Stationary Measures of Three-State Quantum Walks with Defect on the One-Dimension Lattice

In this paper, we focus on the space-inhomogeneous three-state on the one-dimension lattice, a one-phase model and a two-phase model include. By using the transfer matrices method by Endo et al., we calculate the stationary measure for initial state concrete eigenvalue. Finally we found the transfer matrices method is more effective for the three-state quantum walks than the method obtained by Kawai et al.

Efficient quantum private comparison protocol based on one direction discrete quantum walks on the circle

We propose an efficient quantum private comparison protocol firstly based on one direction quantum walks.With the help of one direction quantum walk,we develop a novel method that allows the semi-honest third party to set a flag to judge the comparing result,which improves the qubit efficiency and the maximum quantity of the participants’secret messages.Besides,our protocol can judge the size of the secret messages,not only equality.Furthermore,the quantum walks particle is disentangled in the initial state.It only requires a quantum walks operator to move,making our proposed protocol easy to implement and reducing the quantum resources.Through security analysis,we prove that our protocol can withstand well-known attacks and brute-force attacks.Analyses also reveal that our protocol is correct and practical.

The entanglement of deterministic aperiodic quantum walks

We study the entanglement between the internal(coin)and the external(position)degrees of freedom in the dynamic and the static deterministic aperiodic quantum walks(QWs).For the dynamic(static)aperiodic QWs,the coin depends on the time(position)and takes two coins C(α)and C(β)arranged in the two classes of generalized Fibonacci(GF)and the Thue–Morse(TM)sequences.We found that for the dynamic QWs,the entanglement of three kinds of the aperiodic QWs are close to the maximal value,which are all much larger than that of the homogeneous QWs.Further,the first class of GF(1st GF)QWs can achieve the maximum entangled state,which is similar to that of the dynamic disordered QWs.And the entanglement of 1st GF QWs is greater than that of the TM QWs,being followed closely by the entanglement of the second class of GF(2nd GF)QWs.For the static QWs,the entanglement of three kinds of the aperiodic QWs are also close to the maximal value and 1st GF QWs can achieve the maximum entangled state.The entanglement of the TM QWs is between1st GF QWs and 2nd GF QWs.However,the entanglement of the static disordered QWs is less than that of three kinds of the aperiodic QWs.This is different from those of the dynamic QWs.From these results,we can conclude that the dynamic and static 1st GF QWs can also be considered as maximal entanglement generators.

Localization of quantum walks on finite graphs

We analyze the localization of quantum walks on a one-dimensional finite graph using vector-distance. We first vectorize the probability distribution of a quantum walker in each node. Then we compute out the probability distribution vectors of quantum walks in infinite and finite graphs in the presence of static disorder respectively, and get the distance between these two vectors. We find that when the steps taken are small and the boundary condition is tight, the localization between the infinite and finite cases is greatly different. However, the difference is negligible when the steps taken are large or the boundary condition is loose. It means quantum walks on a one-dimensional finite graph may also suffer from localization in the presence of static disorder. Our approach and results can be generalized to analyze the localization of quantum walks in higher-dimensional cases.

Disorder in parity-time symmetric quantum walks

We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity-time(PT)-symmetric quantum walks.Via temporally alternating photon losses in an interferometric network,we realize the passive PT-symmetric quantum dynamics for single photons.Controllable coin operations allow us to simulate different environmental influences,which result in three different behaviors of quantum walkers:a standard ballistic spread,a diffusive behavior,and a localization,respectively,in a PT-symmetric quantum walk architecture.

Non-Markovian decoherent quantum walks

Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker’s behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.

A Controlled Quantum Dialogue Protocol Based on Quantum Walks

In order to enable two parties to exchange their secret information equally,we propose a controlled quantum dialogue protocol based on quantum walks,which implements the equal exchange of secret information between the two parties with the help of the controller TP.The secret information is transmitted via quantum walks,by using this method,the previously required entangled particles do not need to be prepared in the initial phase,and the entangled particles can be produced spontaneously via quantum walks.Furthermore,to resist TP’s dishonest behavior,we use a hash function to verify the correctness of the secret information.The protocol analysis shows that it is safe and reliable facing some attacks,including intercept-measure-resend attack,entanglement attack,dishonest controller’s attack and participant attack.And has a slightly increasing efficiency comparing with the previous protocols.Note that the proposed protocol may be feasible because quantum walks prove to be implemented in different physical systems and experiments.

Quantum walks with coins undergoing different quantum noisy channels

Quantum walks have significantly different properties compared to classical random walks, which have potential applications in quantum computation and quantum simulation. We study Hadamard quantum walks with coins undergoing different quantum noisy channels and deduce the analytical expressions of the first two moments of position in the long- time limit. Numerical simulations have been done, the results are compared with the analytical results, and they match extremely well. We show that the variance of the position distributions of the walks grows linearly with time when enough steps are taken and the linear coefficient is affected by the strength of the quantum noisy channels.

Effect of Quantum Coins on Two-Particle Quantum Walks

We numerically study the effect of the quantum coins on the two-particle quantum walks on an infinite line. Both non-interacting and interacting particles are considered. The joint probability as well as the bunching or anti- bunching behavior are greatly affected by the phase factors in the coin operation. Further, the spatial correlation can be maximized by choosing appropriate coin parameters. The entanglement between the two particles can be adjusted in the same manner.

Probe of topological invariants using quantum walks of a trapped ion in coherent state space

We present a protocol to realize topological discrete-time quantum walks,which comprise a sequence of spindependent flipping displacement operations and quantum coin tossing operations,with a single trapped ion.It is demonstrated that the information of bulk topological invariants can be extracted by measuring the average projective phonon number when the walk takes place in coherent state space.Interestingly,the specific chiral symmetry owned by our discrete-time quantum walks simplifies the measuring process.Furthermore,we prove the robustness of such bulk topological invariants by introducing dynamical disorder and decoherence.Our work provides a simple method to measure bulk topological features in discrete-time quantum walks,which can be experimentally realized in the system of single trapped ions.

Experimental realization of one-dimensional optical quantum walks

We analyze the process of a discrete-time quantum walk over 4 steps and 5 positions with linear optics elements. The quantum walk is characterized by a ballistic spread of wavepackets along 4 steps. By employing different initial coin states, we observe non-Gaussian distribution of the walkers＇ finial position, which characterizes a quadratic enhancement of the spread of photon wavepackets compared to a classical random walk. By introducing controllable decoherence, we observe the quantum-to-classical transmission in a quantum walk architecture.

Disordered quantum walks in two-dimensional lattices

The properties of the two-dimensional quantum walk with point, line, and circle disorders in phase are reported.Localization is observed in the two-dimensional quantum walk with certain phase disorder and specific initial coin states.We give an explanation of the localization behavior via the localized stationary states of the unitary operator of the walker＋ coin system and the overlap between the initial state of the whole system and the localized stationary states.

One-dimensional lazy quantum walks and occupancy rate

In this paper, we discuss the properties of lazy quantum walks. Our analysis shows that the lazy quantum walks have O（tn） order of the n-th moment of the corresponding probability distribution, which is the same as that for normal quantum walks. The lazy quantum walk with a discrete Fourier transform （DFT） coin operator has a similar probability distribution concentrated interval to that of the normal Hadamard quantum walk. Most importantly, we introduce the concepts of occupancy number and occupancy rate to measure the extent to which the walk has a （relatively） high probability at every position in its range. We conclude that the lazy quantum walks have a higher occupancy rate than other walks such as normal quantum walks, classical walks, and lazy classical walks.

Search algorithm on strongly regular graphs based on scattering quantum walks

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs（SRGs） with parameters（N, k, λ, μ） achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm＇s performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs.The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as√N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O（√N） time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.

Disorder and decoherence in coined quantum walks

This article aims to provide a review on quantum walks. Starting form a basic idea of discrete-time quantum walks, we will review the impact of disorder and decoherence on the properties of quantum walks. The evolution of the standard quantum walks is deterministic and disorder introduces randomness to the whole system and change interference pattern leading to the localization effect. Whereas, decoherence plays the role of transmitting quantum walks to classical random walks.

State transfer on two-fold Cayley trees via quantum walks

Perfect state transfer(PST)has great significance due to its applications in quantum information processing and quantum computation.The main problem we study in this paper is to determine whether the two-fold Cayley tree,an extension of the Cayley tree,admits perfect state transfer between two roots using quantum walks.We show that PST can be achieved by means of the so-called nonrepeating quantum walk[Phys.Rev.A 89042332(2014)]within time steps that are the distance between the two roots;while both the continuous-time quantum walk and the typical discrete-time quantum walk with Grover coin approaches fail.Our results suggest that in some cases the dynamics of a discrete-time quantum walk may be much richer than that of the continuous-time quantum walk.

Average position in quantum walks with a U(2) coin

We investigated discrete-time quantum walks with an arbitary unitary coin.Here we discover that the average position x=max（x） sin（α＋γ）,while the initial state is 1/2～（1/2）（｜0L＋i｜0R）.We verify the result,and obtain some symmetry properties of quantum walks with a U（2） coin with ｜0L and ｜0R as the initial state.

Mobility edges and localization characteristics in one-dimensional quasiperiodic quantum walk

We construct a one-dimensional quasiperiodic quantum walk to investigate the localization–delocalization transition.The inverse participation ratio and Lyapunov exponent are employed as two indexes to determine the mobility edge, a critical energy to distinguish the energy regions of extended and localized states. The analytical solution of mobility edge is obtained by the Lyapunov exponents in global theory, and the consistency of the two indexes is confirmed. We further study the dynamic characteristics of the quantum walk and show that the probabilities are localized to some specific lattice sites with time evolution. This phenomenon is explained by the effective potential of the Hamiltonian which corresponds to the phase in the coin operator of the quantum walk.

Photonic discrete-time quantum walks [Invited]

Quantum walks, a counterpart of classical random walks, have many applications due to their neoteric features.Since they were first proposed, quantum walks have been explored in many fields theoretically and have also been demonstrated experimentally in various physical systems. In this paper, we review the experimental realizations of discrete-time quantum walks in photonic systems with different physical structures, such as bulk optics and time-multiplexed framework. Then, some typical applications using quantum walks are introduced. Finally, the advantages and disadvantages of these physical systems are discussed.

One-dimensional quantum walks with a position-dependent coin

We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin.The rotation angle,which depends upon the position of a quantum particle,parameterizes the coin operator.For different values of the rotation angle,we observe that such a coin leads to a variety of probability distributions,e.g.localized,periodic,classicallike,semi-classical-like,and quantum-like.Further,we study the Shannon entropy associated with position and the coin space of a quantum particle,and compare them with the case of the position-independent coin.Our results show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin.We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy by considering a quantum walk with two identical position-dependent entangled coins.We observe that in general,a wave function becomes more localized as compared to the case of the positionindependent coin and hence the corresponding Shannon entropy is lower.Our results show that a position-dependent coin can be used as a controlling tool of quantum walks.