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Quantum error reduction with deep neural network applied at the post-processing stage

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Abstract

Deep neural networks (DNN) can be applied at the post-processing stage for the improvement of the results of quantum computations on noisy intermediate-scale quantum (NISQ) processors. Here, we propose a method based on this idea, which is most suitable for digital quantum simulation characterized by the periodic structure of quantum circuits consisting of Trotter steps. A key ingredient of our approach is that it does not require any data from a classical simulator at the training stage. The network is trained to transform data obtained from quantum hardware with artificially increased Trotter steps number (noise level) toward the data obtained without such an increase. The additional Trotter steps are fictitious, i.e., they contain negligibly small rotations and, in the absence of hardware imperfections, reduce essentially to the identity gates. This preserves, at the training stage, information about relevant quantum circuit features. Two particular examples are considered that are the dynamics of the transverse-field Ising chain and XY spin chain, which were implemented on two real five-qubit IBM Q processors. A significant error reduction is demonstrated as a result of the DNN application that allows us to effectively increase quantum circuit depth in terms of Trotter steps.

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Data availability

Raw data for this study were generated by running quantum circuits via QISKIT on real quantum processors IBM Athens and Bogota or simulator. The datasets generated during the current study are available from the corresponding author on request.

References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549(7671), 242–246 (2017)

    Article  ADS  Google Scholar 

  3. Barends, R., Lamata, L., Kelly, J., García-Álvarez, L., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y.: Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6(1), 1–7 (2015)

    Article  Google Scholar 

  4. Mattle, K., Weinfurter, H., Kwiat, P.G., Zeilinger, A.: Dense coding in experimental quantum communication. Phys. Rev. Lett. 76(25), 4656 (1996)

    Article  ADS  Google Scholar 

  5. Zhukov, A.A., Kiktenko, E.O., Elistratov, A.A., Pogosov, W.V., Lozovik, Y.E.: Quantum communication protocols as a benchmark for programmable quantum computers. Quantum Inf. Proc. 18(1), 1–23 (2019)

    Article  ADS  MATH  Google Scholar 

  6. Georgescu, I.M., Ashhab, S., Nori, F.: Quantum simulation. Rev. Mod. Phys. 86(1), 153 (2014)

    Article  ADS  Google Scholar 

  7. Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P.J., Aspuru-Guzik, A., O’brien, J.L.: A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5(1), 1–7 (2014)

    Article  Google Scholar 

  8. O’Malley, P.J., Babbush, R., Kivlichan, I.D., Romero, J., McClean, J.R., Barends, R., Kelly, J., Roushan, P., Tranter, A., Ding, N.: Scalable quantum simulation of molecular energies. Phys. Rev. X 6(3), 031007 (2016)

    Google Scholar 

  9. Mohseni, M., Read, P., Neven, H., Boixo, S., Denchev, V., Babbush, R., Fowler, A., Smelyanskiy, V., Martinis, J.: Commercialize quantum technologies in five years. Nat. News 543(7644), 171 (2017)

    Article  Google Scholar 

  10. Li, Y., Benjamin, S.C.: Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7(2), 021050 (2017)

    Google Scholar 

  11. Preskill, J.: Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018)

    Article  Google Scholar 

  12. Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., Bengtsson, A., Boixo, S., Broughton, M., Buckley, B.B., Buell, D.A., Burkett, B., Bushnell, N., Chen, Y., Chen, Z., Chen, Y.-A., Chiaro, B., Collins, R., Cotton, S.J., Courtney, W., Demura, S., Derk, A., Dunsworth, A., Eppens, D., Eckl, T., Erickson, C., Farhi, E., Fowler, A., Foxen, B., Gidney, C., Giustina, M., Graff, R., Gross, J.A., Habegger, S., Harrigan, M.P., Ho, A., Hong, S., Huang, T., Huggins, W., Ioffe, L.B., Isakov, S.V., Jeffrey, E., Jiang, Z., Jones, C., Kafri, D., Kechedzhi, K., Kelly, J., Kim, S., Klimov, P.V., Korotkov, A.N., Kostritsa, F., Landhuis, D., Laptev, P., Lindmark, M., Lucero, E., Marthaler, M., Martin, O., Martinis, J.M., Marusczyk, A., McArdle, S., McClean, J.R., McCourt, T., McEwen, M., Megrant, A., Mejuto-Zaera, C., Mi, X., Mohseni, M., Mruczkiewicz, W., Mutus, J., Naaman, O., Neeley, M., Neill, C., Neven, H., Newman, M., Niu, M.Y., O’Brien, T.E., Ostby, E., Pató, B., Petukhov, A., Putterman, H., Quintana, C., Reiner, J.-M., Roushan, P., Rubin, N.C., Sank, D., Satzinger, K.J., Smelyanskiy, V., Strain, D., Sung, K.J., Schmitteckert, P., Szalay, M., Tubman, N.M., Vainsencher, A., White, T., Vogt, N., Yao, Z.J., Yeh, P., Zalcman, A., Zanker, S.: Observation of separated dynamics of charge and spin in the Fermi-Hubbard model (2020)

  13. Cappellaro, P., Viola, L., Ramanathan, C.: Coherent-state transfer via highly mixed quantum spin chains. Phys. Rev. A 83, 032304 (2011)

    Article  ADS  Google Scholar 

  14. Zhukov, A.A., Remizov, S.V., Pogosov, W.V., Lozovik, Y.E.: Algorithmic simulation of far-from-equilibrium dynamics using quantum computer. Quantum Inf. Proc. 17(9), 1–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Babukhin, D.V., Zhukov, A.A., Pogosov, W.V.: Hybrid digital-analog simulation of many-body dynamics with superconducting qubits. Phys. Rev. A 101, 052337 (2020)

    Article  ADS  Google Scholar 

  16. Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature 549(7671), 195–202 (2017)

    Article  ADS  Google Scholar 

  17. Dunjko, V., Briegel, H.J.: Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep. Prog. Phys. 81(7), 074001 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. Perdomo-Ortiz, A., Benedetti, M., Realpe-Gómez, J., Biswas, R.: Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers. Quantum Sci. Technol. 3(3), 030502 (2018)

    Article  ADS  Google Scholar 

  19. Ciliberto, C., Herbster, M., Ialongo, A.D., Pontil, M., Rocchetto, A., Severini, S., Wossnig, L.: Quantum machine learning: a classical perspective. Proc. Math. Phys. Eng. Sci. 474(2209), 20170551 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Schuld, M., Sinayskiy, I., Petruccione, F.: An introduction to quantum machine learning. Contemp. Phys. 56(2), 172–185 (2015)

    Article  ADS  MATH  Google Scholar 

  21. Benedetti, M., Lloyd, E., Sack, S., Fiorentini, M.: Parameterized quantum circuits as machine learning models. Quantum Sci. Technol. 4(4), 043001 (2019)

    Article  ADS  Google Scholar 

  22. Nielsen, M.A.: Neural Networks and Deep Learning, vol. 25 (2015)

  23. Carleo, G., Cirac, I., Cranmer, K., Daudet, L., Schuld, M., Tishby, N., Vogt-Maranto, L., Zdeborová, L.: Machine learning and the physical sciences. Rev. Mod. Phys. 91(4), 045002 (2019)

    Article  ADS  Google Scholar 

  24. Lennon, D., Moon, H., Camenzind, L., Yu, L., Zumbühl, D., Briggs, G., Osborne, M., Laird, E., Ares, N.: Efficiently measuring a quantum device using machine learning. Npj Quantum Inf. 5(1), 1–8 (2019)

    Google Scholar 

  25. Nautrup, H.P., Delfosse, N., Dunjko, V., Briegel, H.J., Friis, N.: Optimizing quantum error correction codes with reinforcement learning. Quantum 3, 215 (2019)

    Article  Google Scholar 

  26. Baireuther, P., O’Brien, T.E., Tarasinski, B., Beenakker, C.W.J.: Machine-learning-assisted correction of correlated qubit errors in a topological code. Quantum 2, 48 (2018)

    Article  Google Scholar 

  27. Andreasson, P., Johansson, J., Liljestrand, S., Granath, M.: Quantum error correction for the toric code using deep reinforcement learning. Quantum 3, 183 (2019)

    Article  Google Scholar 

  28. Kalantre, S.S., Zwolak, J.P., Ragole, S., Wu, X., Zimmerman, N.M., Stewart, M.D., Taylor, J.M.: Machine learning techniques for state recognition and auto-tuning in quantum dots. Npj Quantum Inf. 5(1), 1–10 (2019)

    Article  ADS  Google Scholar 

  29. Vozhakov, V., Bastrakova, M.V., Klenov, N.V., Soloviev, I.I., Pogosov, W.V., Babukhin, D.V., Zhukov, A.A., Satanin, A.M.: State control in superconducting quantum processors. Physics–Uspekhi (2021)

  30. Bukov, M., Day, A.G., Sels, D., Weinberg, P., Polkovnikov, A., Mehta, P.: Reinforcement learning in different phases of quantum control. Phys. Rev. X 8(3), 031086 (2018)

    Google Scholar 

  31. Niu, M.Y., Boixo, S., Smelyanskiy, V.N., Neven, H.: Universal quantum control through deep reinforcement learning. Npj Quantum Inf. 5(1), 1–8 (2019)

    Article  Google Scholar 

  32. Babukhin, D.V., Zhukov, A.A., Pogosov, W.V.: Nondestructive classification of quantum states using an algorithmic quantum computer. Quantum Mach. Intell. 1(3), 87–96 (2019)

    Article  Google Scholar 

  33. Carrasquilla, J., Melko, R.G.: Machine learning phases of matter. Nat. Phys. 13(5), 431–434 (2017)

    Article  Google Scholar 

  34. Altepeter, J.B., Jeffrey, E.R., Kwiat, P.G.: Photonic state tomography. Adv. At. Mol. Opt. Phys. 52, 105–159 (2005)

    Article  ADS  Google Scholar 

  35. Torlai, G., Mazzola, G., Carrasquilla, J., Troyer, M., Melko, R., Carleo, G.: Neural-network quantum state tomography. Nat. Phys. 14(5), 447–450 (2018)

    Article  Google Scholar 

  36. Neugebauer, M., Fischer, L., Jäger, A., Czischek, S., Jochim, S., Weidemüller, M., Gärttner, M.: Neural-network quantum state tomography in a two-qubit experiment. Phys. Rev. A 102(4), 042604 (2020)

    Article  ADS  Google Scholar 

  37. Lohani, S., Kirby, B.T., Brodsky, M., Danaci, O., Glasser, R.T.: Machine learning assisted quantum state estimation. Mach. Learn. Sci. Technol. 1(3), 035007 (2020)

    Article  Google Scholar 

  38. Sehayek, D., Golubeva, A., Albergo, M.S., Kulchytskyy, B., Torlai, G., Melko, R.G.: Learnability scaling of quantum states: restricted Boltzmann machines. Phys. Rev. B 100(19), 195125 (2019)

    Article  ADS  Google Scholar 

  39. Palmieri, A.M., Kovlakov, E., Bianchi, F., Yudin, D., Straupe, S., Biamonte, J.D., Kulik, S.: Experimental neural network enhanced quantum tomography. Npj Quantum Inf. 6(1), 1–5 (2020)

    Article  Google Scholar 

  40. Teo, Y.S., Shin, S., Jeong, H., Kim, Y., Kim, Y.-H., Struchalin, G.I., Kovlakov, E.V., Straupe, S.S., Kulik, S.P., Leuchs, G., et al.: Benchmarking quantum tomography completeness and fidelity with machine learning. arXiv:2103.01535 (2021)

  41. Czarnik, P., Arrasmith, A., Coles, P.J., Cincio, L.: Error mitigation with clifford quantum-circuit data. arXiv:2005.10189 (2020)

  42. Strikis, A., Qin, D., Chen, Y., Benjamin, S.C., Li, Y.: Learning-based quantum error mitigation. arXiv:2005.07601 (2020)

  43. Kim, C., Park, K.D., Rhee, J.-K.: Quantum error mitigation with artificial neural network. IEEE Access 8, 188853–188860 (2020)

    Article  Google Scholar 

  44. Proctor, T., Rudinger, K., Young, K., Nielsen, E., Blume-Kohout, R.: Measuring the Capabilities of Quantum Computers (2020)

  45. Temme, K., Bravyi, S., Gambetta, J.M.: Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017)

  46. Fauseweh, B., Zhu, J.-X.: Digital quantum simulation of non-equilibrium quantum many-body systems. Quantum Inf. Proc. 20, 138 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  47. Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017)

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Acknowledgements

We acknowledge use of the IBM Quantum Experience for this work. The viewpoints expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience team. A. A. Zh. acknowledges a support from RFBR (project no. 20-37-70028). W. V. P. acknowledges a support from RFBR (project no. 19-02-00421).

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Authors and Affiliations

  1. Dukhov Research Institute of Automatics (VNIIA), Moscow, 127055, Russia

    Andrey Zhukov & Walter Pogosov

  2. Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412, Russia

    Walter Pogosov

  3. HSE University, Moscow, 109028, Russia

    Walter Pogosov

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  1. Andrey Zhukov
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  2. Walter Pogosov
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Zhukov, A., Pogosov, W. Quantum error reduction with deep neural network applied at the post-processing stage. Quantum Inf Process 21, 93 (2022). https://doi.org/10.1007/s11128-022-03433-9

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