Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

Mohammadsadegh Khazali

Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck, Austria

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

This article proposes the first discrete-time implementation of Rydberg quantum walk in multi-dimensional spatial space that could ideally simulate different classes of topological insulators. Using distance-selective exchange-interaction between Rydberg excited atoms in an atomic-array with dual lattice-constants, the new setup operates both coined and coin-less models of discrete-time quantum walk (DTQW). Here, complicated coupling tessellations are performed by global laser that exclusively excite the site at the anti-blockade region. The long-range interaction provides a new feature of designing different topologically ordered periodic boundary conditions. Limiting the Rydberg population to two excitations, coherent QW over hundreds of lattice sites and steps are achievable with the current technology. These features would improve the performance of this quantum machine in running the quantum search algorithm over topologically ordered databases as well as diversifying the range of topological insulators that could be simulated.

Topological insulators are a new class of quantum materials that insulate in the bulk but exhibit robust topologically protected current-carrying edge states. These materials are challenging to synthesize, and limited in topological phases accessible with solid-state materials. This has motivated the search for topological phases on the systems that simulate the same principles underlying topological insulators. Discrete-time quantum-walk (DTQW) has been proposed for making Floquet topological insulators. The topological properties of these systems are controlled via an external periodic drive rather than an external magnetic field. Topological edge states have been realized exclusively in photonic DTQW with limited sites and steps.

Atomic lattices are the pioneering quantum platform in terms of scalability. An ideal tool to perform complicated quantum operations in an atomic system is provided by the laser-excited Rydberg atoms. This paper proposes the implementation of multiple time-dependent connectivity tessellations in a 3D lattice of dimers that run coined and coinless DTQW. The scheme operates by a simple Rydberg population rotation scheme via a global laser, shining on the lattice. The desired operation is performed via distance-selective spin-exchange interaction between Rydberg atoms in a holographically designed lattice structure. Switching among different coupling tessellations is controlled by tuning the laser’s energy to form a resonant interaction exclusively between the targeted sites. The long-range interaction allows engineering the periodic boundary conditions resulting in DTQW over topological surfaces like torus and Mobius stripe.

Studying a wide range of decoherence sources, the proposed scheme preserves the coherence over hundreds of lattice sites and steps. This major improvement is obtained by suppressing the population of the short-lived Rydberg states to two atoms. Finally, based on the mentioned advances, the paper address different classes of Rydberg Floquet topological insulators with an experimental breakthrough on the scale that these quantum materials could be realized.

► BibTeX data

► References

[1] Y. Aharonov, L. Davidovich, and N. Zagury. Quantum random walks. Phys. Rev. A 48, 1687 (1993).
https:/​/​doi.org/​10.1103/​PhysRevA.48.1687

[2] E. Farhi and S. Gutmann. Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998).
https:/​/​doi.org/​10.1103/​PhysRevA.58.915

[3] J. Kempe. Quantum random walks: an introductory overview. Contemp. Phys. 50, 339 (2009).
https:/​/​doi.org/​10.1080/​00107510902734722

[4] S. Dadras, A. Gresch, C.Groiseau, S. Wimberger, and G. S Summy. Quantum walk in momentum space with a Bose-Einstein condensate. Phys. Rev. Lett. 121, 070402 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.070402

[5] G. Summy and S. Wimberger. Quantum random walk of a Bose-Einstein condensate in momentum space. Phys. Rev. A 93, 023638 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.023638

[6] P. M Preiss, et al., Strongly correlated quantum walks in optical lattices. Science 347, 1229 (2015).
https:/​/​doi.org/​10.1126/​science.1260364

[7] R. Portugal. Quantum walks and search algorithms. Springer, (2013).
https:/​/​doi.org/​10.1007/​978-1-4614-6336-8

[8] N. Shenvi, J. Kempe, and K B. Whaley. Quantum random-walk search algorithm. Phys. Rev. A, 67, 052307 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.052307

[9] A. M Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A Spielman. Exponential algorithmic speedup by a quantum walk. Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 59 (2003).
https:/​/​doi.org/​10.1145/​780542.780552

[10] A. M Childs and J. Goldstone. Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.70.022314

[11] R. Portugal and T. D Fernandes. Quantum search on the two-dimensional lattice using the staggered model with hamiltonians. Phys. Rev. A 95, 042341 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.042341

[12] A. M Childs. Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.180501

[13] V. Kendon, How to compute using quantum walks. EPTCS 315, 1 (2020).
https:/​/​doi.org/​10.4204/​EPTCS.315.1

[14] N. B Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon. Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.042330

[15] A. M Childs, D. Gosset, and Z. Webb. Universal computation by multiparticle quantum walk. Science 339, 791 (2013).
https:/​/​doi.org/​10.1126/​science.1229957

[16] S. Elías V.-Andraca. Quantum walks: a comprehensive review. Quantum Inf. Process 11, 1015 (2012).
https:/​/​doi.org/​10.1007/​s11128-012-0432-5

[17] T. Kitagawa, M. S Rudner, E. Berg, and E. Demler. Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.82.033429

[18] H. Schmitz, et al., Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103, 090504 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.090504

[19] F. Zähringer, et al., Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.100503

[20] M. Karski, et al., Quantum walk in position space with single optically trapped atoms. Science 325, 174 (2009).
https:/​/​doi.org/​10.1126/​science.1174436

[21] C. Weitenberg, et al., Single-spin addressing in an atomic Mott insulator. Nature 471, 319 (2011).
https:/​/​doi.org/​10.1038/​nature09827

[22] T. Fukuhara, et al., Microscopic observation of magnon bound states and their dynamics. Nature 502, 76 (2013).
https:/​/​doi.org/​10.1038/​nature12541

[23] Ji. Wang and K. Manouchehri. Physical implementation of quantum walks. Springer (2013).
https:/​/​doi.org/​10.1007/​978-3-642-36014-5

[24] H. Bernien, et al., Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579 (2017).
https:/​/​doi.org/​10.1038/​nature24622

[25] A. Omran, et al., Generation and manipulation of Schrödinger cat states in Rydberg atom arrays. Science 365, 570 (2019).
https:/​/​doi.org/​10.1126/​science.aax9743

[26] S Zhang, F Robicheaux, and M Saffman. Magic-wavelength optical traps for Rydberg atoms. Phys. Rev. A 84, 043408 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.043408

[27] MJ Piotrowicz, et al., Two-dimensional lattice of blue-detuned atom traps using a projected gaussian beam array. Phys. Rev. A 88, 013420 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.013420

[28] F. Nogrette, et al., Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries. Phys. Rev. X 4, 021034 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.88.013420

[29] T Xia, et al., Randomized benchmarking of single-qubit gates in a 2d array of neutral-atom qubits. Phys. Rev. Lett. 114, 100503 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.100503

[30] J. Zeiher, et al., Many-body interferometry of a Rydberg-dressed spin lattice. Nature Physics 12, 1095 (2016).
https:/​/​doi.org/​10.1038/​nphys3835

[31] V. Lienhard, et al., Observing the space-and time-dependent growth of correlations in dynamically tuned synthetic ising models with antiferromagnetic interactions. Phys. Rev. X 8, 021070 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.021070

[32] MA Norcia, AW Young, and AM Kaufman. Microscopic control and detection of ultracold strontium in optical-tweezer arrays. Phys. Rev. X 8, 041054 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041054

[33] D. D. Yavuz, N. A. Proite, and J. T. Green, Nanometer-scale optical traps using atomic state localization, Phys. Rev. A 79, 055401 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.055401

[34] A. Cooper, et al., Alkaline-earth atoms in optical tweezers. Phys. Rev. X 8, 041055 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041055

[35] S. Hollerith, et al., Quantum gas microscopy of Rydberg macrodimers. Science 364, 664 (2019).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041055

[36] S. Saskin, JT Wilson, B. Grinkemeyer, and J. D. Thompson. Narrow-line cooling and imaging of ytterbium atoms in an optical tweezer array. Phys. Rev. Lett. 122, 143002 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.143002

[37] Y. Wang, A. Kumar, T.-Y. Wu, and D. S Weiss. Single-qubit gates based on targeted phase shifts in a 3D neutral atom array. Science 352, 1562 (2016).
https:/​/​doi.org/​10.1126/​science.aaf2581

[38] D. Barredo, V. Lienhard, S. De Leseleuc, T. Lahaye, and A. Browaeys. Synthetic three-dimensional atomic structures assembled atom by atom. Nature 561, 79 (2018).
https:/​/​doi.org/​10.1038/​s41586-018-0450-2

[39] H. Levine, et al., High-fidelity control and entanglement of Rydberg-atom qubits. Phys. Rev. Lett. 121, 123603 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.123603

[40] M. Saffman, T. G Walker, and K. Mølmer. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313 (2010).
https:/​/​doi.org/​10.1103/​RevModPhys.82.2313

[41] CS Adams, JD Pritchard, and JP Shaffer. Rydberg atom quantum technologies. J. Phys. B 53, 012002 (2019).
https:/​/​doi.org/​10.1088/​1361-6455/​ab52ef

[42] M. Khazali, K. Heshami, and C. Simon. Photon-photon gate via the interaction between two collective Rydberg excitations. Phys. Rev. A, 91, 030301 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.030301

[43] M. Khazali and K. Mølmer. Fast multiqubit gates by adiabatic evolution in interacting excited-state manifolds of Rydberg atoms and superconducting circuits. Phys. Rev. X, 10, 021054 (2020).
https:/​/​doi.org/​10.1103/​PhysRevX.10.021054

[44] M. Khazali, and W. Lechner. "Electron cloud design for Rydberg multi-qubit gates." arXiv:2111.01581 (2021).
arXiv:2111.01581

[45] M. Khazali. Rydberg noisy-dressing and applications in making soliton-molecules and droplet quasi-crystals. Phys. Rev. Research 3, L032033 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.L032033

[46] Khazali, Mohammadsadegh. Applications of Atomic Ensembles for Photonic Quantum Information Processing and Fundamental Tests of Quantum Physics. Diss. University of Calgary (Canada), (2016).
https:/​/​www.proquest.com/​openview/​b35882136d01bc6643040d8c66c410e6/​1?cbl=18750&pq-origsite=gscholar

[47] Khazali, M. Quantum Information and Computation with Rydberg Atoms. Iranian Journal of Applied Physics 10, 19 (2021).
https:/​/​doi.org/​10.22051/​IJAP.2021.34445.1188

[48] M. Khazali, C. R Murray, and T. Pohl. Polariton exchange interactions in multichannel optical networks. Phys. Rev. Lett., 123, 113605 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.113605

[49] M. Khazali, H. W. Lau, A. Humeniuk, and C. Simon. Large energy superpositions via Rydberg dressing. Phys. Rev. A, 94, 023408 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.023408

[50] M. Khazali, K. Heshami, and C. Simon. Single-photon source based on Rydberg exciton blockade. J. Phys. B: At. Mol. Opt. Phys. 50, 215301, (2017).
https:/​/​doi.org/​10.1088/​1361-6455/​aa8d7c

[51] M. Khazali. Progress towards macroscopic spin and mechanical superposition via Rydberg interaction. Phys. Rev. A 98, 043836 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.043836

[52] R. Côté, A. Russell, E. E Eyler, and P. L Gould. Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys. 8, 156 (2006).
https:/​/​doi.org/​10.1088/​1367-2630/​8/​8/​156

[53] de Leseleuc, Sylvain, et al. "Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms." Science 365, 775 (2019).
https:/​/​doi.org/​10.1126/​science.aav9105

[54] D. Barredo, et al., Coherent excitation transfer in a spin chain of three Rydberg atoms. Phys. Rev. Lett. 114, 113002 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.113002

[55] DW Schönleber, A. Eisfeld, M. Genkin, S Whitlock, and S. Wüster. Quantum simulation of energy transport with embedded Rydberg aggregates. Phys. Rev. Lett. 114, 123005 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.123005

[56] A Pineiro Orioli, et al., Relaxation of an isolated dipolar-interacting Rydberg quantum spin system. Phys. Rev. Lett. 120, 063601 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.063601

[57] G Günter, et al., Observing the dynamics of dipole-mediated energy transport by interaction-enhanced imaging. Science 342, 954 (2013).
https:/​/​doi.org/​10.1126/​science.1244843

[58] H Schempp, et al., Correlated exciton transport in Rydberg-dressed-atom spin chains. Phys. Rev. Lett. 115, 093002 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.093002

[59] F. Letscher and . Petrosyan. Mobile bound states of Rydberg excitations in a lattice. Phys. Rev. A 97, 043415 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.043415

[60] S Wüster, C Ates, A Eisfeld, and JM Rost. Excitation transport through Rydberg dressing. New J. Phys. 13, 073044 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​7/​073044

[61] A. Dauphin, M. Müller, and M. A. Martin-Delgado. Quantum simulation of a topological Mott insulator with Rydberg atoms in a Lieb lattice. Phys. Rev. A 93, 043611 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.043611

[62] Y. Ando. Topological insulator materials. J. Phys. Soc. Jpn 82, 102001 (2013).
https:/​/​doi.org/​10.7566/​JPSJ.82.102001

[63] J. Cayssol, B. Dóra, F. Simon, and R. Moessner. Floquet topological insulators. pss (RRL) 7, 101 (2013).
https:/​/​doi.org/​10.1002/​pssr.201206451

[64] A. Kitaev. Periodic table for topological insulators and superconductors. AIP Conf Proc 1134, 22 (2009).
https:/​/​doi.org/​10.1063/​1.3149495

[65] S Panahiyan and S Fritzsche. Toward simulation of topological phenomenas with one-, two-and three-dimensional quantum walks. Phys. Rev. A 103, 012201 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.012201

[66] Mikael C Rechtsman, et al., Photonic floquet topological insulators. Nature 496, 196 (2013).
https:/​/​doi.org/​10.1038/​nature12066

[67] L Xiao, et al., Observation of topological edge states in parity-time-symmetric quantum walks. Nature Phys. 13, 1117 (2017).
https:/​/​doi.org/​10.1038/​nphys4204

[68] S. Mukherjee, et al., Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nature Com. 8, 1 (2017).
https:/​/​doi.org/​10.1038/​ncomms13918

[69] A. Ambainis, R. Portugal, and N. Nahimov, Spatial search on grids with minimum memory, Quantum Information and Computation 15, 1233 (2015).
https:/​/​doi.org/​10.26421/​QIC15.13-14-9

[70] R. Portugal, S. Boettcher, and S. Falkner, One-dimensional coinless quantum walks, Phys. Rev. A 91, 052319 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052319

[71] R. Portugal, R. A. M. Santos, T. D. Fernandes and D. N. Goncalves, The staggered quantum walk model, Quantum Information Processing 15, 85 (2016).
https:/​/​doi.org/​10.1007/​s11128-015-1149-z

[72] R. Portugal, Staggered quantum walks on graphs, Phys. Rev. A 93, 062335 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.062335

[73] R. Portugal, M. C. de Oliveira, and J. K. Moqadam, Staggered quantum walks with Hamiltonians, Phys. Rev. A 95, 012328 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.012328

[74] W. P. Su, JR Schrieffer, and Ao J Heeger. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979).
https:/​/​doi.org/​10.1103/​PhysRevLett.42.1698

[75] T M Michelitsch et al., Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices, J. Phys. A: Math. Theor. 50, 505004 (2017).
https:/​/​doi.org/​10.1088/​1751-8121/​aa9008

[76] Moqadam, J. Khatibi, and Ali T. Rezakhani. Boundary-induced coherence in the staggered quantum walk on different topologies. Phys. Rev. A 98, 012123 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.012123

[77] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
http:/​/​10.1103/​RevModPhys.82.1959

[78] F. Grusdt and M. Honing, Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons, Phys. Rev. A 90, 053623 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.053623

[79] M. Lacki, et al., Quantum Hall physics with cold atoms in cylindrical optical lattices, Phys. Rev. A 93, 013604 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.013604

[80] M. A Schlosshauer. Decoherence: and the quantum-to-classical transition. Springer (2007).
https:/​/​doi.org/​10.1007/​978-3-540-35775-9

[81] Alberti, Andrea, et al. "Decoherence models for discrete-time quantum walks and their application to neutral atom experiments." New Journal of Physics 16, 123052 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​12/​123052

[82] R. B Hutson, Aet al., Engineering quantum states of matter for atomic clocks in shallow optical lattices. Phys. Rev. Lett. 123, 123401 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.123401

[83] A. M Kaufman, B. J Lester, and C. A Regal. Cooling a single atom in an optical tweezer to its quantum ground state. Phys. Rev. X 2, 041014 (2012).
https:/​/​doi.org/​10.1103/​PhysRevX.2.041014

[84] J. D. Thompson, et al., Coherence and raman sideband cooling of a single atom in an optical tweezer. Phys. Rev. Lett. 110, 133001 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.133001

[85] N. Belmechri, et al., Microwave control of atomic motional states in a spin-dependent optical lattice. J. Phys. B 46, 104006 (2013).
https:/​/​doi.org/​10.1088/​0953-4075/​46/​10/​104006

[86] Wang, Kunpeng, et al. "Preparation of a heteronuclear two-atom system in the three-dimensional ground state in an optical tweezer." Phys. Rev. A 100, 063429 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.063429

[87] D. Barredo, et al., Three-Dimensional Trapping of Individual Rydberg Atoms in Ponderomotive Bottle Beam Traps, Phys. Rev. Lett. 124, 023201 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.023201

[88] T. M. Graham, et al., Rydberg Mediated Entanglement in a Two-Dimensional Neutral Atom Qubit Array, Phys. Rev. Lett. 123, 230501 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.230501

[89] Wilson, J., et al., Trapped arrays of alkaline earth Rydberg atoms in optical tweezers. arXiv:1912.08754 (2019).
arXiv:1912.08754

[90] II Beterov, II Ryabtsev, DB Tretyakov, and VM Entin. Quasiclassical calculations of blackbody-radiation-induced depopulation rates and effective lifetimes of Rydberg ns, np, and nd alkali-metal atoms with n<80. Phys. Rev. A 79, 052504 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.052504

[91] Signoles, A., et al., Coherent transfer between low-angular-momentum and circular Rydberg states, Phys. Rev. Lett. 118, 253603 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.253603

[92] R. Cardman and G. Raithel, Circularizing Rydberg atoms with time-dependent optical traps, Phys. Rev. A 101, 013434 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.013434

[93] T. Long Nguyen, et al. Towards quantum simulation with circular Rydberg atoms. Phys. Rev. X 8, 011032, (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011032

[94] M. Kwon, M. F Ebert, T. G Walker, and M Saffman. Parallel low-loss measurement of multiple atomic qubits. Phys. Rev. Lett. 119, 180504 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180504

[95] J. P Covey, I. S Madjarov, A. Cooper, and M. Endres. 2000-times repeated imaging of strontium atoms in clock-magic tweezer arrays. Phys. Rev. Lett. 122, 173201 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.173201

[96] B J Wieder and CL Kane. Spin-orbit semimetals in the layer groups. Phys. Rev. B 94, 155108 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.94.155108

[97] M. Sajid, et al., Creating Floquet Chern insulators with magnetic quantum walks. Phys. Rev. B 99 214303 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.214303

[98] S. Neil, J. Kempe, and K. Whaley. "Quantum random-walk search algorithm." Phys. Rev. A 67, 052307 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.052307

[99] Luc Segoufin, Victor Vianu, Querying Spatial Databases via Topological Invariants, Journal of Computer and System Sciences 61, 270 (2000).
https:/​/​doi.org/​10.1006/​jcss.2000.1712.

[100] Clementini, E. "Topological relations in spatial databases", Intelligent Systems: Technology and Applications 4, 47 (2002).
https:/​/​doi.org/​10.1201/​9781420040814.Ch3d

[101] M. S Rudner, N. H Lindner, E. Berg, and M. Levin. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).
https:/​/​doi.org/​10.1103/​PhysRevX.3.031005

Cited by

[1] Ceren B. Dag and Aditi Mitra, "Floquet topological systems with flat bands: Edge modes, Berry curvature, and orbital magnetization", Physical Review B 105 24, 245136 (2022).

[2] Ying Lei, Xi-Wang Luo, and Shaoliang Zhang, "Second-order topological insulator in periodically driven optical lattices", Optics Express 30 13, 24048 (2022).

[3] Aaron W. Young, William J. Eckner, Nathan Schine, Andrew M. Childs, and Adam M. Kaufman, "Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice", Science 377 6608, 885 (2022).

[4] Kai Li, Jiong-Hao Wang, Yan-Bin Yang, and Yong Xu, "Symmetry-Protected Topological Phases in a Rydberg Glass", Physical Review Letters 127 26, 263004 (2021).

[5] Mohammadsadegh Khazali, "Rydberg noisy dressing and applications in making soliton molecules and droplet quasicrystals", Physical Review Research 3 3, L032033 (2021).

[6] Haihu Cui and Wenxi Lai, "Free-space quantum walk of neutral atoms", arXiv:2204.02864.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 09:26:02) and SAO/NASA ADS (last updated successfully 2022-10-04 09:26:03). The list may be incomplete as not all publishers provide suitable and complete citation data.