Skip to main content
SpringerLink
Log in

Algorithmic simulation of far-from-equilibrium dynamics using quantum computer

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We point out that superconducting quantum computers are prospective for the simulation of the dynamics of spin models far from equilibrium, including nonadiabatic phenomena and quenches. The important advantage of these machines is that they are programmable, so that different spin models can be simulated in the same chip, as well as various initial states can be encoded into it in a controllable way. This opens an opportunity to use superconducting quantum computers in studies of fundamental problems of statistical physics such as the absence or presence of thermalization in the free evolution of a closed quantum system depending on the choice of the initial state as well as on the integrability of the model. In the present paper, we performed proof-of-principle digital simulations of two spin models, which are the central spin model and the transverse-field Ising model, using 5- and 16-qubit superconducting quantum computers of the IBM Quantum Experience. We found that these devices are able to reproduce some important consequences of the symmetry of the initial state for the system’s subsequent dynamics, such as the excitation blockade. However, lengths of algorithms are currently limited due to quantum gate errors. We also discuss some heuristic methods which can be used to extract valuable information from the imperfect experimental data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Schoelkopf, R.J., Girvin, S.M.: Wiring up quantum systems. Nature 451(7179), 664 (2008)

    Article  ADS  Google Scholar 

  2. Blatt, R., Roos, C.F.: Quantum simulations with trapped ions. Nat. Phys. 8(4), 277 (2012)

    Article  Google Scholar 

  3. Turner, C.J., Michailidis, A.A., Abanin, D.A., Serbyn, M., Papic, Z.: Quantum many-body scars. arXiv:1711.03528 (2017)

  4. Harty, T.P., Allcock, D.T.C., Ballance, C.J., Guidoni, L., Janacek, H.A., Linke, N.M., Stacey, D.N., Lucas, D.M.: High-fidelity preparation, gates, memory, and readout of a trapped-ion quantum bit. Phys. Rev. Lett. 113, 220501 (2014)

    Article  ADS  Google Scholar 

  5. Lu, D., Li, H., Trottier, D.A., Li, J., Brodutch, A., Krismanich, A.P., Ghavami, A., Dmitrienko, G.I., Long, G., Baugh, J., Laflamme, R.: Experimental estimation of average fidelity of a clifford gate on a 7-qubit quantum processor. Phys. Rev. Lett. 114, 140505 (2015)

    Article  ADS  Google Scholar 

  6. Maller, K.M., Lichtman, M.T., Xia, T., Sun, Y., Piotrowicz, M.J., Carr, A.W., Isenhower, L., Saffman, M.: Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits. Phys. Rev. A 92, 022336 (2015)

    Article  ADS  Google Scholar 

  7. Zwanenburg, F.A., Dzurak, A.S., Morello, A., Simmons, M.Y., Hollenberg, L.C.L., Klimeck, G., Rogge, S., Coppersmith, S.N., Eriksson, M.A.: Silicon quantum electronics. Rev. Mod. Phys. 85, 961 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Kelly, J., Barends, R., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Hoi, I.C., Neill, C., OMalley, P.J.J., Quintana, C., Roushan, P., Vainsencher, A., Wenner, J., Cleland, A.N., Martinis, J.M.: State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519(7541), 66 (2015)

    Article  ADS  Google Scholar 

  10. Ristè, D., Poletto, S., Huang, M.Z., Bruno, A., Vesterinen, V., Saira, O.P., DiCarlo, L.: Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6(1), 6983 (2015)

    Article  Google Scholar 

  11. Córcoles, A., Magesan, E., Srinivasan, S.J., Cross, A.W., Steffen, M., Gambetta, J.M., Chow, J.M.: Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6(1), 6979 (2015)

    Article  Google Scholar 

  12. Gambetta, J.M., Chow, J.M., Steffen, M.: Building logical qubits in a superconducting quantum computing system. npj Quant. Inf. 3(1), 2 (2017)

    Article  ADS  Google Scholar 

  13. 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 (2017)

    Article  ADS  Google Scholar 

  14. Barends, R., Lamata, L., Kelly, J., García-Álvarez, L., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Hoi, I.C., Neill, C., OMalley, P.J.J., Quintana, C., Roushan, P., Vainsencher, A., Wenner, J., Solano, E., Martinis, J.M.: Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6(1), 7654 (2015)

    Article  Google Scholar 

  15. Langford, N.K., Sagastizabal, R., Kounalakis, M., Dickel, C., Bruno, A., Luthi, F., Thoen, D.J., Endo, A., DiCarlo, L.: Experimentally simulating the dynamics of quantum light and matter at ultrastrong coupling. Nat. Commun. 8, 1715 (2017)

    Article  ADS  Google Scholar 

  16. Roushan, P., Neill, C., Tangpanitanon, J., Bastidas, V.M., Megrant, A., Barends, R., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Fowler, A., Foxen, B., Giustina, M., Jeffrey, E., Kelly, J., Lucero, E., Mutus, J., Neeley, M., Quintana, C., Sank, D., Vainsencher, A., Wenner, J., White, T., Neven, H., Angelakis, D.G., Martinis, J.: Spectral signatures of many-body localization with interacting photons. arXiv:1709.07108 (2017)

  17. Ristè, D., da Silva, M.P., Ryan, C.A., Cross, A.W., Córcoles, A.D., Smolin, J.A., Gambetta, J.M., Chow, J.M., Johnson, B.R.: Demonstration of quantum advantage in machine learning. npj Quant. Inf. 3(1), 16 (2017)

    Article  ADS  Google Scholar 

  18. Reagor, M., Osborn, C.B., Tezak, N., Staley, A., Prawiroatmodjo, G., Scheer, M., Alidoust, N., Sete, E.A., Didier, N., da Silva, M.P., Acala, E., Angeles, J., Bestwick, A., Block, M., Bloom, B., Bradley, A., Bui, C., Caldwell, S., Capelluto, L., Chilcott, R., Cordova, J., Crossman, G., Curtis, M., Deshpande, S., El Bouayadi, T., Girshovich, D., Hong, S., Hudson, A., Karalekas, P., Kuang, K., Lenihan, M., Manenti, R., Manning, T., Marshall, J., Mohan, Y., O’Brien, W., Otterbach, J., Papageorge, A., Paquette, J.P., Pelstring, M., Polloreno, A., Rawat, V., Ryan, C.A., Renzas, R., Rubin, N., Russel, D., Rust, M., Scarabelli, D., Selvanayagam, M., Sinclair, R., Smith, R., Suska, M., To, T.W., Vahidpour, M., Vodrahalli, N., Whyland, T., Yadav, K., Zeng, W., Rigetti, C.T.: Demonstration of universal parametric entangling gates on a multi-qubit lattice. Sci. Adv. 4(2), eaao3603 (2018)

    Article  Google Scholar 

  19. Devoret, M., Girvin, S., Schoelkopf, R.: Circuit-qed: how strong can the coupling between a josephson junction atom and a transmission line resonator be? Ann. Phys. (Leipzig) 16(10–11), 767 (2007)

    Article  ADS  MATH  Google Scholar 

  20. Astafiev, O., Zagoskin, A.M., Abdumalikov, A.A., Pashkin, Y.A., Yamamoto, T., Inomata, K., Nakamura, Y., Tsai, J.S.: Resonance fluorescence of a single artificial atom. Science 327(5967), 840 (2010)

    Article  ADS  Google Scholar 

  21. Macha, P., Oelsner, G., Reiner, J.M., Marthaler, M., André, S., Schön, G., Hübner, U., Meyer, H.G., Il’ichev, E., Ustinov, A.V.: Implementation of a quantum metamaterial using superconducting qubits. Nat. Commun. 5, 5146 (2014)

    Article  Google Scholar 

  22. Oelsner, G., Macha, P., Astafiev, O.V., Il’ichev, E., Grajcar, M., Hübner, U., Ivanov, B.I., Neilinger, P., Meyer, H.G.: Dressed-state amplification by a single superconducting qubit. Phys. Rev. Lett. 110, 053602 (2013)

    Article  ADS  Google Scholar 

  23. Lähteenmäki, P., Paraoanu, G.S., Hassel, J., Hakonen, P.J.: Dynamical Casimir effect in a Josephson metamaterial. Proc. Natl. Acad. Sci. USA 110(11), 4234 (2013)

    Article  ADS  Google Scholar 

  24. Wilson, C.M., Johansson, G., Pourkabirian, A., Simoen, M., Johansson, J.R., Duty, T., Nori, F., Delsing, P.: Observation of the dynamical Casimir effect in a superconducting circuit. Nature 479(7373), 376 (2011)

    Article  ADS  Google Scholar 

  25. Segev, E., Abdo, B., Shtempluck, O., Buks, E., Yurke, B.: Prospects of employing superconducting stripline resonators for studying the dynamical Casimir effect experimentally. Phys. Lett. A 370(3–4), 202 (2007)

    Article  ADS  Google Scholar 

  26. Remizov, S.V., Zhukov, A.A., Shapiro, D.S., Pogosov, W.V., Lozovik, Y.E.: Parametrically driven hybrid qubit–photon systems: dissipation-induced quantum entanglement and photon production from vacuum. Phys. Rev. A 96, 043870 (2017)

    Article  ADS  Google Scholar 

  27. Zhukov, A.A., Shapiro, D.S., Pogosov, W.V., Lozovik, Y.E.: Dynamical Lamb effect versus dissipation in superconducting quantum circuits. Phys. Rev. A 93, 063845 (2016)

    Article  ADS  Google Scholar 

  28. Shapiro, D.S., Zhukov, A.A., Pogosov, W.V., Lozovik, Y.E.: Dynamical Lamb effect in a tunable superconducting qubit-cavity system. Phys. Rev. A 91, 063814 (2015)

    Article  ADS  Google Scholar 

  29. Neill, C., Roushan, P., Kechedzhi, K., Boixo, S., Isakov, S.V., Smelyanskiy, V., Barends, R., Burkett, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Fowler, A., Foxen, B., Graff, R., Jeffrey, E., Kelly, J., Lucero, E., Megrant, A., Mutus, J., Neeley, M., Quintana, C., Sank, D., Vainsencher, A., Wenner, J., White, T.C., Neven, H., Martinis, J.M.: A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360(6385), 195 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  30. Calabrese, P., Essler, F.H.L., Fagotti, M.: Quantum quench in the transverse-field Ising chain. Phys. Rev. Lett. 106, 227203 (2011)

    Article  ADS  Google Scholar 

  31. Rieger, H., Iglói, F.: Semiclassical theory for quantum quenches in finite transverse Ising chains. Phys. Rev. B 84, 165117 (2011)

    Article  ADS  Google Scholar 

  32. Eisert, J., Friesdorf, M., Gogolin, C.: Quantum many-body systems out of equilibrium. Nat. Phys. 11(2), 124 (2015)

    Article  Google Scholar 

  33. Gogolin, C., Eisert, J.: Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 79(5), 056001 (2016)

    Article  ADS  Google Scholar 

  34. Heyl, M., Polkovnikov, A., Kehrein, S.: Dynamical quantum phase transitions in the transverse-field Ising model. Phys. Rev. Lett. 110, 135704 (2013)

    Article  ADS  Google Scholar 

  35. D’Alessio, L., Kafri, Y., Polkovnikov, A., Rigol, M.: From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65(3), 239 (2016)

    Article  ADS  Google Scholar 

  36. Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T., Schmiedmayer, J.: Non-equilibrium coherence dynamics in one-dimensional Bose gases. Nature 449(7160), 324 (2007)

    Article  ADS  Google Scholar 

  37. Trotzky, S., Chen, Y.A., Flesch, A., McCulloch, I.P., Schollwöck, U., Eisert, J., Bloch, I.: Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys. 8(4), 325 (2012)

    Article  Google Scholar 

  38. Rigol, M., Dunjko, V., Yurovsky, V., Olshanii, M.: Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)

    Article  ADS  Google Scholar 

  39. Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008)

    Article  ADS  Google Scholar 

  40. Polkovnikov, A., Sengupta, K., Silva, A., Vengalattore, M.: Colloquium: nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863 (2011)

    Article  ADS  Google Scholar 

  41. Bañuls, M.C., Cirac, J.I., Hastings, M.B.: Strong and weak thermalization of infinite nonintegrable quantum systems. Phys. Rev. Lett. 106, 050405 (2011)

    Article  ADS  Google Scholar 

  42. Gogolin, C., Müller, M.P., Eisert, J.: Absence of thermalization in nonintegrable systems. Phys. Rev. Lett. 106, 040401 (2011)

    Article  ADS  Google Scholar 

  43. Rosner, H., Singh, R.R.P., Zheng, W.H., Oitmaa, J., Pickett, W.E.: High-temperature expansions for the \({J}_{1}{-}{J}_{2}\) Heisenberg models: applications to ab initio calculated models for \({\text{ Li }}_{2}{\text{ VOSiO }}_{4}\) and \({\text{ Li }}_{2}{\text{ VOGeO }}_{4}\). Phys. Rev. B 67, 014416 (2003)

    Article  ADS  Google Scholar 

  44. Barends, R., Shabani, A., Lamata, L., Kelly, J., Mezzacapo, A., Heras, U.L., Babbush, R., Fowler, A.G., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Jeffrey, E., Lucero, E., Megrant, A., Mutus, J.Y., Neeley, M., Neill, C., OMalley, P.J.J., Quintana, C., Roushan, P., Sank, D., Vainsencher, A., Wenner, J., White, T.C., Solano, E., Neven, H., Martinis, J.M.: Digitized adiabatic quantum computing with a superconducting circuit. Nature 534(7606), 222 (2016)

    Article  ADS  Google Scholar 

  45. Das, A., Chakrabarti, B.K.: Quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Albash, T., Lidar, D.A.: Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  47. Tanaka, S., Tamura, R., Chakrabarti, B.K.: Quantum Spin Glasses, Annealing and Computation. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  48. Prokof’ev, N.V., Stamp, P.C.E.: Theory of the spin bath. Rep. Prog. Phys. 63(4), 669 (2000)

    Article  ADS  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

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

    A. A. Zhukov, S. V. Remizov, W. V. Pogosov & Yu. E. Lozovik

  2. National Research Nuclear University (MEPhI), Moscow, Russia, 115409

    A. A. Zhukov

  3. Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia, 125009

    S. V. Remizov

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

    W. V. Pogosov

  5. Institute of Spectroscopy, Russian Academy of Sciences, Moscow, Russia, 142190

    Yu. E. Lozovik

  6. Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia, 101000

    Yu. E. Lozovik

Authors
  1. A. A. Zhukov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Remizov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. V. Pogosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yu. E. Lozovik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. V. Pogosov.

Additional information

W. V. Pogosov acknowledges a support from RFBR (Project No. 15-02-02128). Yu. E. Lozovik acknowledges a support from RFBR (Project No. 17-02-01134) and the Program of Basic Research of HSE.

Appendices

Appendix A: Preparation of three-particle entangled state

Quantum circuit used to prepare three-particle entangled state (5) is shown in Fig. 19. Single-qubit gates \(A_{\varphi }\) and B are constructed from the standard IBMqx4 gate \(U_3\) as \(A_{\varphi }=U_3(\theta = 2 \arccos \frac{1}{\sqrt{3}}, \varphi , \lambda = 0)\), \(B=U_3(\theta = \frac{\pi }{4}, \varphi =0, \lambda = 0)\); Z is Pauli-Z gate.

Fig. 19
figure 19

Quantum circuit for the preparation of three-qubit excited state

Full size image

Appendix B: Full quantum circuits for the central spin model

Figures 20, 21, and 22 show full quantum circuits for three different situation within the central spin Hamiltonian. For the sake of simplicity, we restrict ourselves to the single Trotter number, \(N=1\). The generalization to the circuits with \(N > 1\) is straightforward.

Fig. 20
figure 20

Quantum circuit for the evolution of the system starting from the initial state of two-particle entangled state of the bath and unexcited central spin at the Trotter number \(N=1\)

Full size image
Fig. 21
figure 21

Quantum circuit for the evolution of the system starting from the initial state of three-particle entangled state of the bath and unexcited central spin at the Trotter number \(N=1\)

Full size image
Fig. 22
figure 22

Quantum circuit for the evolution of the system starting from the initial state of excited central spin and four unexcited spins of the bath at the Trotter number \(N=1\)

Full size image

Appendix C: Raw results for the 8-spin transverse-field Ising chain

Figure 23 provides raw data for the occupations of the upper levels of the 8-spin transverse-field Ising chain simulated in the real device in comparison with the theoretical results. In both cases, Trotter number is \(N=2\).

Fig. 23
figure 23

(Color online) The results of our experiment (solid blue lines) and theory (dashed brown lines) for the occupations of the upper levels of 8-spin transverse Ising chain at \(\alpha =J\) (a), \(\alpha =2J\) (b), \(\alpha =5J\) (c) as a function of the dimensionless time \(\tau \) for the Trotter number \(N=2\)

Full size image

Appendix D: Raw results for the 16-spin transverse-field Ising ladder

Figure 24 provides raw data for the occupations of the upper levels of 16-spin transverse-field Ising ladder simulated in the real device in comparison with the theoretical results. In both cases, Trotter number is \(N=1\).

Fig. 24
figure 24

(Color online) The results of our experiment (solid blue lines) and theory (dashed brown lines) for the occupations of the upper levels of 16-spin transverse Ising ladder at \(\alpha =J\) (a), \(\alpha =2J\) (b), \(\alpha =5J\) (c) as a function of the dimensionless time \(\tau \) for the Trotter number \(N=1\)

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhukov, A.A., Remizov, S.V., Pogosov, W.V. et al. Algorithmic simulation of far-from-equilibrium dynamics using quantum computer. Quantum Inf Process 17, 223 (2018). https://doi.org/10.1007/s11128-018-2002-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-2002-y

Keywords

Search

Navigation