Pattern Recognition Techniques
for Boson Sampling Validation
Iris Agresti1, Niko Viggianiello1, Fulvio Flamini1,
Nicolรฒ Spagnolo1, Andrea Crespi2,3, Roberto Osellame2,3,
Nathan Wiebe4, Fabio Sciarrino1
www.quantumlab.it
1. Dipartimento di fisica, Sapienza Universitร 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
|๐๐โฆ ๐๐ >
OUTPUT
๐=๐
๐ต๐บ
ฮณ๐บ |๐๐๐บโฆ ๐๐
๐บ >
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
๐11 โฏ ๐15โฎ โฑ โฎ๐51 โฏ ๐55
OUTPUT
|ฯ >๐ถ๐ผ๐ป = ฯ๐=๐๐ต๐บ ฮณ๐บ |๐๐
๐บโฆ ๐๐๐บ >
Probability Distribution:
|๐ธ๐บ|๐ โ |๐ท๐๐ ๐ผ๐บ |
๐
๐๐๐ ๐ =เท
๐(๐)
๐=1
๐
๐๐๐(๐)
PERMANENT
๐2โ3โ5 =
๐21 ๐22 ๐23๐31 ๐32 ๐33๐51 ๐52 ๐53
Boson Sampling Problemin classical computation
๐11 โฏ ๐15โฎ โฑ โฎ๐51 โฏ ๐55
OUTPUT
|ฯ >๐ถ๐ผ๐ป = ฯ๐=๐๐ต๐บ ฮณ๐บ |๐๐
๐บโฆ ๐๐๐บ >
Probability Distribution:
|๐ธ๐บ|๐ โ |๐ท๐๐ ๐ผ๐บ |
๐
๐๐๐ ๐ =เท
๐(๐)
๐=1
๐
๐๐๐(๐)
PERMANENT
๐2โ3โ5 =
๐21 ๐22 ๐23๐31 ๐32 ๐33๐51 ๐52 ๐53
Boson Sampling Problemin classical computation
๐11 โฏ ๐15โฎ โฑ โฎ๐51 โฏ ๐55
OUTPUT
|ฯ >๐ถ๐ผ๐ป = ฯ๐=๐๐ต๐บ ฮณ๐บ |๐๐
๐บโฆ ๐๐๐บ >
Probability Distribution:
|๐ธ๐บ|๐ โ |๐ท๐๐ ๐ผ๐บ |
๐
๐๐๐ ๐ =เท
๐(๐)
๐=1
๐
๐๐๐(๐)
PERMANENT
๐2โ3โ5 =
๐21 ๐22 ๐23๐31 ๐32 ๐33๐51 ๐52 ๐53
The dimension of the outputsโ space has
an exponential growth in (N,m)
Boson Sampling Problemin classical computation
proposal distribution:
๐๐ =๐๐๐(|๐ด๐ |
2)
๐ 1! โฆ ๐ ๐!
Markov Chain Monte Carlo Independent Sampler
โข Brute force sampling: n = ( ๐2
๐)
โข Rejection Sampling: n= O(๐2)โข 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:
๐๐ =๐๐๐(|๐ด๐ |
2)
๐ 1! โฆ ๐ ๐!
Markov Chain Monte Carlo Independent Sampler
โข Brute force sampling: n = ( ๐2
๐)
โข Rejection Sampling: n= O(๐2)โข 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 canโt 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 couldnโt check its correctness (permanent is hard
also to verify)
Is our device samplingfrom the correct distribution?
Raising the number of photons and modes, we canโt 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 couldnโt check its correctness (permanent is hard
also to verify)
Boson Sampling Validationthrough MACHINE LEARNING
โข 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
โข 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
โข 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
โข 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 ++โฆ)
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 doesnโt require any PERMANENT computation.
2) the trained algorithm is effective also for larger systems and for different
alternative samplers.
3) The required sample size doesnโt grow in the range (102 - 108),
showing a good scalability.