Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops
Topological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy’s quantum search on a one-dimensional (1D) cycle with self-loops, where the search operator can be formulated as an open boundary condition. By est...
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MDPI AG
2025-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/27/6/623 |
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| author | Mengke Xu Xi Li Xunan Wang Wanglei Mi Xiao Chen |
| author_facet | Mengke Xu Xi Li Xunan Wang Wanglei Mi Xiao Chen |
| author_sort | Mengke Xu |
| collection | DOAJ |
| description | Topological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy’s 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’s quantum search. Interestingly, novel gapless three-band structures are observed, featuring a flat band and three-fold degenerate points. We identify the topological phases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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’s 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. |
| format | Article |
| id | doaj-art-1ca7a433c98c49e48e297db6cb7f7c73 |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-1ca7a433c98c49e48e297db6cb7f7c732025-08-20T03:27:07ZengMDPI AGEntropy1099-43002025-06-0127662310.3390/e27060623Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-LoopsMengke Xu0Xi Li1Xunan Wang2Wanglei Mi3Xiao Chen4School of Information Engineering, China Jiliang University, Hangzhou 310018, ChinaSchool of Software, Henan University, Zhengzhou 450046, ChinaSchool of Information Engineering, China Jiliang University, Hangzhou 310018, ChinaSchool of Information Engineering, China Jiliang University, Hangzhou 310018, ChinaSchool of Information Engineering, China Jiliang University, Hangzhou 310018, ChinaTopological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy’s 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’s quantum search. Interestingly, novel gapless three-band structures are observed, featuring a flat band and three-fold degenerate points. We identify the topological phases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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’s 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.https://www.mdpi.com/1099-4300/27/6/623topological phasesedge statesthree-fold degenerate pointsquantum searchChern number |
| spellingShingle | Mengke Xu Xi Li Xunan Wang Wanglei Mi Xiao Chen Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops Entropy topological phases edge states three-fold degenerate points quantum search Chern number |
| title | Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops |
| title_full | Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops |
| title_fullStr | Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops |
| title_full_unstemmed | Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops |
| title_short | Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops |
| title_sort | edge states bulk spectra and topological phases of szegedy s quantum search on a one dimensional cycle with self loops |
| topic | topological phases edge states three-fold degenerate points quantum search Chern number |
| url | https://www.mdpi.com/1099-4300/27/6/623 |
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