# Edge States, Bulk Spectra, and Topological Phases of Szegedy’s Quantum Search on a One-Dimensional Cycle with Self-Loops

## TL;DR

This paper studies quantum search on a 1D cycle with self-loops, revealing new topological phases and edge states that improve search efficiency.

## Contribution

The paper introduces novel gapless three-band structures and identifies topological phases ±2 using chiral symmetry.

## Key findings

- Gapless three-band structures with a flat band and three-fold degenerate points are observed.
- Topological phases ±2 are identified as the Chern number using chiral symmetry.
- Edge states enhance search efficiency on marked vertices, while nontrivial bulk spectra enable ballistic spread.

## Abstract

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 ±2 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.