We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We are mainly focused on the completion of a table for the protocols of the quantum walk that could simulate different family of the topological phases in one, two dimensions and take the first initiatives to build necessary protocols for three-dimensional cases. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences or absences. To further enrich the simulation of the topological phenomenas, we include step-dependent coins in the evolution operators of the quantum walks. Consequently, this leads to step-dependency of the simulated topological phenomenas and their properties which in turn introduce dynamicality as a feature to simulated topological phases and boundary states. This dynamicality provides the step-number of the quantum walk as a mean to control and engineer the number of topological phases and boundary states, their populations, types and even occurrences.
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 t…
We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary o…
The quantum walk was originally proposed as a quantum mechanical analogue 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 investigation. In particular, we demonstrate that recent experimental realizations of quantum walks simulate a non-trivial one dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of…
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed 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 gapped quantum walks, topological invariants at quasienergies 0 and π probe directly the existence of protected boundary states, while quantum walks with a non-trivial quasienergy winding have a discrete number of perfectly transmistting unidirectional modes. Our classificatio…
We study the quantum walk of two interacting particles on a line with an interface separating two topologically distinct regions. The interaction induces a localization-delocalization transition of the edge state at the interface. We characterize the transition through the entanglement between the two particles.
We simulate various topological phenomena in condense matter, such as formation of different topological phases, boundary and edge states, through two types of quantum walk with step-dependent coins. Particularly, we show that one-dimensional quantum walk with step-dependent coin simulates all types of topological phases in BDI family, as well as all types of boundary and edge states. In addition, we show that step-dependent coins provide the number of steps as a controlling factor over the simulations. In fact, with tuning number of steps, we can determine the occurrences of boundary, edge st…
The quantum-mechanical counterpart of a classical random walk offers a rich dynamics that has recently been shown to include topologically protected bound states (zero-modes) at boundaries or domain walls. Here we show that a topological zero-mode may acquire a dynamical role in the presence of nonlinearities. We consider a one-dimensional discrete-time quantum walk that combines zero-modes with a particle-conserving nonlinear relaxation mechanism. The presence of both particle-hole and chiral symmetry converts two zero-modes of opposite chirality into an attractor-repeller pair of the nonline…
Quantum walks are promising for information processing tasks because on regular graphs they spread quadratically faster than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, whereby the spread of the walker stays within a finite region even in the infinite time limit. It is therefore important to understand when we can expect a quantum walk to be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a generic one-dimensional qua…
The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here we motivate a toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Sc…
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 Z2 \times Z2 topological invariant in a simple way that is consistent with a general definition for 1-dime…
Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-s…
Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases …
Inspired by recent breakthroughs with topological quantum materials, which pave the way to novel, high-efficiency, low-energy magnetoelectric devices and fault-tolerant quantum information processing, inter alia, topological quantum walks have emerged as an exciting topic in its own right, especially due to the theoretical and experimental simplifications this approach offers. Motivated by impressive progress in topological quantum walks, we provide a perspective on theoretical studies and experimental investigations of topological quantum walks focusing on current explorations of topological …
We extend non-Hermitian topological quantum walks on a Su-Schrieffer-Heeger (SSH) lattice [M. S. Rudner and L. Levitov, Phys. Rev. Lett. 102, 065703 (2009)] to the case of non-Markovian evolution. This non-Markovian model is established by coupling each unit cell in the SSH lattice to a reservoir formed by a quasi-continuum of levels. We find a topological transition in this model even in the case of non-Markovian evolution, where the walker may visit the reservoir and return to the SSH lattice at a later time. The existence of a topological transition does, however, depend on the low-frequenc…
Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathc…
The split step quantum walk for two noninteracting particles is numerically simulated. The entropy of entanglement and spatial particle distributions are calculated for a range of initial states and for a range of disorder. The impact of varying the topological phase on these quantities is discussed.
Higher-order topological insulators not only exhibit exotic bulk-boundary correspondence principle, but also have an important application in quantum computing. However, they have never been achieved in quantum walk. In this paper, we construct a two-dimensional coinless discrete-time quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after multi-step discrete-time quantum walks. Furthermore, we demonstrate the robustness of the topolo…
We derive the continuous-time limit of discrete quantum walks with topological phases. We show the existence of a continuous-time limit that preserves their topological phases. We consider both simple-step and split-step walks, and derive analytically equations of motion governing their behavior. We obtain simple analytical solutions showing the existence of bound states at the boundary of two phases, and solve the equations of motion numerically in the bulk.
The symmetries associated with discrete-time quantum walks (DTQWs) and the flexibilities in controlling their dynamical parameters allow to create a large number of topological phases. An interface in position space, which separates two regions with different topological numbers, can, for example, be effectively modelled using different coin parameters for the walk on either side of the interface. Depending on the neighbouring numbers, this can lead to localized states in one-dimensional configurations and here we carry out a detailed study into the strength of such localized states. We show t…
Quantum walks are powerful tools for quantum applications and for designing topological systems. Although they are simulated in a variety of platforms, genuine two-dimensional realizations are still challenging. Here we present an innovative approach to the photonic simulation of a quantum walk in two dimensions, where walker positions are encoded in the transverse wavevector components of a single light beam. The desired dynamics is obtained by means of a sequence of liquid-crystal devices, which apply polarization-dependent transverse "kicks" to the photons in the beam. We engineer our quant…
We present a protocol to implement discrete-time quantum walks and simulate topological insulator phases in cavity-based quantum networks, where the single photon is the quantum walker and the cavity input-output process is employed to realize the state-dependent translation operation. Different topological phases can be simulated through tuning the single-photon polarization rotation angles. We show that both the topological boundary states and topological phase transitions can be directly observed via measuring the final photonic density distribution. Moreover, we also demonstrate that these…
The topology of a single-particle band structure plays a fundamental role in understanding a multitude of physical phenomena. Motivated by the connection between quantum walks and such topological band structures, we demonstrate that a simple time-dependent, Bloch-oscillating quantum walk enables the direct measurement of topological invariants. We consider two classes of one-dimensional quantum walks and connect the global phase imprinted on the walker with its refocusing behavior. By disentangling the dynamical and geometric contributions to this phase we describe a general strategy to measu…
We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a non-unitary quantum walk. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that the quantum walk can indefinitely sustain hybrid entanglement in the unbroken symmetry phase even when gain and loss mechanisms are present. However, when the gain-loss strength is too large, the PT-symmetry of the model is spontaneously broken and entanglement vanishes.…
Bulk-boundary correspondence is a fundamental principle for topological phases where bulk topology determines gapless boundary states. On the other hand, it has been known that corner or hinge modes in higher order topological insulators may appear due to "extrinsic" topology of the boundaries even when the bulk topological numbers are trivial. In this paper, we find that Floquet anomalous boundary states in quantum walks have similar extrinsic topological natures. In contrast to higher order topological insulators, the extrinsic topology in quantum walks is manifest even for first-order topol…
The appearance of topological effects in systems exhibiting a non-trivial topological band structure strongly relies on the coherent wave nature of the equations of motion. Here, we reveal topological dynamics in a classical stochastic random walk version of the Su-Schrieffer-Heeger model with no relation to coherent wave dynamics. We explain that the commonly used topological invariant in the momentum space translates into an invariant in a counting-field space. This invariant gives rise to clear signatures of the topological phase in an associated escape time distribution.
We construct a two-dimensional, discrete-time quantum walk exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states are consistent with Floquet winding numbers calculated using a non-Bloch band theory invoking time-dependent generalized Billouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined through local biorthogonal eigen wave functions of the no…
Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this pap…
I propose to study the complex physics of topological phases by an all optical implementation of a discrete-time quantum walk. The main novel ingredient is the use of parametric amplifiers in the random network which can in turn be used to emulate intra-atomic interactions and thus analyze many-body effects in topological phases even when using light as the quantum walker. I plan to characterize the intensity probability distribution and the spatial correlations of the output interference pattern for different input states, as well the robustness of localized boundary states characterizing top…
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 topolo…
Discrete-time quantum walks allow Floquet topological insulator materials to be explored using controllable systems such as ultracold atoms in optical lattices. By numerical simulations, we study the robustness of topologically protected edge states in the presence of decoherence in one- and two-dimensional discrete-time quantum walks. We also develop a simple analytical model quantifying the robustness of these edge states against either spin or spatial dephasing, predicting an exponential decay of the population of topologically protected edge states. Moreover, we present an experimental pro…
We discuss an efficient physical realization of topological quantum walks on a finite lattice. The $N$-point lattice is realized with $\log_2 N$ qubits, and the quantum circuit utilizes a number of quantum gates which is polynomial in the number of qubits. In a certain scaling limit, we show that a large number of steps is implemented with a number of quantum gates which is independent of the number of steps. We ran the quantum algorithm on the IBM-Q five-qubit quantum computer, thus experimentally demonstrating topological features, such as boundary bound states, on a lattice with $N=4$ point…
The topological phases in one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasi-energy $E=0,π$. These modes are qualitatively different for different boundary conditions used. For two-boundary system with symmetric boundary conditions, the interaction energy between two Majorana bound states can be computed, as in the case of a finite wire. Suggestion of observing these m…
We extend the non-Hermitian one-dimensional quantum walk model [Phys. Rev. Lett. 102, 065703 (2009)] by taking the dephasing effect into account. We prove that the feature of topological transition does not change even when dephasing between the sites within units is present. The potential experimental observation of our theoretical results in the circuit QED system consisting of superconducting qubit coupled to a superconducting resonator mode is discussed and numerically simulated. The results clearly show a topological transition in quantum walk and display the robustness of such a system t…
We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. We approximate the symmetry protected edge states with high similarities and read out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absen…
Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed matter physics. Here, we utilize a step dependent coin in quantum walks and investigate what topological phases we can simulate with it, their topological invariants, bound states and possibility of phase transitions. These quantum walks simulate non-trivial phases characterized by topological invariants (winding number) $\pm 1$ which are similar to the ones observed in topological insulators and polyacetylene. We confirm that the number of phases and their corresponding bound states inc…
Transport phenomena play a crucial role in modern physics and applied sciences. Examples include the dissipation of energy across a large system, the distribution of quantum information in optical networks, and the timely modeling of spreading diseases. In this work, we experimentally prove the feasibility of disordered quantum walks to realize a quantum simulator that is able to model general subdiffusive phenomena, exhibiting a sublinear spreading in space over time. Our experiment simulates such phenomena by means of a finely controlled insertion of various levels of disorder during the evo…
Non-Hermitian topological edge states have many intriguing properties, but have so far mainly been discussed in terms of bulk-boundary correspondence. Here we propose to use a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient is found to show unique features with the topological phase transitions driven by paritytime( PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite artificial boundaries are not constructed by inhomogeneous quantum walk. For a Hermitian system, a turnin…
We consider one-dimensional quantum walks in optical linear networks with synthetically introduced disorder and tunable system parameters allowing for the engineered realization of distinct topological phases. The option to directly monitor the walker's probability distribution makes this optical platform ideally suited for the experimental observation of the unique signatures of the one-dimensional topological Anderson transition. We analytically calculate the probability distribution describing the quantum critical walk in terms of a (time staggered) spin polarization signal and propose a co…
We investigate one-dimensional (1D) discrete time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at non-zero energy in their dynamics. Contrary to the previous studies, therefore, the spatially disordered QWs can avoid complete l…
The quantum walk is a dynamical protocol which describes the motion of spinful particles on a lattice. Also, it has been demonstrated to be a powerful platform to explore topological quantum matter. Recently, the quantum walk in coherent state space has been proposed theoretically and realized experimentally. However, due to the inherent characteristics of coherent states, it is challenging to control the number of photons when we need the coherent space to be a nearly orthogonal space in practice. Here, we demonstrate that the nonorthogonality of coherent sates, on the one hand can be cancell…
We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the…
The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity suscep…
We perform the fidelity analysis for Boltzmann-Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e., the topological behaviour is washed out gradually. We also show that …
Phases of matter with non-trivial topological order are predicted to exhibit a variety of exotic phenomena, such as the existence robust localized bound states in 1D systems, and edge states in 2D systems, which are expected to display spin-helicity, immunity to back-scattering, and weak anti-localization. In this Letter, we present an experimental observation of topological structures generated via the controlled implementation of two consecutive non-commuting rotations in photonic discrete-time quantum walks. The second rotation introduces valley-like Dirac points in the system, allowing to …
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like arc-reversal walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that arc-reversal walks and verte…
We review methods for using time-periodic fields (e.g., laser or microwave fields) to induce non-equilibrium topological phenomena in quantum many-body systems. We discuss how such fields can be used to change the topological properties of the single particle spectrum, and key experimental demonstrations in solid state, cold atomic, and photonic systems. The single particle Floquet band structure provides a stage on which the system's dynamics play out; the crucial question is then how to obtain robust topological behaviour in the many-particle setting. In the regime of mesoscopic transport, w…
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 up to date have been limited to single particles, with only a few i…
Topologically ordered materials may serve as a platform for new quantum technologies such as fault-tolerant quantum computers. To fulfil this promise, efficient and general methods are needed to discover and classify new topological phases of matter. We demonstrate that deep neural networks augmented with external memory can use the density profiles formed in quantum walks to efficiently identify properties of a topological phase as well as phase transitions. On a trial topological ordered model, our method's accuracy of topological phase identification reaches 97.4%, and is shown to be robust…
Topological matter exhibits exotic properties yet phases characterized by large topological invariants are difficult to implement, despite rapid experimental progress. A promising route toward higher topological invariants is via engineered Floquet systems, particularly in photonics, where flexible control holds the potential of extending the study of conventional topological matter to novel regimes. Here we implement a one-dimensional photonic quantum walk to explore large winding numbers. By introducing partial measurements and hence loss into the system, we detect winding numbers of three a…
Signaled by non-analyticities in the time evolution of physical observables, dynamic quantum phase transitions (DQPTs) emerge in quench dynamics of topological systems and possess an interesting geometric origin captured by dynamic topological order parameters (DTOPs). In this work, we report the experimental study of DQPTs using discrete-time quantum walks of single photons. We simulate quench dynamics between distinct Floquet topological phases using quantum-walk dynamics, and experimentally characterize DQPTs and the underlying DTOPs through interference-based measurements. The versatile ph…
