{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:23:53Z","timestamp":1760059433711,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T00:00:00Z","timestamp":1749686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Zhejiang Province, China","award":["Q24A050004"],"award-info":[{"award-number":["Q24A050004"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Topological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy\u2019s quantum search on a one-dimensional (1D) cycle with self-loops, where the search operator can be formulated as an open boundary condition. By establishing an equivalence with coined quantum walks (QWs), we analytically derive and numerically illustrate the quasienergies dispersion relations of bulk spectra and edge states for Szegedy\u2019s quantum search. Interestingly, novel gapless three-band structures are observed, featuring a flat band and three-fold degenerate points. We identify the topological phases \u00b12 as the Chern number. This invariant is computed by leveraging chiral symmetry in zero diagonal Hermitian Hamiltonians that satisfy our quasienergies constraints. Furthermore, we demonstrate that the edge states enhance searches on the marked vertices, while the nontrivial bulk spectra facilitate ballistic spread for Szegedy\u2019s quantum search. Crucially, we find that gapless topological phases arise from three-fold degenerate points and are protected by chiral symmetry, distinguishing ill-defined topological transition boundaries.<\/jats:p>","DOI":"10.3390\/e27060623","type":"journal-article","created":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T03:59:17Z","timestamp":1749700757000},"page":"623","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Edge States, Bulk Spectra, and Topological Phases of Szegedy\u2019s Quantum Search on a One-Dimensional Cycle with Self-Loops"],"prefix":"10.3390","volume":"27","author":[{"given":"Mengke","family":"Xu","sequence":"first","affiliation":[{"name":"School of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xi","family":"Li","sequence":"additional","affiliation":[{"name":"School of Software, Henan University, Zhengzhou 450046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xunan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wanglei","family":"Mi","sequence":"additional","affiliation":[{"name":"School of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,12]]},"reference":[{"key":"ref_1","unstructured":"Ambainis, A., Kempe, J., and Rivosh, A. (2005, January 23\u201325). Coins make quantum walks faster. Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Vancouver, BC, Canada."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2014.08.033","article-title":"Quantum artificial neural networks with applications","volume":"290","author":"Cao","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"052323","DOI":"10.1103\/PhysRevA.69.052323","article-title":"Localization of two-dimensional quantum walks","volume":"69","author":"Inui","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"033429","DOI":"10.1103\/PhysRevA.82.033429","article-title":"Exploring topological phases with quantum walks","volume":"82","author":"Kitagawa","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"126828","DOI":"10.1016\/j.physleta.2020.126828","article-title":"Controllable simulation of topological phases and edge states with quantum walk","volume":"384","author":"Panahiyan","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"052307","DOI":"10.1103\/PhysRevA.67.052307","article-title":"Quantum random-walk search algorithm","volume":"67","author":"Shenvi","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"069501","DOI":"10.1088\/1751-8121\/aa8f54","article-title":"Grover search with lackadaisical quantum walks","volume":"51","author":"Wong","year":"2018","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","unstructured":"Szegedy, M. (2004, January 17\u201319). Quantum speed-up of Markov chain based algorithms. Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy."},{"key":"ref_9","first-page":"1281","article-title":"Localization and discrete probability function of Szegedy\u2019s quantum search on the one-dimensional cycle with self-loops","volume":"20","author":"Xu","year":"2020","journal-title":"Quantum Inf. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1109\/LWC.2023.3281423","article-title":"RIS-Based Opposite Phase Index Modulation and Its DNN-Assisted Joint Sequence Detection","volume":"12","author":"Jin","year":"2023","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"195414","DOI":"10.1103\/PhysRevB.86.195414","article-title":"Symmetries, topological phases, and bound states in the one-dimensional quantum walk","volume":"86","author":"Asboth","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"195139","DOI":"10.1103\/PhysRevB.84.195139","article-title":"Topological phases and delocalization of quantum walks in random environments","volume":"84","author":"Obuse","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"045424","DOI":"10.1103\/PhysRevB.92.045424","article-title":"Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk","volume":"92","author":"Obuse","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_14","first-page":"1373","article-title":"Limit theorems of a two-phase quantum walk with one defect","volume":"15","author":"Edo","year":"2015","journal-title":"Quantum Inf. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"065703","DOI":"10.1103\/PhysRevLett.102.065703","article-title":"Topological Transition in a Non-Hermitian Quantum Walk","volume":"102","author":"Rudner","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhou, L. (2024). Entanglement Phase Transitions in Non-Hermitian Kitaev Chains. Entropy, 26.","DOI":"10.3390\/e26030272"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"235114","DOI":"10.1103\/PhysRevB.82.235114","article-title":"Topological characterization of periodically driven quantum systems","volume":"82","author":"Kitagawa","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"155106","DOI":"10.1103\/PhysRevB.85.155106","article-title":"Berry phase and anomalous Hall effect in a three-orbital tight-binding Hamiltonian","volume":"85","author":"He","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"052311","DOI":"10.1103\/PhysRevA.92.052311","article-title":"Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk","volume":"92","author":"Rakovszky","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104202","DOI":"10.1103\/PhysRevB.91.104202","article-title":"Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks","volume":"91","author":"Edge","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"022324","DOI":"10.1103\/PhysRevA.91.022324","article-title":"Edge-state-enhanced transport in a two-dimensional quantum walk","volume":"91","author":"Asboth","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s11128-017-1667-y","article-title":"Equivalence of Szegedy\u2019s and coined quantum walks","volume":"16","author":"Wong","year":"2017","journal-title":"Quantum Inf. Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"033634","DOI":"10.1103\/PhysRevA.96.033634","article-title":"Emergent pseudospin-1 Maxwell fermions with a threefold degeneracy in optical lattices","volume":"96","author":"Zhu","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"130503","DOI":"10.1103\/PhysRevLett.120.130503","article-title":"Topological Maxwell Metal Bands in a Superconducting Qutrit","volume":"120","author":"Tan","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"045017","DOI":"10.1088\/2058-9565\/abb028","article-title":"Enhanced repulsively bound atom pairs in topological optical lattice ladders","volume":"5","author":"Stuart","year":"2020","journal-title":"Quantum Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"056112","DOI":"10.1103\/PhysRevE.72.056112","article-title":"One-dimensional three-state quantum walk","volume":"72","author":"Inui","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"455302","DOI":"10.1088\/1751-8121\/aa8c5e","article-title":"Sensitivity of quantum walks to a boundary of two-dimensional lattices: Approaches based on the CGMV method and topological phases","volume":"50","author":"Endo","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s11128-021-02994-5","article-title":"Probability and Entanglement Evolutions for Szegedy\u2019s Quantum Search on the One-dimensional Cycle with Self-loops","volume":"20","author":"Xu","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"121406","DOI":"10.1103\/PhysRevB.88.121406","article-title":"Bulk-boundary correspondence for chiral symmetric quantum walks","volume":"88","author":"Asboth","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"125143","DOI":"10.1103\/PhysRevB.90.125143","article-title":"Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems","volume":"90","author":"Asboth","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"161104","DOI":"10.1103\/PhysRevB.96.161104","article-title":"Chiral flat bands: Existence, engineering, and stability","volume":"96","author":"Ramachandran","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"505302","DOI":"10.1088\/1751-8121\/aa9235","article-title":"One-dimensional lackadaisical quantum walks","volume":"50","author":"Wang","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_33","first-page":"052115","article-title":"Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems","volume":"97","author":"Yin","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"146402","DOI":"10.1103\/PhysRevLett.120.146402","article-title":"Topological Band Theory for Non-Hermitian Hamiltonians","volume":"120","author":"Shen","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"133903","DOI":"10.1103\/PhysRevLett.116.133903","article-title":"Anomalous Edge State in a Non-Hermitian Lattice","volume":"116","author":"Lee","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"040401","DOI":"10.1103\/PhysRevLett.118.040401","article-title":"Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems","volume":"118","author":"Leykam","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"086803","DOI":"10.1103\/PhysRevLett.121.086803","article-title":"Edge States and Topological Invariants of Non-Hermitian Systems","volume":"121","author":"Yao","year":"2018","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/623\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:50:34Z","timestamp":1760032234000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/623"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,12]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["e27060623"],"URL":"https:\/\/doi.org\/10.3390\/e27060623","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,6,12]]}}}