Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1505.03264 · Physical Review B

Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

Hideaki Obuse, János K. Asbóth, Yuki Nishimura, Norio Kawakami

Published Jul 23, 2015• 79 citations
arXivHTMLPDF

Abstract

Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator, is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment; we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.

DOI
10.1103/physrevb.92.045424

Related papers

Symmetries, topological phases, and bound states in the one-dimensional quantum walk

János K. Asbóth

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.

arXiv:1208.2143•Physical Review B•Nov 9, 2012•207 citations

Scattering theory of topological phases in discrete-time quantum walks

Brian Tarasinski, János K. Asbóth, Jan P. Dahlhaus

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.

arXiv:1401.2673•Physical Review A•Apr 25, 2014•69 citations

Measuring topological invariants in disordered discrete-time quantum walks

Barkhofen, S., Nitsche, T., Elster, F. +2 more

Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.

•Jan 1, 2017•73 citations

Measuring topological invariants in disordered discrete-time quantum walks

Sonja Barkhofen, Thomas Nitsche, Fabian Elster +4 more

Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.

arXiv:1606.00299•Physical Review A•Sep 27, 2017•79 citations

Exploring topological phases in quantum walks of twisted light

Filippo Cardano

Emerged as the quantum counterpart of classical random walks, quantum walks are established precious resources in a variety of quantum sciences. Recent studies have shown that quantum walks may be characterized by topological invariants, in close analogy to condensed matter systems exhibiting topological order. Exploiting these features, quantum walks are currently used to simulate topological systems and to probe their exotic features. Here we present the implementation of a one-dimensional quantum walk protocol based on the orbital angular momentum of light, manifesting the topological phases that characterize time-periodic systems (Floquet topological insulators) showing chiral symmetry. By considering the orbital angular momentum spectrum of a light beam undergoing this quantum evolution, we show that the associated statistical moments have marked differences in distinct phases and contain information on the system topology. While varying a control parameter determining the value of the invariants, these moments in the large step-number limit exhibit a sharp variation at the phase changes. We show that these phenomena arise from the singular behavior of the dispersion relation at the transition points. The extension of our results to systems featuring different symmetries, or characterized by higher spatial dimensions, may unveil novel intriguing features associated with these complex systems.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 27, 2017•0 citations

Topological quantum walk with discrete time-glide symmetry

Ken Mochizuki, Takumi Bessho, Masatoshi Sato +1 more

Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.

arXiv:2004.09332•Physical review. B./Physical review. B•Jul 13, 2020•16 citations
Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator, is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment; we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.
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.
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.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Emerged as the quantum counterpart of classical random walks, quantum walks are established precious resources in a variety of quantum sciences. Recent studies have shown that quantum walks may be characterized by topological invariants, in close analogy to condensed matter systems exhibiting topological order. Exploiting these features, quantum walks are currently used to simulate topological systems and to probe their exotic features. Here we present the implementation of a one-dimensional quantum walk protocol based on the orbital angular momentum of light, manifesting the topological phases that characterize time-periodic systems (Floquet topological insulators) showing chiral symmetry. By considering the orbital angular momentum spectrum of a light beam undergoing this quantum evolution, we show that the associated statistical moments have marked differences in distinct phases and contain information on the system topology. While varying a control parameter determining the value of the invariants, these moments in the large step-number limit exhibit a sharp variation at the phase changes. We show that these phenomena arise from the singular behavior of the dispersion relation at the transition points. The extension of our results to systems featuring different symmetries, or characterized by higher spatial dimensions, may unveil novel intriguing features associated with these complex systems.
Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.