Download Boson sampling

Boson sampling,
where quantum optics meet computer science
Raul Garcia-Patron
QuIC Université Libre de Bruxelles (Brussels)
DIQIP & Qalgo Joint meeting
Brussels, May 13, 2014
Motivation
New look at “old stuff”:
regime of large-size (complex) liner-optics
Test-bed for protocols of certification
of large-size quantum systems
Better understanding of quantum optics
Evidence for the separation between quantum
and classical computation
Inspire new algorithms for permanent?
Boson-sampling
Boson Sampling “Gedankenexperiment”
RELATIVELY “easy” to implement
Single-photons detectors
Single-photons input
Linear optics interferometer
● Beamsplitter
● Phase-shift
Statistic are given by PRMANENTS: HARD to calculate ( #P-complete)
Efficient weak simulation (sampling) on a classical computer HARD
Linear optics representations
1 PHOTON=1 QUDIT
Dimension D = # Modes
Path encoding
|x ⟩ =|0,0,1,0,0,. .. , 0 ⟩
Interferometer is a unitary
Preserving the # photons
| y ⟩ =U D|x ⟩
Creation operator notation
●
|0,0,1,0,0,. .. ,0 ⟩ =a↑i |0 ⟩
Evolution through linear interferometer
̄b =Λ ā
b̂i =∑ j Λi , j â j
Linear optics and Symmetric subspace
N PHOTON=N QUDIT
Dimension D = # Modes
√
1
|1,1,2 ⟩
|ψ ⟩ 0 =|2,1,0 ⟩ =
∑
3−tuples
3
Mode 1
Photon 1
Linear optics and Symmetric subspace
N PHOTON=N QUDIT
Dimension D = # Modes
√
1
|ψ ⟩ 0 =|t 1, t 2, t 3,. .. , t D ⟩=
∑
BT X
BT =
N!
t 1 !t 2 ! ... t D !
T ∈B T
|x1 , x2 ,... , x N ⟩
Basis of the symmetric subspace of dimension
(
)
(N + D−1)!
Λ DN = N + D−1 =
N
N !( D−1)!
Evolution by interferometer (non-interacting)
Interferometer is a unitary
Preserving the # photons
|ψ ⟩ f =U
| y ⟩ =U D|x ⟩
∘N
D |
ψ⟩f
Boson sampling permanent rule
INPUT: N single-photons distributed over D modes
|T ⟩=|1,1,1,0,. .. ,0 ⟩ =
√
1
|π(1), π (2) ,... , π (N ) ⟩
∑
π∈S
N!
P (S∥T )=|⟨ s 1 , s2 , ... , s M|U
D
∘N
D
2
|t 1 ,t 2 ,... ,t M ⟩|
Modes with
detected photons
2
P(S∥T )=|Per (Λ S ,T )|
S. Scheel quant-ph/0406127
Per ( A)=∑σ ∈S
n
n
∏i
A i ,σ (i )
Input
modes
(
U 1,1 U 1,2
U 2,1 U 2,2
⋮
⋮
U D ,1 U D ,2
… U 1, D
… U 2, D
⋱
⋮
… U D, D
Because D=O(N^2): no-collision regime, random Gaussian matrix,
Sub-matrix of UD of dimension NxN
As opposed to Determinant, Permanent is hard to calculate: #P-Hard (Valiant 1979)
)
Hardness of simulating
boson-sampling
Hardness of Exact Sampling
KLM proof
Single photons+linear optics
+post-selection
Exact classical algorithm for BOSON-SAMPLING
Universal quantum computation
Collapse PH to 3rd order
An artist vision on the collapses of the PH to the 3rd order...
An the FULL collapse N=NP
Hardness of Exact Sampling
KLM proof
Single photons+linear optics
+post-selection
Universal quantum computation
PP
PH ⊆ P =P
Post −BQP
Toda 91
Exact classical algorithm for BOSON-SAMPLING
P
Post −BPP
⊆Δ3
Han 96
Collapse PH to 3rd order
Hardness of Exact Sampling
χ
+ post-selection
Universal quantum computation
Non-universal
set of gates and states
Exact classical algorithm for χ-SAMPLING
Collapse PH to 3rd order
Approximate Sampling
BOSON-SAMPLING(N,M,Λ)
1
‖P (S)−~
P(S)‖= ∑ s |P (s)−~
P(s)|<ϵ
2
Appr. classical algorithm for BOSON-SAMPLING
●
●
Conjectured
Collapse PH to 3rd order
Approximate Permanent of matrix of independent Gaussian entries hard (#P-hard)
Permanent anti-concentration conjecture
√ n!
∣Per (Λ S , T )∣≥ poly
(n)
Experiments
Experiments
3 to 4 photons over 8 modes
“Our results demonstrate that boson sampling
is related to the computation of matrix permanents,
a problem believed to be classically hard”
Spring et al. Science 339, 798 (2013)
"The bottom line is that we have built
boson sampling machines,
that they work and that the errors are not fatal"
Ian Walmsley
We want the simplest and most robust
experimental setup
that can be proven to be hard to sample
Open Questions
Can we prove “supremacy”?
No experiment will provide a “mathematical proof” of “quantum supremacy”
What shall we aim to?
In practice it will look more like a competition between
Classical Team
Quantum Team
VS
Classical Sampler
Bosonic Sampler
Verifier
Can we prove “supremacy”?
No experiment will provide a “mathematical proof” of “quantum supremacy”
What shall we aim to?
Build a Boson Sampler big enough that a classical computer can not efficiently simulate
But that it can be certified in practice (regime ~ 30 photons)
In practice it will look more like a competition between...
Quantum Team
Classical Team
VS
Classical Sampler
Bosonic Sampler
Can we certify a
Boson-sampling device?
Is BS far from uniform distribution?
●
●
●
Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently
sample from the uniform distribution.
If one chooses U at random, the chances of being caught cheating becomes large only after exponentially
many samples.
The findings of any experimental realization of Boson-Sampling have to be interpreted with great
care, as far as the notion “quantum supremacy” is concerned.
Is BS far from uniform distribution?
●
●
●
Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently
sample from the uniform distribution.
If one chooses U at random, the chances of being caught cheating becomes large only after exponentially
many samples.
The findings of any experimental realization of Boson-Sampling have to be interpreted with great
care, as far as the notion “quantum supremacy” is concerned.
BS is far from uniform distribution
●
●
●
Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently
sample from the uniform distribution.
If one chooses U at random, the chances of being caught cheating becomes large only after exponentially
many samples.
The findings of any experimental realization of Boson-Sampling have to be interpreted with great
care, as far as the notion “quantum supremacy” is concerned.
●
●
1 October 2013
We prove that the BosonSampling distribution is far from the
uniform distribution in total variation distance.
We give an efficient algorithm that distinguishes these two
distributions with constant bias.”
Nonetheless, certifying a BosonSampling distribution might be classically hard
“BosonSampling device faster than its fastest classical simulation,
but without classical certification being practically impossible (regime n = 30 photons)”
Verification
Verification of output data is Hard
For the same reason sampling is hard...
Build a Boson Sampler big enough that a classical computer can not efficiently simulate
But that it can be certified in practice (regime ~ 30 photons)
SOLUTIONS?
Find a set of matrices that are hard to sample,
but easy to verify. Probably not...
Use RUV Classical Command
Nature 496, 456–460 (25 April 2013)
Yes, but.....
What about an honest
experimentalist?
What about if we assume the circuit to be linear optics?
Tomography of UD
Self-Testing (Mosca, ...)
+ random circuits
+ two boxes
Boson-sampling variations
Quantum harmonic oscillators
Intermezzo
Fock vs Phase-space
Fock Representation
Phase-space Representation
P
X
ρ=∑ ρn ,m|n ⟩ ⟨ m|
[ a,a ↓] =1
a|n ⟩ = √ n|n−1 ⟩
x= ( a↓ +a ) / √ 2
p=i ( a↓−a ) / √ 2
[ x,p ] =i
Wigner Function: quasi-probability distribution
N |n ⟩=a a|n ⟩=n|n ⟩
a|0 ⟩ =0
1
H^ = N^ +
↓
2
a |n ⟩= √ n+1|n+1 ⟩
↓
↓
D (α)=exp (α a −ᾱ a)
Characteristic function:χ ( ξ ) =Tr
[ ρD ( ξ ) ]
COHERENT STATE
∞
∣α 〉 =e−∣α ∣/2 ∑n= 0
2
α 〉 =D (α)∣0 〉
∣
( )
αn
∣n 〉
√n !
Coherent State
Vacuum
p
Poisson Distribution
Quadratures first moments
d i=Tr [ ρ x̂ i ] ̄d =√ 2(ℜ α , ℑ α )
Quadratures second moments
γi , j =Tr [ ρ { x̂ i−d i , x̂ j−d j } ]
( )
1 0
γ=
0 1
x
SQUEEZED STATE
Vacuum
∣r 〉 =S (r )∣0 〉
S (r )=exp ( r (a↓ a↓−aa ) )
∞
|r ⟩ =∑n=0
(√
√(2n)!
cosh (r )2n n !
)
Quadratures second moments
γ=
(
−2r
e
0
0
2r
e
n
(tanh (r)) |2 n ⟩
)
Displaced Squeezed State
∣α , r 〉 =D(α)S (r )∣0 〉
Squeezed Vacuum State
ENTANGLEMENT
Two-Mode Squeezing (OPA)
↓
G
↓
A( τ)=exp ( τ( a b −ab) )
Two-mode Squeezed Vacuum State
∞
1
n
(tanh
(r))
|n ⟩|n ⟩
|ψ(r) ⟩=
∑
n=0
cosh (r )
Beam-splitter operator
↓
↓
B(τ )=exp ( τ (a b −a b) )
THERMAL STATES
Two-mode Squeezed Vacuum State
∞
1
n
(tanh
(r))
|n ⟩|n ⟩
|ψ(r) ⟩=
∑
n=0
cosh (r )
Thermal State
( )
∞
̄
1
N
ρN̄ =
∣n 〉 〈 n∣
∑
n=0
̄ +1
̄ +1
N
N
Partial Trace
Phase-space
P
ρN̄
2
̄
N =sinh (r )
n
α〉
∣
Maximally mixed state for fixed energy
X
g ( N̄ )=( N̄ +1)log ( N̄ +1)− N̄ log ( N̄ )
MEASUREMENT
Homodyne:
measurement of x
Heterodyne:
measurement of x and p
Gaussian POVM
H
∣0 〉
Variations
Measurement
Coherent States
Gaussian States
Single-photon
Adaptive
QND
BQP
BQP
BQP
Adaptive
NOT-QND
P
BQP
BQP [KLM]
NOT-Adaptive
QND
Sampling Hard
Sampling Hard
Sampling Hard
NOT-Adaptive
NOT-QND
P
Sampling Hard
Sampling Hard
Homodyne
(Gaussian)
P
P
?
Gaussian Fock-basis sampling
Gaussian Boson Sampling
Sample on the photon-number basis a n-mode Gaussian state
Pure states
∣0 〉
∣G 〉=U G∣0 〉 =e −i τ H ∣0 〉
∣0 〉
Quadratic Hamiltonian in
creation and annihilation operators.
H=i(α a+ a F a↓ + a G a)+h.c.
H
∣0 〉
Euler decomposition:
S(r1)
H
K
S(r2)
S(r3)
S(r1)
L
S(r2)
S(r3)
L
Gaussian Boson Sampling
Sample on the photon-number basis a n-mode Gaussian state
I
Hardness of Exact Gaussian BS
χ
+ post-selection
Universal quantum computation
+ Complexity Theory tricks
Non-universal
set of gates and states
Exact classical algorithm for χ-SAMPLING
Collapse PH to 3rd order
Gaussian states
Post-selection
+
Two-mode squeezer
+
Linear optics
∣1×1∣
∣0 〉
TMS
∣0 〉
∣1 〉
Approximate Gaussian BS
?
Scattershot boson-sampling
Scattershot boson-sampling
Two-mode squeezer
SCATTERSHOT-BS(N,M,Λ)
G
Lund et al. arXiv:1305.4346
Scattershot boson-sampling permanent rule
INPUT: N single-photons distributed over D modes
|T ⟩=|1,1,1,0,. .. ,0 ⟩ =
√
1
|π(1), π (2) ,... , π (N ) ⟩
∑
π∈S
N!
P (S∥T )=|⟨ s 1 , s2 , ... , s M|U
D
∘N
D
2
|t 1 ,t 2 ,... ,t M ⟩|
Modes with
detected photons
2
P(S∥T )=|Per (Λ S ,T )|
S. Scheel quant-ph/0406127
Per ( A)=∑σ ∈S
n
n
∏i
A i ,σ (i )
Input
modes
(
U 1,1 U 1,2
U 2,1 U 2,2
⋮
⋮
U D ,1 U D ,2
Because D=O(N^2): no-collision regime, random Gaussian matrix,
Sub-matrix of UD of dimension NxN
… U 1, D
… U 2, D
⋱
⋮
… U D, D
)
Approximate Scattershot-BS
SCATTERSHOT-BS(N,M,Λ)
Appr. classical algorithm for SCATTERSHOT-BS
1
̃
̃ s)∣<ϵ
∥P(S )− P(S)∥=
∣P( s)− P(
2 ∑s
Hardness of approximate
Sampling of Gaussian-sampling
Conjectured
Collapse PH to 3rd order
What if we pump more photons in?
More photons per input?
Can we get rid of the
“collision-free” condition?
2
P( S∥T )=
∣ Per (Λ S ,T )∣
∏i (si !)∏ j (t j !)
S. Scheel quant-ph/0406127
Much easier for experimentalists!
Example: Hong Ou Mandel
2
P( S∥T )=
∣ Per (Λ S ,T )∣
∏i ( si !)∏ j (t j !)
One photon per output
(
1 1 1
Λ S ,T =Λ=
√ 2 −1 1
Per (Λ)=0
)
2
P((1,1)∥(1,1))=
∣ Per (Λ )∣
2
2
(1!) (1!)
=0
Example: Hong Ou Mandel
2
P( S∥T )=
∣ Per (Λ S ,T )∣
∏i ( si !)∏ j (t j !)
Two photons on one output
(
1
1
√ 2 Λ=
−1 1
) (
1 1
−1 1
) (
1 1
−1 1
)
( )
1 1 = 2Λ
√ S ,T
1 1
2
∣Per (Λ S , T )∣
1
P((2,0)∥(1,1))=
=
2
2
2 ! 0 ! (1 !)
Per (Λ S , T )=1
General permanent rule
Boson Sampling “Gedankenexperiment”
M
∣T 〉 =∣t 1 , t2 , ... ,t M 〉 =∏ i=1
●
N photons @ input of M modes
●
Evolution through linear interferometer
●
Photon counting @ output
P (S∥T )=|⟨ s 1 , s2 , ... , s M|U
∘N
( )
(a↑ )t
∣0 〉
√ ti
i
b̂i =∑ j Λi , j â j
̄b =Λ ā
2
|t 1 ,t 2 ,... ,t M ⟩|
2
P (S∥T )=
|Per (Λ S ,T )|
∏i (si !)∏ j (t j !)
S. Scheel quant-ph/0406127
Per ( A)=∑σ ∈S
n
n
∏i
Ai , σ (i)
As opposed to Determinant, Permanent is hard to calculate: #P-Hard (Valiant 1979)
Homodyne detection sampling
Homodyne sampling
Gaussian states+Gaussian operations+homodyne measurement
Is classically simulatable
what about non-Gaussian states + homodyne?
Full quantum modeling of homodyne measurement
(Fock basis)...
Implementations
and
tolerance to noise
Effect of errors
Showing results 1 through 7 (of 7 total) for (au:Rohde AND all:(boson AND sampling))
1. arXiv:1403.4007 [pdf, other]
Scalable boson-sampling with time-bin encoding using a loop-based architecture
Keith R. Motes, Alexei Gilchrist, Jonathan P. Dowling, Peter P. Rohde
Comments: 7 pages, 7 fgures
Subjects: Quantum Physics (quant-ph)
2. arXiv:1402.0531 [pdf, other]
Boson sampling with photon-added coherent states
Kaushik P. Seshadreesan, Jonathan P. Olson, Keith R. Motes, Peter P. Rohde, Jonathan P. Dowling
Comments: 5 pages, 2 fgures, Comments are welcome
Subjects: Quantum Physics (quant-ph)
3. arXiv:1401.2199 [pdf, other]
Will boson-sampling ever disprove the Extended Church-Turing thesis?
Peter P. Rohde, Keith R. Motes, Paul A. Knott, William J. Munro
Comments: 3 pages, 0 fgures
Subjects: Quantum Physics (quant-ph)
4. arXiv:1310.0297 [pdf, other]
Sampling generalized cat states with linear optics is probably hard
Peter P. Rohde, Keith R. Motes, Paul Knott, William Munro, Jonathan P. Dowling
Comments: 8 pages, 2 fgures
Subjects: Quantum Physics (quant-ph)
5. arXiv:1307.8238 [pdf, other]
Spontaneous parametric down-conversion photon sources are scalable in the asymptotic limit for boson-sampling
Keith R. Motes, Jonathan P. Dowling, Peter P. Rohde
Comments: 6 pages, 5 fgures; added reference to Meany et al
Journal-ref: Phys. Rev. A 88, 063822 (2013)
Subjects: Quantum Physics (quant-ph)
6. arXiv:1208.2475 [pdf, other]
Optical quantum computing with photons of arbitrarily low fdelity and purity
Peter P. Rohde
s
Comments: Version submitted to Phys. Rev. A
Journal-ref: Phys. Rev. A 86, 052321 (2012)
Subjects: Quantum Physics (quant-ph)
̃ S)∥= 1 ∑ ∣P(s)− P(s)
̃ ∣<ϵ
∥P(S )− P(
2
Anthony Leverrier, R. G.-P., arXiv:1309.4687
Future directions
Future Directions
Improve the Hardness proof
●
●
●
Hardness for approximated Boson Sampling (proof the conjecture)
Hardness for larger input photons (collisions)
Hardness for realistic experimental conditions (losses, detector efficiency,...)
Verification of Boson Sampling
Variants of Boson Sampling
Qudit repr of active transformations
Classical Simulation Boson Sampling
●
●
Optimize classical sampling algorithm
Does exist families of states with efficient classical sampling? (bosonic “stabilizer states”)
THANK YOU!
@QuantumCrypto