Pattern Recognition Techniques for Boson Sampling...

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.

Can quantum computation

outperform classical computation?

vs.

EXTENDED CHURCH TURING THESIS

A probabilistic Turing machine can

EFFICIENTLY simulate any

realistic model of 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


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)

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].



STEP 2

N

N

N



STEP 3

n

n

n



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.


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.

www.quantumlab.it

Thank you for the attention

Pattern Recognition Techniques for Boson Sampling...

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