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Quantum learning with noise and decoherence: a robust quantum neural network

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Abstract

Noise and decoherence are two major obstacles to the implementation of large-scale quantum computing. Because of the no-cloning theorem, which says we cannot make an exact copy of an arbitrary quantum state, simple redundancy will not work in a quantum context, and unwanted interactions with the environment can destroy coherence and thus the quantum nature of the computation. Because of the parallel and distributed nature of classical neural networks, they have long been successfully used to deal with incomplete or damaged data. In this work, we show that our model of a quantum neural network (QNN) is similarly robust to noise, and that, in addition, it is robust to decoherence. Moreover, robustness to noise and decoherence is not only maintained but improved as the size of the system is increased. Noise and decoherence may even be of advantage in training, as it helps correct for overfitting. We demonstrate the robustness using entanglement as a means for pattern storage in a qubit array. Our results provide evidence that machine learning approaches can obviate otherwise recalcitrant problems in quantum computing.

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References

  • Aizenberg I (2011) Complex-valued neural networks with multi-valued neurons. Springer

  • Albash T, Lidar DA (2015) Decoherence in adiabatic quantum computation. Phys Rev A 91:062320

    Article  Google Scholar 

  • Allauddin R, Gaddam K, Boehmer S, Behrman EC, Steck JE (2008) Quantum simultaneous recurrent networks for content addressable memory. In: Nedjah N, dos Santos Coelho L, de Macedo Mourelle L (eds) Quantum-Inspired Intelligent Systems. Springer, Verlag

    MATH  Google Scholar 

  • Bang J, Dutta A, Lee S-W, Kim J (2019) Optimal usage of quantum random access memory in quantum machine learning. Phys Rev A 99:012326

    Article  Google Scholar 

  • Behrman EC, Steck JE (2013) Multiqubit entanglement of a general input state. Quantum Inf. Comput. 13:36–53

    MathSciNet  Google Scholar 

  • Behrman E.C, Niemel J, Steck J.E, and Skinner S.R, “A quantum dot neural network,” Proceedings of the Fourth Workshop on Physics and Computation (PhysComp96), 22 (1996)

  • Behrman EC, Nash LR, Steck JE, Chandrashekar V, Skinner SR (2000) Simulations of quantum neural networks. Inf Sci 128:257

    Article  MathSciNet  Google Scholar 

  • Behrman E.C, Chandrashekar V, Wang Z, Belur C.K, Steck J.E, and Skinner S.R, “A quantum neural network computes entanglement,” arXiv:quant-ph/0202131 (2002)

  • Behrman EC, Steck JE, Kumar P, Walsh KA (2008) Quantum algorithm design using dynamic learning. Quantum Inf. Comput. 8:12–29

    MathSciNet  MATH  Google Scholar 

  • Behrman EC, Nguyen NH, Steck JE, McCann M (2016) Quantum neural computation of entanglement is robust to noise and decoherence. In: Bhattacharyya S (ed) Quantum Inspired Computational Intelligence: Research and Applications. Elsevier, Morgan Kauffman, pp 3–33

    Google Scholar 

  • Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK (1996) Mixed-state entanglement and quantum error correction. Phys Rev A 54:3824–3851

    Article  MathSciNet  Google Scholar 

  • Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S (2017) Quantum machine learning. Nature 549:195–202

    Article  Google Scholar 

  • Bishop CM (1995a) Neural networks for pattern recognition. Oxford Univ Press

  • Bishop CM (1995b) Training with noise is equivalent to Tikhonov regularization. Neural Comp 7:108–116

    Article  Google Scholar 

  • Coffman V, Kundu J, Wootters WK (2000) Distributed entanglement. Phys Rev A 61:052306

    Article  Google Scholar 

  • Cross AW, Smith G, Smolin JA (2015) Quantum learning robust to noise. Phys Rev A 92:012327

    Article  Google Scholar 

  • Dong D, Mabrok M.A, Petersen I.R, Qi B, Chen C, and Rabitz H, “Sampling-based learning control for quantum systems with uncertainties,” IEEE Transactions on Control Systems Technology 23, pp. 2155–2166 (2015)

  • Dunjiko V, Taylor JM, Briegel HJ (2016) Quantum-enhanced machine learning. Phys Rev Lett 117:130501

    Article  MathSciNet  Google Scholar 

  • Efron B, Tibshirani RJ (1994) An introduction to the bootstrap. Chapman and Hall/CRC, Boca Raton, FL

    Book  Google Scholar 

  • Fausett L.V, Fundamentals of neural networks. Pearson, (1993)

    MATH  Google Scholar 

  • Glickman Y, Kotler S, Akerman N, Ozeri R (2012) Emergence of a measurement basis in atom-photon scattering. Science 339:1187–1191

    Article  Google Scholar 

  • Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. Johns Hopkins University Press

  • Goodfellow I, Bengio Y, Courville A, Bach F (2016) Deep learning. MIT Press

  • Graves A, Mohamed A, and Hinton G, “Speech recognition with deep recurrent neural networks,” arXiv:1303.5778 (2013)

  • Hartmann MJ, Carleo G (2019) Neural network approach to dissipative quantum many-body dynamics. Phys Rev Lett 122:250502

    Article  Google Scholar 

  • Holmstrom L, Koistinen P (1992) Using additive noise in back-propagation training. IEEE Trans. Neural Networks 3:24–38

    Article  Google Scholar 

  • Kalai G, “Quantum computers: noise propagation and adversarial noise models,” arXiv:0904.3265 (2009)

  • Kalai G, “How quantum computer fail: quantum codes, correlations in physical systems, and noise accumulation,” arXiv:1106.0485 (2011)

  • Kalai G, “The quantum computer puzzle,” arXiv:1605.00992 (2016)

  • Lloyd S, “The universe as quantum computer,” arXiv:1312.4455 (2013)

  • Mehta P, Bukov M, Wang C-H, Day AGR, Richardson C, Fisher CK, Schwab DJ (2019) A high-bias, low-variance introduction to machine learning for physicists. Phys Rep 810:1–124

    Article  MathSciNet  Google Scholar 

  • Microsoft Quantum Development Kit. https://docs.microsoft.com/en-us/quantum/?view=qsharp-preview, 2018

  • Nagy A, Savona V (2019) Variational quantum Monte Carlo method with a neural network ansatz for open quantum systems. Phys Rev Lett 122:250501

    Article  MathSciNet  Google Scholar 

  • Neelakantan A, Vilnis L, Le Q.V, Sutskever I, Kaiser L, Kurach K, and Martens J, “Adding gradient noise improves learning for very deep networks,” arXiv:1511.06807 (2015)

  • Neuralware getting started: a tutorial in NeuralWorks Professional II/Plus, (2000)

  • Nguyen N.H, Behrman E.C, Moustafa M.A and Steck J.E, “Benchmarking neural networks for quantum computations,” IEEE Transactions on Neural Networks and Learning Systems (to appear, 2019a); also at arXiv:1807.03253

  • Nguyen N.H, Samarakoon B, Behrman E.C, and Steck J.E, “Pattern storage in qubit arrays using entanglement,” in preparation (2019b)

  • Nielsen MA, Chuang IL (2001) Quantum computation and quantum information. Cambridge University Press

  • Paetznick A, Reichardt BW (2013) Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code. Quantum Inf Comput 12(11–12):1034–1080

    MathSciNet  MATH  Google Scholar 

  • Piquero-Zulaica I, Lobo-Checa J, Sadeghi A, Abd El-Fattah ZM, Mitsui C, Okamoto T, Pawlak R, Meier T, Arnau A, Ortega JE, Takeya J, Goedecker S, Meyer E, Kawai S (2017) Precise engineering of quantum dot array coupling through their barrier widths. Nature Commun 8:787

    Article  Google Scholar 

  • Reed R.D, Neural smithing. Bradford, 1999

  • Rethinam MJ, Javali AK, Hart AE, Behrman EC, Steck JE (2011) A genetic algorithm for finding pulse sequences for nmr quantum computing. Paritantra – Journal of Systems Science and Engineering 20:32–42

    Google Scholar 

  • Roszak K, Filip R, and Novotny T, “Decoherence control by decoherence itself,” Scientific Reports 5, Article number: 9796 (2014)

  • Schuld M, Sinayskiy I, Petruccione F (2014) An introduction to quantum machine learning. Contemp Phys 56:172–185

    Article  Google Scholar 

  • Shor PW (1995) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput 26:1484–1509

    Article  MathSciNet  Google Scholar 

  • Takahasi S, Tupitsyn IS, van Tol J, Beedle CC, Hendrickson DN, Stamp PCE (2011) Decoherence in crystals of quantum molecular magnets. Nature 476:76–79

    Article  Google Scholar 

  • The IBM quantum experience. https://quantumexperience.ng.bluemix.net/qx, 2018

  • Thompson N.L, Nguyen N.H, Behrman E.C, and Steck J.E, “Experimental pairwise entanglement estimation for an N-qubit system: a machine learning approach for programming quantum hardware,” submitted to Quantum Information Processing (2019); available at arXiv:1902.07754

  • Vicentini F, Biella A, Regnault N, Ciuti C (2019) Variational neural network ansatz for steady states in open quantum systems. Phys Rev Lett 122:250503

    Article  Google Scholar 

  • Wax N (2014) Selected papers on noise and stochastic process. Dover

  • Wiebe N, Kapoor A, Svore K (2014) Quantum algorithms for nearest neighbor methods for supervised and unsupervised learning. Quantum Inf. Comput. 15:318–358

    MathSciNet  Google Scholar 

  • Wootters WK (1998) Entanglement of formation of an arbitrary state of two qubits. Phys Rev Lett 80:2245

    Article  Google Scholar 

  • Zurek WH (2009) Quantum Darwinism. Nat Phys 5:181–188

    Article  Google Scholar 

Download references

Acknowledgments

We thank Mohammed Moustafa and Henry Elliott for helpful discussions and for providing the NeuralWorks data and the six-qubit array symbol data, respectively.

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Correspondence to Elizabeth C. Behrman.

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Nguyen, N.H., Behrman, E.C. & Steck, J.E. Quantum learning with noise and decoherence: a robust quantum neural network. Quantum Mach. Intell. 2, 1 (2020). https://doi.org/10.1007/s42484-020-00013-x

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  • DOI: https://doi.org/10.1007/s42484-020-00013-x

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