classical computers very likely can not efficiently simulate multimode linear optical...
TRANSCRIPT
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
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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:1304.4206Why 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!
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Andrew White
Experiments
With Permanents!
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Why We Thought Linear Optics Sucks at Quantum ComputingWhy We Thought Linear Optics Sucks at Quantum Computing
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BlowUpIn
Energy!
BlowUpIn
Time!
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Why We Thought Linear Optics Sucks at Quantum ComputingWhy We Thought Linear Optics Sucks at Quantum Computing
BlowUpIn
Space!
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Why We Thought Linear Optics Sucks at Quantum ComputingWhy We Thought Linear Optics Sucks at Quantum Computing
Linear Optics Alone Can NOTIncrease Entanglement—
Even withSqueezed-State Inputs!
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Why We Thought Linear Optics Sucks at Quantum ComputingWhy We Thought Linear Optics Sucks at Quantum Computing
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
Two photons interfere with each other!(Take that, and that, Dirac!)
HOM effect in two-photon coincidences
Schrödinger 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
Schrödinger Picture: Feynman Paths
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!
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 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.
What ? The Fock ! — Heisenberg Picture
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.
What ? The Fock ! — Heisenberg Picture
What ? The Fock ! — Heisenberg Picture
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
What ? The Fock ! — Heisenberg Picture
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.
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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
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!
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