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Experimental quantum fast hitting on hexagonal graphs


Quantum walks are powerful kernels in quantum computing protocols, and possess strong capabilities in speeding up various simulation and optimization tasks. One striking example is provided by quantum walkers evolving on glued trees1, which demonstrate faster hitting performances than classical random walks. However, their experimental implementation is challenging, as this involves highly complex arrangements of an exponentially increasing number of nodes. Here, we propose an alternative structure with a polynomially increasing number of nodes. We successfully map such graphs on quantum photonic chips using femtosecond-laser direct writing techniques in a geometrically scalable fashion. We experimentally demonstrate quantum fast hitting by implementing two-dimensional quantum walks on graphs with up to 160 nodes and a depth of eight layers, achieving a linear relationship between the optimal hitting time and the network depth. Our results open up a scalable path towards quantum speed-up in classically intractable complex problems.

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Fig. 1: Theoretical graphs and their implementation on photonic chips.
Fig. 2: Fast hitting on a two-layered hexagonal graph.
Fig. 3: Increasing the complexity of hexagonal graphs.
Fig. 4: Comparison between quantum hitting and classical hitting.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


  1. Childs, A. M., Farhi, E. & Gutmann, S. An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1, 35–43 (2002).

    MathSciNet  Article  Google Scholar 

  2. Mohseni, M. et al. Commercialize quantum technologies in five years. Nature 543, 171–174 (2017).

    ADS  Article  Google Scholar 

  3. Childs, A. M. & van Dam, W. Quantum algorithms for algebraic problems. Rev. Mod. Phys. 82, 1–52 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  4. Szegedy, M. in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 32–41 (IEEE Computer Society, Washington, 2004).

  5. Childs, A. M. & Goldstone, J. Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004).

    ADS  Article  Google Scholar 

  6. Aaronson, S. & Shi, Y. Quantum lower bounds for the collision and the element distinctness problems. J. ACM 51, 595–605 (2004).

    MathSciNet  Article  Google Scholar 

  7. Buhrman, H. & Špalek, R. in Proceedings of the Seventeenth Annual ACM–SIAM Symposium on Discrete Algorithm 880–889 (SIAM, Philadelphia, 2006).

  8. Farhi, E., Goldstone, J. & Gutmann, S. A quantum algorithm for the Hamiltonian NAND tree. Theory Comput. 4, 169–190 (2008).

    MathSciNet  Article  Google Scholar 

  9. Douglas, B. L. & Wang, J. B. A classical approach to the graph isomorphism problem using quantum walks. J. Phys. A 41, 075303 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  10. Bruderer, M. & Plenio, M. B. Decoherence enhances performance of quantum walks applied to graph isomorphism testing. Phys. Rev. A 94, 062317 (2016).

    ADS  Article  Google Scholar 

  11. Childs, A. M. et al. in Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing 59–68 (ACM, New York, 2003).

  12. Farhi, E. & Gutmann, S. Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  13. Douglas, B. L. & Wang, J. B. Efficient quantum circuit implementation of quantum walks. Phys. Rev. A 79, 052335 (2009).

    ADS  Article  Google Scholar 

  14. Carneiro, I. et al. Entanglement in coined quantum walks on regular graphs. New J. Phys. 7, 156 (2005).

    ADS  Article  Google Scholar 

  15. Schmitz, H. et al. Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103, 090504 (2009).

    ADS  Article  Google Scholar 

  16. Karski, M. et al. Quantum walk in position space with single optically trapped atoms. Science 325, 174–177 (2009).

    ADS  Article  Google Scholar 

  17. Broome, M. et al. Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010).

    ADS  Article  Google Scholar 

  18. Sansoni, L. et al. Two-particle bosonic-fermionic quantum walk via integrated photonics. Phys. Rev. Lett. 108, 010502 (2012).

    ADS  Article  Google Scholar 

  19. Cardano, F. et al. Quantum walks and wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 1, e1500087 (2015).

    ADS  Article  Google Scholar 

  20. Du, J. et al. Experimental implementation of the quantum random-walk algorithm. Phys. Rev. A 67, 042316 (2003).

    ADS  Article  Google Scholar 

  21. Perets, H. B. et al. Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100, 170506 (2008).

    ADS  Article  Google Scholar 

  22. Peruzzo, A. et al. Quantum walks of correlated photons. Science 329, 1500–1503 (2010).

    ADS  Article  Google Scholar 

  23. Preiss, P. M. et al. Strongly correlated quantum walks in optical lattices. Science 347, 1229–1233 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  24. Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012).

    ADS  Article  Google Scholar 

  25. Jeong, Y. C., Di Franco, C., Lim, H. T., Kim, M. S. & Kim, Y. H. Experimental realization of a delayed-choice quantum walk. Nat. Commun. 4, 2471 (2013).

    Article  Google Scholar 

  26. Tang, H. et al. Experimental two-dimensional quantum walk on a photonic chip. Sci. Adv. 4, eaat3174 (2018).

    ADS  Article  Google Scholar 

  27. Gao, J. et al. Non-classical photon correlation in a two-dimensional photonic lattice. Opt. Express 24, 12607–12616 (2016).

    ADS  Article  Google Scholar 

  28. Szameit, A., Dreisow, F., Pertsch, T., Nolte, S. & Tünnermann, A. Control of directional evanescent coupling in fs laser written waveguides. Opt. Express 15, 1579–1587 (2007).

    ADS  Article  Google Scholar 

  29. Izaac, J. A. & Wang, J. B. pyctqw: a continuous-time quantum walk simulator on distributed memory computers. Comput. Phys. Commun. 186, 81–92 (2015).

    ADS  Article  Google Scholar 

  30. Whitfield, J. D., Rodríguez-Rosario, C. A. & Aspuru-Guzik, A. Quantum stochastic walks: a generalization of classical random walks and quantum walks. Phys. Rev. A 81, 022323 (2010).

    ADS  Article  Google Scholar 

  31. Sánchez-Burillo, E., Duch, J., Gómez-Gardenes, J. & Zueco, D. Quantum navigation and ranking in complex networks. Sci. Rep. 2, 605 (2012).

    ADS  Article  Google Scholar 

  32. Crespi, A. et al. Integrated multimode interferometers with arbitrary designs for photonic boson sampling. Nat. Photon. 7, 545–549 (2013).

    ADS  Article  Google Scholar 

  33. Chaboyer, Z., Meany, T., Helt, L. G., Withford, M. J. & Steel, M. J. Tunable quantum interference in a 3D integrated circuit. Sci. Rep. 5, 9601 (2015).

    ADS  Article  Google Scholar 

  34. Feng, Z. et al. Invisibility cloak printed on a photonic chip. Sci. Rep. 6, 28527 (2016).

    ADS  Article  Google Scholar 

  35. Darázs, Z., Anishchenko, A., Kiss, T., Blumen, A. & Mülken, O. Transport properties of continuous-time quantum walks on sierpinski fractals. Phys. Rev. E 90, 032113 (2014).

    ADS  Article  Google Scholar 

  36. Kim, Y. H. Quantum interference with beamlike type-II spontaneous parametric down-conversion. Phys. Rev. A 68, 013804 (2003).

    ADS  Article  Google Scholar 

  37. Sun, K. et al. Mapping and measuring large-scale photonic correlation with single-photon imaging. Preprint at (2018).

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The authors thank J. D. Whitfield for a very useful conversation on numerical methods for the quantum stochastic walks, and J.-W. Pan for helpful discussions. This research is supported by the National Key R&D Program of China (2017YFA0303700), the National Natural Science Foundation of China (11690033, 61734005, 11761141014, 11374211), the Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405, 17JC1400403), Shanghai Municipal Education Commission (SMEC) (16SG09, 2017-01-07-00-02-E00049) and the open fund from the State Key Laboratory of High Performance Computing (HPCL) (no. 201511-01). C.D.F. is funded by the Singapore National Research Foundation (Fellowship NRF-NRFF2016-02). M.S.K. is supported by the Samsung Global Research Outreach (GRO) project, the Korea Institute of Science and Technology (KIST) Institutional Program (2E26680-18-P025), the Engineering and Physical Sciences Research Council (EPSRC) (EP/K034480/1) and the Royal Society. X.-M.J. acknowledges support from the National Young 1000 Talents Plan.

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Authors and Affiliations



X.-M.J. and M.S.K. conceived and supervised the project. H.T. and X.-M.J. designed the experiment. C.D.F., M.S.K. and T.-S.H. conducted the theoretical work. H.T., Z.-Y.S., J.G., K.S., Z.-Q.J. and X.-M.J. performed the single-photon experiment. H.T., Z.-M.L. and T.-Y.W. analysed the experimental data. Z.F. and Z.-Y.S. conducted chip fabrication. H.T., C.D.F., M.S.K. and X.-M.J. wrote the paper, with input from all the other authors.

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Correspondence to Xian-Min Jin.

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Supplementary Video 1

Quantum fast hitting dynamics.

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Tang, H., Di Franco, C., Shi, ZY. et al. Experimental quantum fast hitting on hexagonal graphs. Nature Photon 12, 754–758 (2018).

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