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arXiv:2609.05823

Topologically Protected Edge States in One-Dimensional Quantum Walks

Emily Maxey, Jacob Mansfield, Beth Thacker, Wade DeGottardi

quant-phSubmitted Sep 5, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a $\mathbb{Z}$- to $\mathbb{Z}_2$-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.

Comments
9 pages, 5 figures

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Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a Z\mathbb{Z}Z- to Z2\mathbb{Z}_2Z2​-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator in momentum space. In this work, we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and $\ensuremath{\pi}$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one-dimensional discrete-time quantum walks and is very well suited for the analysis of finite-size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum-walk experiment.
Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an ``inversion symmetric'' pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}\mathbb{Z}$ topological invariant that controls the number of topologically protected 0 and $\ensuremath{\pi}$ energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the ``4-step quantum walk,'' where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Quantum walks are discrete-time evolutions of single particles on lattices that transfer the concept of classical random walks to quantum theory. Topological insulators are objects that promise symmetry-protected edge-states for transportation along the edge. Together, both subjects lay the foundation to a description of new materials that may be used e.g. for quantum computation or transport. In this thesis, we study topological phases in quantum walks. Given a representation of the discrete symmetry groups of the tenfold way, we develop a topological classification which distinguishes three symmetry indices whose values lie in the group of integers, the group of two elements, or the trivial group, depending on the symmetry type under consideration. The classification is applicable to quantum walks that are gapped at the symmetry-protected points. All symmetry indices are proven to be stable under norm-continuous perturbations, but only two of the three indices are invariant under compact non-continuous perturbations. These two indices describe the asymptotic behaviour far to the left and far to the right, respectively. The third index reflects whether a compact perturbation can be performed in arbitrarily small steps without violating the symmetries. Given two walks in the same phase (i.e. the symmetry indices coincide), we show that there is a norm-continuous path of walks in that phase that connects them. This renders the set of invariants we introduced complete. Our theory covers translation invariant bulks as well, where we prove that the third index vanishes and the left- and right indices add up to zero. Hence these systems are fully classified by the right index alone. Joining two bulks in different phases (one left, one right), we show that eigenvalues emerge in the gap (bulk-boundary correspondence), whose eigenfunctions decay exponentially away from the boundary. Since our theory does not demand translation invariance at all, these composed systems are still described by our general classification. Restricting to the translation invariant case, we express the symmetry index as a winding number of a loop in momentum space. Within this restricted class, we prove that our classification is complete as well.
Discrete-time quantum walks (QWs) represent robust and versatile platforms for the controlled engineering of single particle quantum dynamics, and have attracted special attention due to their algorithmic applications in quantum information science. Even in their simplest 1D architectures, they display complex topological phenomena, which can be employed in the systematic study of topological quantum phase transitions [1]. Due to the exponential scaling in the number of resources required, most experimental realizations of QWs to date have been limited to single particles, with only a few implementations involving correlated quantum pairs. In this paper we study applications of QWs in the controlled dynamical engineering of entanglement in bipartite bosonic systems. We show that QWs can be employed in the transition from mode entanglement, where indistinguishability of the quantum particles plays a key role, to the standard type of entanglement associated with distinguishable particles. We also show that by carefully tailoring the steps in the QWs, as well as the initial state for the quantum walker, it is possible to preserve the entanglement content by topological protection. The underlying mechanism that allows for the possibility of both entanglement engineering and entanglement protection is the strong ‘spin–orbit’ coupling induced by the QW. We anticipate that the results reported here can be employed for the controlled emulation of quantum correlations in topological phases.
The quantum walk was originally proposed as a quantum-mechanical analog of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete-time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigations. In particular, we demonstrate that recent experimental realizations of quantum walks with cold atoms, photons, and ions simulate a nontrivial one-dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of the topological phases, which have been classified in one and two dimensions. We further discuss the existence of robust edge modes at phase boundaries, which provide experimental signatures for the nontrivial topological character of the system.