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.
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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: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
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What? The Fock!
Slater Determinant vs. Slater Permanent
Andrew White
Why We Thought Linear Optics Sucks at Quantum Computing
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Blow
Up
In
Energy!
Why We Thought Linear Optics Sucks at Quantum Computing
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Blow
Up
In
Time!
Why We Thought Linear Optics Sucks at Quantum Computing
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Blow
Up
In
Space!
Why We Thought Linear Optics Sucks at Quantum Computing
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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

l 1
3
l
1
0
0
2
 
0† N
1
 aˆ
aˆ  iraˆ  taˆ
†0
1
BS XFMRS
6
0

†1
1
†1
2


0
0
1
0
0 2 / N!
aˆ  taˆ  iraˆ
†0
2
†1
1
1

irt 2 aˆ1† 3  r 2taˆ2† 3  ir t 2  r 2 aˆ3† 3  2r 2taˆ4† 3  irt 2 aˆ5† 3  t 3aˆ6† 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 aˆk
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!

Dˆ 10    exp aˆ1†0   *aˆ10
Dˆ 10   0
0
1
0
0

Displacement Operator
Input State
2
  n  N  2L  1
2

L
 2L L †L

 exp     l aˆl  H.c. 0
 l 1

 exp 
2L
l 1
2L

 Dˆ 
l 1
L
l
L
l

aˆ
 H.c. 0
0
L
L †L
l l
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!

Sˆ10    exp  * aˆ10


0
1
0
0
2

2
  / 2
  aˆ1†0
 Sˆ10   0
0
1
0
2
Squeezed Vacuum Operator
Input State
0
2
n  N  2L 1

L
2
2
2L
  2 L
 



 exp   *   l*aˆlL      l aˆl† 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.”
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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

Ain

Bin
 
Ain
Bin
Aout

Ain


Bout
Spatial WF Symmetric (Bosonic)

Fermions (Total WF AntiSymmetric)
Aout


Bin
Bout
Spatial WF AntiSymmetric (Fermionic)
Bin


Ain
Bin
 
Ain

Bin
Effect Explains Bound
State Of Neutral
Hydrogen Molecule!

Aout

Bout
 
Aout

Spatial WF AntiSymmetric (Fermionic)
Bout

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
 M1M2M3 


3
5
 53
f /b
in
2
0
4
B33

2
l 2
5
3
6
 63
l  3 L
M1
M2
M3
Efficient!!!
O(L3)
Must Properly Symmetrize Input State!


f
in
b
in
 TotallyÊAnti-Symmetric  Slater Determinant of Matrix
 TotallyÊSymmetricÊ '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
 TotallyÊAnti-Symmetric
 TotallyÊSymmetric
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 ™emiTk ciuQ
ro s serpmo ced
.e rut cip sih t ees ot dedeen era
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