Abstract We study the quantum walk on the off-diagonal Aubry-André-Harper (AAH) lattice with periodic modulation using a digital quantum computer. We investigate various initial states at the single-particle level, considering different hopping modulation strengths and phase factors. Initiating the quantum walk with a particle at the lattice edge reveals the robustness of the edge state, attributed to the topological nature of the AAH model, and displays the influence of the phase factor on this edge state. On the other hand, when the quantum walk begins with a particle in the lattice bulk, we observe a repulsion of the bulk walker from the edge, especially under strong hopping modulation. Furthermore, we extend our investigation to the quantum walk of two particles with nearest-neighbour (NN) interaction. We show the repulsion effect in the quantum walk when two walkers originate from the edge and bulk of the lattice due to the interaction. Additionally, when two particles are positioned at nearest-neighbor sites with strong hopping modulation, they unexpectedly form a local bound state at very small interaction strength, highlighting the unique interplay between hopping modulation and interaction in our quantum walk setup. We analyze these phenomena by examining physical quantities such as density evolution, two-particle correlation, and participation entropy, and discuss their potential applications in quantum technologies.
Abstract We study the quantum walk on the off-diagonal Aubry-André-Harper (AAH) lattice with periodic modulation using a digital quantum computer. We investigate various initial states at the single-particle level, considering different hopping modulation strengths and phase factors. Initiating the quantum walk with a particle at the lattice edge reveals the robustness of the edge state, attributed to the topological nature of the AAH model, and displays the influence of the phase factor on this edge state. On the other hand, when the quantum walk begins with a particle in the lattice bulk, we observe a repulsion of the bulk walker from the edge, especially under strong hopping modulation. Furthermore, we extend our investigation to the quantum walk of two particles with nearest-neighbour (NN) interaction. We show the repulsion effect in the quantum walk when two walkers originate from the edge and bulk of the lattice due to the interaction. Additionally, when two particles are positioned at nearest-neighbor sites with strong hopping modulation, they unexpectedly form a local bound state at very small interaction strength, highlighting the unique interplay between hopping modulation and interaction in our quantum walk setup. We analyze these phenomena by examining physical quantities such as density evolution, two-particle correlation, and participation entropy, and discuss their potential applications in quantum technologies.
Due to the topological nature of the Aubry-Andr\'e-Harper (AAH) model, interesting edge states have been found existing in one-dimensional periodic and quasiperiodic lattices. In this article, we investigate continuous-time quantum walks of identical particles initially located on either edge of commensurate AAH lattices in detail. It is shown that the quantum walker is delocalized among the whole lattice until the strength of periodic modulation is strong enough. The inverse participation ratios (IPRs) for all of the eigenstates are calculated. It is found that the localization properties of the quantum walker is mainly determined by the IPRs of the topologically protected edge states. More interestingly, the edge states are shown to have an evident ``repulsion'' effect on quantum walkers initiated from the lattice sites inside the bulk. Furthermore, we examine the role of nearest-neighbor interaction on the quantum walks of two identical fermions. Clear enhancement of the repulsion effect by strong interaction has been shown.
We investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.
The field of quantum computation and quantum simulation has been recently driven to a new rising edge by the experimental realization of quantum walks in various setups, highlighting that different physical systems can be adapted for the implementation of these models. In particular, optical systems have shown their full potential, allowing the experimental demonstration of two-dimensional quantum walks for the first time [1, 2], even if several further progresses will sur ely be obtained also in the other physical scenarios that have been already exploited for the one-dimensional case [3, 4]. From the theoretical point of view, the interest in twodimensional quantum walks has been boosted by the fact that, differently from the one-dimensional version, highe rdimensional schemes (i.e., walkers moving on structures with dimension larger than one) can be exploited for the efficient implementation of quantum search algorithms [5]. In particular, the Grover walk has been intensively studied due to its localization feature [6]. It has been proved that the non localized case of the Grover walk can be simulated by a walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions i n which the walker can move [7]. In the quest for more feasible quantum walk models, an important step forward is presented in this paper: a strong localization-like effect can be indeed obtained in the modified version of the alternate quantum walk presented here, reducing the gap with the standard two-dimensional version. This result could pave the way for adapting this scheme to the realization of quantum algorithms, providing a clear advantage in terms of experimental resources. After a short introduct ion on the quantum walk studied here, we characterize it in terms of its relevant parameters and show that a proper choice can guarantee a non-negligible probability of finding the walke r at the origin even for large times. We then hint at a qualitative relation between the ability of the system to localize and th e behavior of coherences established in the state of the parti cles performing a two-particle equivalent scheme of the walk, which in turns provides information on the way correlations are set up between them. We discuss the robustness of this model to imperfections and its possible experimental realization with the current state-of-the-art settings in linear o ptics and cold-atom devices. Let us consider a quantum system with two degrees of freedom, and thus described by a vector in the composite Hilbert space H = HW ⊗ HC . The coin space HC is a two-dimensional Hilbert space spanned by {|0i,|1i} and the walker space HW is an infinite-dimensional Hilbert space spanned by {|x,yi}, with x and y assuming all possible integer values. We take as a basis of this space H the set {|x,y,ci}, with |x,y,ci= |x,yiW ⊗ |ciC ; x and y could denote, for instance, the position of a particle ( walker) along the x and y directions, respectively, while |ciC is an internal twolevel degree of freedom. From now on, we consider|0,0iW as the initial state of our walker, and |+yiC = (|0iC+i|1iC)/ √ 2 as the coin one. The evolution of the system is given by a sequence of conditional shift, coin operations, and phase gates. The effec t of the two different conditional shift operations
Abstract In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys. Rev . A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly, we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.
In this paper, we present a simulator for two-particle quantum walks on the line and one-particle on a two-dimensional squared lattice. It can be used to investigate the equivalence between the two cases (one- and two-particle walks) for various boundary conditions (open, circular, reflecting, absorbing and their combinations). For the case of a single walker on a two-dimensional lattice, the simulator can also implement the Möbius strip. Furthermore, other topologies for the walker are also simulated by the proposed tool, like certain types of planar graphs with degree up to 4, by considering missing links over the lattice. The main purpose of the simulator is to study the genuinely quantum effects on the global properties of the two-particle joint probability distribution on the entanglement between the walkers/axis. For that purpose, the simulator is designed to compute various quantities such as: the entanglement and classical correlations, (classical and quantum) mutual information, the average distance between the two walkers, different hitting times and quantum discord. These quantities are of vital importance in designing possible algorithmic applications of quantum walks, namely in search, 3-SAT problems, etc. The simulator can also implement the static partial measurements of particle(s) positions and dynamic breaking of the links between certain nodes, both of which can be used to investigate the effects of decoherence on the walker(s). Finally, the simulator can be used to investigate the dynamic Anderson-like particle localization by varying the coin operators of certain nodes on the line/lattice. We also present some illustrative and relevant examples of one- and two-particle quantum walks in various scenarios. The tool was implemented in C and is available on-line at http://qwsim.weebly.com/ .