pattern recognition techniques for boson sampling...
TRANSCRIPT
Pattern Recognition Techniques
for Boson Sampling Validation
Iris Agresti1, Niko Viggianiello1, Fulvio Flamini1,
Nicolò Spagnolo1, Andrea Crespi2,3, Roberto Osellame2,3,
Nathan Wiebe4, Fabio Sciarrino1
www.quantumlab.it
1. Dipartimento di fisica, Sapienza Università di Roma
2. Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR)
3. Dipartimento di Fisica, Politecnico di Milano
4. Station Q Quantum Architectures and Computation Group, Microsoft Research
Verona QML 20176/11/2017
vs.
EXTENDED CHURCH TURING THESIS
A probabilistic Turing machine can
EFFICIENTLY simulate any
realistic model of computation
Can quantum computation
outperform classical computation?
Boson Sampling Problem
Unitary
Tranformation
INPUT
|𝒏𝟏… 𝒏𝒎 >
OUTPUT
𝒊=𝟏
𝑵𝑺
γ𝑺 |𝒏𝟏𝑺… 𝒏𝒎
𝑺 >
We want to sample the output probability
Proposed in 2010
by Scott Aaronson and Alex Arkhipov
The computational complexity of linear optics, 2011,
Proceedings of the 43rd Annual ACM Symposium on Theory of Computing
(ACM, New York), 333
Boson Sampling Problemin classical computation
𝑈11 ⋯ 𝑈15⋮ ⋱ ⋮𝑈51 ⋯ 𝑈55
OUTPUT
|ψ >𝑶𝑼𝑻 = σ𝒊=𝟏𝑵𝑺 γ𝑺 |𝒏𝟏
𝑺… 𝒏𝒎𝑺 >
Probability Distribution:
|𝜸𝑺|𝟐 ∝ |𝑷𝒆𝒓 𝑼𝑺 |
𝟐
𝑃𝑒𝑟 𝑈 =ෑ
𝜎(𝑖)
𝑖=1
𝑁
𝑈𝑖𝜎(𝑖)
PERMANENT
𝑈2−3−5 =
𝑈21 𝑈22 𝑈23𝑈31 𝑈32 𝑈33𝑈51 𝑈52 𝑈53
Boson Sampling Problemin classical computation
𝑈11 ⋯ 𝑈15⋮ ⋱ ⋮𝑈51 ⋯ 𝑈55
OUTPUT
|ψ >𝑶𝑼𝑻 = σ𝒊=𝟏𝑵𝑺 γ𝑺 |𝒏𝟏
𝑺… 𝒏𝒎𝑺 >
Probability Distribution:
|𝜸𝑺|𝟐 ∝ |𝑷𝒆𝒓 𝑼𝑺 |
𝟐
𝑃𝑒𝑟 𝑈 =ෑ
𝜎(𝑖)
𝑖=1
𝑁
𝑈𝑖𝜎(𝑖)
PERMANENT
𝑈2−3−5 =
𝑈21 𝑈22 𝑈23𝑈31 𝑈32 𝑈33𝑈51 𝑈52 𝑈53
Boson Sampling Problemin classical computation
𝑈11 ⋯ 𝑈15⋮ ⋱ ⋮𝑈51 ⋯ 𝑈55
OUTPUT
|ψ >𝑶𝑼𝑻 = σ𝒊=𝟏𝑵𝑺 γ𝑺 |𝒏𝟏
𝑺… 𝒏𝒎𝑺 >
Probability Distribution:
|𝜸𝑺|𝟐 ∝ |𝑷𝒆𝒓 𝑼𝑺 |
𝟐
𝑃𝑒𝑟 𝑈 =ෑ
𝜎(𝑖)
𝑖=1
𝑁
𝑈𝑖𝜎(𝑖)
PERMANENT
𝑈2−3−5 =
𝑈21 𝑈22 𝑈23𝑈31 𝑈32 𝑈33𝑈51 𝑈52 𝑈53
The dimension of the outputs’ space has
an exponential growth in (N,m)
Boson Sampling Problemin classical computation
proposal distribution:
𝑞𝑠 =𝑃𝑒𝑟(|𝐴𝑠|
2)
𝑠1! … 𝑠𝑁!
Markov Chain Monte Carlo Independent Sampler
• Brute force sampling: n = ( 𝑁2
𝑁)
• Rejection Sampling: n= O(𝑁2)• MCMC independent sampling: O(100)
Neville et al., Classical boson sampling algorithms with superior performance to
near-term experiments, Nat. Phys., advance online publication, http://dx.doi.org/10.1038/nphys4270
Boson Sampling Problemin classical computation
proposal distribution:
𝑞𝑠 =𝑃𝑒𝑟(|𝐴𝑠|
2)
𝑠1! … 𝑠𝑁!
Markov Chain Monte Carlo Independent Sampler
• Brute force sampling: n = ( 𝑁2
𝑁)
• Rejection Sampling: n= O(𝑁2)• MCMC independent sampling: O(100)
Neville et al., Classical boson sampling algorithms with superior performance to
near-term experiments, Nat. Phys., advance online publication, http://dx.doi.org/10.1038/nphys4270
Boson Sampling Problemin quantum computation
Prepare the photonic state
Implement
the unitary operator
Sample
Crespi A. et al., (2013), Experimental boson sampling in arbitrary integrated photonic circuits,
Nature Photonics 7, 545.
Tillmann M., (2013), Experimental Boson Sampling, Nature Photonics, 7, 540.
Broome M. A. et al., (2013), Photonic Boson Sampling in a tunable circuit, Science 339.
Spring J.B. et al., (2013), Boson Sampling on a Photonic chip, Science 330.
Bentivegna M. et al., (2017), Experimental Scattershot Boson Sampling, Science Advances, e1400255.
Boson Sampling Problemin quantum computation
Crespi A. et al., (2013), Experimental boson sampling in arbitrary integrated photonic circuits, Nature Photonics 7, 545.
Tillmann M., (2013), Experimental Boson Sampling, Nature Photonics, 7, 540.
Broome M. A. et al., (2013), Photonic Boson Sampling in a tunable circuit, Science 339.
Spring J.B. et al., (2013), Boson Sampling on a Photonic chip, Science 330.
Bentivegna M. et al., (2017), Experimental Scattershot Boson Sampling, Science Advances, e1400255.
Is our device samplingfrom the correct distribution?
Raising the number of photons and modes, we can’t evaluate the theoretical probability distribution
With our measures, we can cover only a limited part of the Hilbert space
Even if we had infinite measures, so that we could have the wholeprobability distribution, we couldn’t check its correctness (permanent is hard
also to verify)
Is our device samplingfrom the correct distribution?
Raising the number of photons and modes, we can’t evaluate the theoretical probability distribution
With our measures, we can cover only a limited part of the Hilbert space
Even if we had infinite measures, so that we could have the wholeprobability distribution, we couldn’t check its correctness (permanent is hard
also to verify)
Boson Sampling Validationthrough MACHINE LEARNING
• We already have a validated device and we want to validate a second one, only relying on experimental data
STEP 1 Coordinates:
Occupation numbers -> Number of bosons in each mode
m-dimensional
geometrical space
Euclidean
Distancesample 1
sample 2
Wang S. -T , Duan L. -M., (2016), Certification of Boson Sampling Devices with Coarse-Grained Measurements,
arXiv:1601.02627 [quant-ph].
Boson Sampling Validationthrough MACHINE LEARNING
• We already have a validated device and we want to validate a second one, only relying on experimental data
STEP 2
N
N
N
Boson Sampling Validationthrough MACHINE LEARNING
• We already have a validated device and we want to validate a second one, only relying on experimental data
STEP 3
n
n
n
Boson Sampling Validationthrough MACHINE LEARNING
• We already have a validated device and we want to validate a second one, only relying on experimental data
STEP 4 COMPATIBILITY TEST
N n
N n
N n
n
nn
N
N
N
1. Choose the number of clusters
2. Initialization of centroids (random/K-means ++…)
3. Every element is assigned to the cluster whose centroid is the
nearest
4. The mean point of each cluster becomes its new centroid
5. IF the new centroids are different from those of the previous
iteration: back to point 2; ELSE the cluster structure is completed.
Boson Sampling Validationthrough MACHINE LEARNING
K-means clustering
RESULTS
The parameters of the test are tuned in the training stage
(sample size and number of clusters)
N= 3 m=13 10 clustersN= 3 m=13 2000 events
RESULTS
The trained algorithm is effective for larger system
and for different alternative samplers
System
dimension
(N, m)
Ind. vs Ind. Ind. vs Dis. Ind. vs M.f. Ind. vs Unif.
3, 13 10/10 10/10 10/10 10/10
5, 50 10/10 10/10 10/10 10/10
6, 50 9/10 10/10 10/10 10/10
7, 50 8/10 10/10 10/10 10/10
Samples of 6000 events and 25 clusters
Tichy et al, Stringent and Efficient Assessment of Boson-Sampling Devices, Phys. Rev. Lett., vol 113, 020502 (2014).
CONCLUSIONS
This validation method brings very encouraging results about the
effectiveness of Machine learning on Boson Sampling:
1) it doesn’t require any PERMANENT computation.
2) the trained algorithm is effective also for larger systems and for different
alternative samplers.
3) The required sample size doesn’t grow in the range (102 - 108),
showing a good scalability.