ψ13 ψ23 ψ33 ψ43 ψ53 ψ63

d 031

ψ

in

f /b

ψ

out

f /b=M1M 2M 3 ψ in

f /b

M1

M2

M3

Efficient!!!O(L3)

Must Properly Symmetrize Input State!

110 020

101 ψ11 ψ21ϕ21ϕ11

102 052

B11

B12 B22

B13 B23 B33

l=0

l=1

l=2

l=3=L∇13 ∇23 ∇33 ∇43 ∇53 ∇63

ϕ22ϕ12 ϕ42ϕ32ψ12 ψ22ψ32 ψ42ψ13 ψ23 ψ33 ψ43 ψ53 ψ63

d 031

ψin

f=TotallyAnti-Symmetric→ SlaterDeterminantofMatrix

ψin

b=TotallySymmetric→ 'Slater'PermanentofMatrix

ψout

f=TotallyAnti-Symmetric

ψout

b=TotallySymmetric Take coherence length >> L

BS XFRMs InsureProper SymmetryAll the Way Down

Input/Output Problem

Laplace Decomposition

+ – +Determinant:

(2L)! Steps

+ + +Permanent: (2L)! Steps

Slater Determinant vs. ‘Slater’ Permanent110 020

101 ψ11 ψ21ϕ21ϕ11

102 052

B11

B12 B22

B13 B23 B33

l=0

l=1

l=2

l=3=L∇13 ∇23 ∇33 ∇43 ∇53 ∇63

ϕ22ϕ12 ϕ42ϕ32ψ12 ψ22ψ32 ψ42ψ13 ψ23 ψ33 ψ43 ψ53 ψ63

d 031Fermions:Dim(H) exponentialAnti-Symmetric WavefunctionSlater Determinant: O(L2)Gaussian Elimination Does Compute!

Hilbert Space Dimension Blow Up Necessary but NOT Sufficient!

Bosons:Dim(H) exponentialSymmetric WavefunctionSlater Permanent: O(22LL2)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:1304.4206Why Linear Optics Should Suck at Quatum Why Linear Optics Should Suck at Quatum

ComputingComputing

Multiphoton Quantum Random WalksMultiphoton Quantum Random Walks

Generalized Hong-Ou-Mandel EffectGeneralized Hong-Ou-Mandel Effect

Chasing Phases with Feynman DiagramsChasing Phases with Feynman Diagrams

Two- and Three- Photon Coincidence Two- and Three- Photon Coincidence

What? The Fock!What? The Fock!

Slater Determinant vs. Slater PermanentSlater Determinant vs. Slater Permanent

This Does Not Compute!This Does Not Compute!



Lee Veronis



Wilde



Dowling



Olson



Sheng



Singh



Xiao



Seshadreesan



Balouchi



Gard



Granier



Jiang



Bardhan



Brown



KimCooney

Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Inputs quantum.phys.lsu.edu Louisiana.

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