Abstract

We report on experimental implementation of a machine-learned quantum gate driven by a classical control. The gate learns optimal phase-covariant cloning in a reinforcement learning scenario having fidelity of the clones as reward. In our experiment, the gate learns to achieve nearly optimal cloning fidelity allowed for this particular class of states. This makes it a proof of present-day feasibility and practical applicability of the hybrid machine learning approach combining quantum information processing with classical control. The quantum information processing performed by the setup is equivalent to boson sampling, which, in complex systems, is predicted to manifest quantum supremacy over classical simulation of linear-optical setups.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

A. Pepper, N. Tischler, and G. J. Pryde, “Experimental realization of a quantum autoencoder: The compression of qutrits via machine learning,” Phys. Rev. Lett. 122(6), 060501 (2019).
[Crossref]

M. Schuld and N. Killoran, “Quantum machine learning in feature hilbert spaces,” Phys. Rev. Lett. 122(4), 040504 (2019).
[Crossref]

2018 (2)

J. Gao, L.-F. Qiao, Z.-Q. Jiao, Y.-C. Ma, C.-Q. Hu, R.-J. Ren, A.-L. Yang, H. Tang, M.-H. Yung, and X.-M. Jin, “Experimental machine learning of quantum states,” Phys. Rev. Lett. 120(24), 240501 (2018).
[Crossref]

K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, “Quantum circuit learning,” Phys. Rev. A 98(3), 032309 (2018).
[Crossref]

2017 (3)

K. Bartkiewicz, A. Černoch, G. Chimczak, K. Lemr, A. Miranowicz, and F. Nori, “Experimental quantum forgery of quantum optical money,” npj Quantum Inf. 3(1), 7 (2017).
[Crossref]

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” Nature 549(7671), 195–202 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11(7), 441–446 (2017).
[Crossref]

2016 (2)

J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,” New J. Phys. 18(2), 023023 (2016).
[Crossref]

M. Schiavon, G. Vallone, and P. Villoresi, “Experimental realization of equiangular three-state quantum key distribution,” Sci. Rep. 6(1), 30089 (2016).
[Crossref]

2015 (1)

X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114(11), 110504 (2015).
[Crossref]

2014 (3)

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor,” Nat. Commun. 5(1), 4213 (2014).
[Crossref]

H. Fan, Y.-N. Wang, L. Jing, J.-D. Yue, H.-D. Shi, Y.-L. Zhang, and L.-Z. Mu, “Quantum cloning machines and the applications,” Phys. Rep. 544(3), 241–322 (2014).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, J. Soubusta, and M. Stobińska, “Efficient amplification of photonic qubits by optimal quantum cloning,” Phys. Rev. A 89(6), 062322 (2014).
[Crossref]

2013 (7)

K. Bartkiewicz, K. Lemr, A. Černoch, J. Soubusta, and A. Miranowicz, “Experimental eavesdropping based on optimal quantum cloning,” Phys. Rev. Lett. 110(17), 173601 (2013).
[Crossref]

N. B. Lovett, C. Crosnier, M. Perarnau-Llobet, and B. C. Sanders, “Differential evolution for many-particle adaptive quantum metrology,” Phys. Rev. Lett. 110(22), 220501 (2013).
[Crossref]

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9(1), 143–252 (2013).
[Crossref]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[Crossref]

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7(7), 545–549 (2013).
[Crossref]

2010 (1)

A. Hentschel and B. C. Sanders, “Machine learning for precise quantum measurement,” Phys. Rev. Lett. 104(6), 063603 (2010).
[Crossref]

2009 (1)

K. Bartkiewicz, A. Miranowicz, and Ş. K. Özdemir, “Optimal mirror phase-covariant cloning,” Phys. Rev. A 80(3), 032306 (2009).
[Crossref]

2008 (2)

J. Soubusta, L. Bartůšková, A. Černoch, M. Dušek, and J. Fiurášek, “Experimental asymmetric phase-covariant quantum cloning of polarization qubits,” Phys. Rev. A 78(5), 052323 (2008).
[Crossref]

J.-S. Xu, C.-F. Li, L. Chen, X.-B. Zou, and G.-C. Guo, “Experimental realization of the optimal universal and phase-covariant quantum cloning machines,” Phys. Rev. A 78(3), 032322 (2008).
[Crossref]

2007 (1)

J. Soubusta, L. Bartůšková, A. Černoch, J. Fiurášek, and M. Dušek, “Several experimental realizations of symmetric phase-covariant quantum cloners of single-photon qubits,” Phys. Rev. A 76(4), 042318 (2007).
[Crossref]

2006 (1)

L. Han and M. Neumann, “Effect of dimensionality on the nelder-mead simplex method,” Optim. Methods Softw. 21(1), 1–16 (2006).
[Crossref]

2005 (1)

F. Buscemi, G. M. D’Ariano, and C. Macchiavello, “Economical phase-covariant cloning of qudits,” Phys. Rev. A 71(4), 042327 (2005).
[Crossref]

2004 (1)

J. M. Renes, “Spherical-code key-distribution protocols for qubits,” Phys. Rev. A 70(5), 052314 (2004).
[Crossref]

2003 (1)

J. Fiurášek, “Optical implementations of the optimal phase-covariant quantum cloning machine,” Phys. Rev. A 67(5), 052314 (2003).
[Crossref]

2000 (1)

D. Bruß, M. Cinchetti, M. G. D’Ariano, and C. Macchiavello, “Phase-covariant quantum cloning,” Phys. Rev. A 62(1), 012302 (2000).
[Crossref]

1983 (1)

S. Wiesner, “Conjugate coding, Original manuscript written circa 1970,” SIGACT News 15(1), 78–88 (1983).
[Crossref]

1982 (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299(5886), 802–803 (1982).
[Crossref]

1979 (1)

L. Valiant, “The complexity of computing the permanent,” Theor. Comput. Sci. 8(2), 189–201 (1979).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Aaronson, S.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9(1), 143–252 (2013).
[Crossref]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[Crossref]

Arkhipov, A.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9(1), 143–252 (2013).
[Crossref]

Aspuru-Guzik, A.

J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,” New J. Phys. 18(2), 023023 (2016).
[Crossref]

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor,” Nat. Commun. 5(1), 4213 (2014).
[Crossref]

Babbush, R.

J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,” New J. Phys. 18(2), 023023 (2016).
[Crossref]

Baehr-Jones, T.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11(7), 441–446 (2017).
[Crossref]

Barbieri, M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[Crossref]

Bartkiewicz, K.

K. Bartkiewicz, A. Černoch, G. Chimczak, K. Lemr, A. Miranowicz, and F. Nori, “Experimental quantum forgery of quantum optical money,” npj Quantum Inf. 3(1), 7 (2017).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, J. Soubusta, and M. Stobińska, “Efficient amplification of photonic qubits by optimal quantum cloning,” Phys. Rev. A 89(6), 062322 (2014).
[Crossref]

K. Bartkiewicz, K. Lemr, A. Černoch, J. Soubusta, and A. Miranowicz, “Experimental eavesdropping based on optimal quantum cloning,” Phys. Rev. Lett. 110(17), 173601 (2013).
[Crossref]

K. Bartkiewicz, A. Miranowicz, and Ş. K. Özdemir, “Optimal mirror phase-covariant cloning,” Phys. Rev. A 80(3), 032306 (2009).
[Crossref]

Bartušková, L.

J. Soubusta, L. Bartůšková, A. Černoch, M. Dušek, and J. Fiurášek, “Experimental asymmetric phase-covariant quantum cloning of polarization qubits,” Phys. Rev. A 78(5), 052323 (2008).
[Crossref]

J. Soubusta, L. Bartůšková, A. Černoch, J. Fiurášek, and M. Dušek, “Several experimental realizations of symmetric phase-covariant quantum cloners of single-photon qubits,” Phys. Rev. A 76(4), 042318 (2007).
[Crossref]

Baudin, M.

M. Baudin, Nelder-Mead User’s Manual (Scilab Consortium, 2010).

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings IEEE International Conference on Computers, Systems and Signal Processing, vol. 1 (IEEE, 1984), pp. 175–179.

Biamonte, J.

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” Nature 549(7671), 195–202 (2017).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings IEEE International Conference on Computers, Systems and Signal Processing, vol. 1 (IEEE, 1984), pp. 175–179.

Brod, D. J.

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7(7), 545–549 (2013).
[Crossref]

Broome, M. A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[Crossref]

Bruß, D.

D. Bruß, M. Cinchetti, M. G. D’Ariano, and C. Macchiavello, “Phase-covariant quantum cloning,” Phys. Rev. A 62(1), 012302 (2000).
[Crossref]

Buscemi, F.

F. Buscemi, G. M. D’Ariano, and C. Macchiavello, “Economical phase-covariant cloning of qudits,” Phys. Rev. A 71(4), 042327 (2005).
[Crossref]

Cai, X.-D.

X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114(11), 110504 (2015).
[Crossref]

Cernoch, A.

K. Bartkiewicz, A. Černoch, G. Chimczak, K. Lemr, A. Miranowicz, and F. Nori, “Experimental quantum forgery of quantum optical money,” npj Quantum Inf. 3(1), 7 (2017).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, J. Soubusta, and M. Stobińska, “Efficient amplification of photonic qubits by optimal quantum cloning,” Phys. Rev. A 89(6), 062322 (2014).
[Crossref]

K. Bartkiewicz, K. Lemr, A. Černoch, J. Soubusta, and A. Miranowicz, “Experimental eavesdropping based on optimal quantum cloning,” Phys. Rev. Lett. 110(17), 173601 (2013).
[Crossref]

J. Soubusta, L. Bartůšková, A. Černoch, M. Dušek, and J. Fiurášek, “Experimental asymmetric phase-covariant quantum cloning of polarization qubits,” Phys. Rev. A 78(5), 052323 (2008).
[Crossref]

J. Soubusta, L. Bartůšková, A. Černoch, J. Fiurášek, and M. Dušek, “Several experimental realizations of symmetric phase-covariant quantum cloners of single-photon qubits,” Phys. Rev. A 76(4), 042318 (2007).
[Crossref]

Chen, L.

J.-S. Xu, C.-F. Li, L. Chen, X.-B. Zou, and G.-C. Guo, “Experimental realization of the optimal universal and phase-covariant quantum cloning machines,” Phys. Rev. A 78(3), 032322 (2008).
[Crossref]

Chen, M.-C.

X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114(11), 110504 (2015).
[Crossref]

Chimczak, G.

K. Bartkiewicz, A. Černoch, G. Chimczak, K. Lemr, A. Miranowicz, and F. Nori, “Experimental quantum forgery of quantum optical money,” npj Quantum Inf. 3(1), 7 (2017).
[Crossref]

Cinchetti, M.

D. Bruß, M. Cinchetti, M. G. D’Ariano, and C. Macchiavello, “Phase-covariant quantum cloning,” Phys. Rev. A 62(1), 012302 (2000).
[Crossref]

Clifford, P.

P. Clifford and R. Clifford, “The classical complexity of boson sampling,” in Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (Society for Industrial and Applied Mathematics, 2018), pp. 146–155.

Clifford, R.

P. Clifford and R. Clifford, “The classical complexity of boson sampling,” in Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (Society for Industrial and Applied Mathematics, 2018), pp. 146–155.

Crespi, A.

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7(7), 545–549 (2013).
[Crossref]

Crosnier, C.

N. B. Lovett, C. Crosnier, M. Perarnau-Llobet, and B. C. Sanders, “Differential evolution for many-particle adaptive quantum metrology,” Phys. Rev. Lett. 110(22), 220501 (2013).
[Crossref]

D’Ariano, G. M.

F. Buscemi, G. M. D’Ariano, and C. Macchiavello, “Economical phase-covariant cloning of qudits,” Phys. Rev. A 71(4), 042327 (2005).
[Crossref]

D’Ariano, M. G.

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Figures (4)

Fig. 1.
Fig. 1. Conceptual scheme of hybrid reinforcement learning of a quantum gate driven by a classical control. The transformation of the quantum register performed by the gate is evaluated by measurement providing a reward to the classical control that iteratively modifies the gate’s parameters. The core of this procedure can be viewed as boson sampling, classical simulation of which is known to be computationally #P-hard in $N$ [8].
Fig. 2.
Fig. 2. Experimental setup. Legend: PBS – polarization beam splitter, PC – polarization controller, BBO – beta barium borate, Det – detector, HWP – half-wave plate, QWP – quarter wave-plate, PS – piezoelectric stage, TAC&SCA – time-to-amplitude converter & single channel analyzer.
Fig. 3.
Fig. 3. Plot of the cost function $C$ for the angles $\phi$ and $\theta$ (corresponding to ${\textrm {HWP}}_{4}$ and ${\textrm {HWP}}_{3}$ in Fig. 2, respectively). The solid yellow lines denote the final triangles reached by the Nelder–Mead simplex minimization [29] algorithm in each of its iteration. The dashed lines mark intermediate steps. The circled numbers stand for gate runs and point F depicts the final state of the gate at the end of training. A simulation was performed prior to the experiment to verify our implementation of the learning algorithm. Note that the starting simplex is chosen in advance and is identical in both cases. Subsequent paths in case of the simulation and actual experiment differ due to setup imperfections.
Fig. 4.
Fig. 4. Plots showing the evolution of fidelity of both the clones (top) as well as the cost function (bottom) throughout the training in case of (a) the first model with two free parameters $\phi$ and $\theta$ and (b) the second model with three free parameters $\phi$, $\theta$ and $\omega$. Fidelity of the first clone $F_{1}$ is visualized by a solid red line and the fidelity of the second clone $F_{2}$ is shown in blue (dashed line). The thick solid black line stands for the theoretical limit of $\approx 0.8535$. This theoretical limit bounds the value of fidelity averaged over both clones and over all equatorial states. It is legitimate and expected for $F_1$ to be close to 1 as the cloner starts in a highly asymmetric regime. Moreover, slight deviation from perfect sampling of equatorial input states can cause the average fidelity to surpass the theoretical limit by a few percent.

Tables (1)

Tables Icon

Table 1. Summary of the Final Values for Both Models. F 2 and F 1 Denote the Mean Fidelities Observed on the Test Sets.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

| ψ s = 1 / 2 ( | 0 + e i η | 1 ) ,
| ψ a = cos 2 ω | H + sin 2 ω | V .
( a ^ H , 1 a ^ V , 1 a ^ H , 2 a ^ V , 2 ) ( cos 2 ϕ 0 sin 2 ϕ 0 0 cos 2 θ 0 sin 2 θ sin 2 ϕ 0 cos 2 ϕ 0 0 sin 2 θ 0 cos 2 θ ) ( a ^ H , 1 a ^ V , 1 a ^ H , 2 a ^ V , 2 ) ,
F G = c c Σ = | 1 H , 1 , 0 V , 1 , 1 H , 2 , 0 V , 1 | S ^ 2 U ^ S ^ 1 ^ | 1 H , 1 , 0 V , 1 , 1 H , 2 , 0 V , 1 | 2 = p e r m 2 [ G ^ o d d ] ,
F 1 = F G + c c Σ , F 2 = F G + c c Σ .
c c Σ = | 1 H , 1 , 0 V , 1 , 0 H , 2 , 1 V , 1 | S ^ 2 U ^ S ^ 1 ^ | 1 H , 1 , 0 V , 1 , 1 H , 2 , 0 V , 1 | 2 = p e r m 2 [ G ^ ] ,
c c Σ = | 0 H , 1 , 1 V , 1 , 1 H , 2 , 0 V , 1 | S ^ 2 U ^ S ^ 1 ^ | 1 H , 1 , 0 V , 1 , 1 H , 2 , 0 V , 1 | 2 = p e r m 2 [ G ^ ] ,

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