Coin Parameter Research Articles

Semi-empirical method for evaluating the robustness of QRWS algorithm for different coin sizes and functional dependence between coin phases

In this work we study the quantum random walk search algorithm when the walk coin is constructed by generalized Householder reflection and a phase multiplier. We focus on the algorithm’s robustness against errors in the phases – how the probability to find solution decreases with increasing the errors in those phases. The robustness depends on the functional dependence between the reflection phase φ and the phase multiplier ζ. Here we use interpolations by non-linear logistic regression functions to study a particular functional dependence between angles ζ = -2φ + 3π + α sin(2φ) for different coin sizes and parameters alpha. The obtained results in this work are discussed, together with their limitations.

Balancing the Objectives of Statistical Efficiency and Allocation Randomness in Randomized Controlled Trials

Various restricted randomization procedures are available to achieve equal (1:1) allocation in a randomized clinical trial. However, for some procedures, there is a nonnegligible probability of imbalance in the final numbers which may result in an underpowered study. It is important to assess such probability at the study planning stage and make adjustments in the design if needed. In this article, we perform a quantitative assessment of the tradeoff between randomness, balance, and power of restricted randomization designs targeting equal allocation. First, we study the small-sample performance of biased coin designs with known asymptotic properties and identify a design with an excellent balance–randomness tradeoff. Second, we investigate the issue of randomization-induced treatment imbalance and the corresponding risk of an underpowered study. We propose two risk mitigation strategies: increasing the total sample size or fine-tuning the biased coin parameter to obtain the least restrictive randomization procedure that attains the target power with a high, user-defined probability for the given sample size. Additionally, we investigate an approach for finding the most balanced design that satisfies a constraint on the chosen measure of randomness. Our proposed methodology is simple and yet generalizable to more complex settings, such as trials with stratified randomization and multi-arm trials with possibly unequal randomization ratios.

A simple coin for a 2d entangled walk

We analyze the effect of a simple coin operator, built out of Bell pairs, in a 2d Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the Entangling Power of the coin operator. Secondly, we compute the Generalized Relative Rényi Entropy between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the Entangling Power and Generalized Relative Rényi Entropy behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the 2d DQRW into two 1d massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.

Monitored recurrence of a one-parameter family of three-state quantum walks

Monitored recurrence of a one-parameter set of three-state quantum walks on a line is investigated. The calculations are considerably simplified by choosing a suitable basis of the coin space. We show that the Polya number (i.e. the site recurrence probability) depends on the coin parameter and the probability that the walker is initially in a particular coin state for which the walk returns to the origin with certainty. Finally, we present a brief investigation of the exact quantum state recurrence.

Localization of two dimensional quantum walks defined by generalized Grover coins

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 paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.

Limit theorems and localization of three-state quantum walks on a line defined by generalized Grover coins

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on one dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position, for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.

Multiparameter quantum metrology with discrete-time quantum walks

We address multi-parameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for unknown parameters encoded on its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical i.e. the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.

One-dimensional quantum walks with a time and spin-dependent phase shift

We investigate the role of a time and spin-dependent phase shift on the evolution of one-dimensional discrete-time quantum walks. By employing Floquet engineering a time and spin-dependent phase shift (ϕ) is stroboscopically imprinted onto the walker's wave function at the end of each step of the walk. For a quantum walk driven by the standard protocol and with rational values of the phase factor, i.e., ϕ/2π=p/q we show with our numerical simulations that revivals with equal periods occur in the probability distribution of the walk. In the split-step quantum walk we employ two different coin operators which are parametrized by a parameter δθ. By making use of δθ as a control parameter we demonstrate that for ϕ/2π=p/q the walk exhibits three major types of dynamics: ballistic spreading, revivals, and localization. For an irrational value of ϕ/2π our results show revivals with unpredictable periods, and the walker remains localized in a small region of the lattice. Furthermore, in view of an experimental realization we investigate the robustness of revivals against noise in the phase shift. Our results show that signatures of revivals persist for smaller values of the noise parameter while these vanish for larger values. Our work is important in the context of quantum computation and simulation with quantum walks where different types of the walk dynamics can be engineered by making use of the time and spin-dependent phases shift, and the coin parameter as control knobs.

High winding number of topological phase in periodic quantum walks

We construct one-dimensional multi-period quantum walks to investigate the topological phases with high winding number. It is found that the maximum of winding number is determined by the number of shift operation in one-step evolution only under an appropriate time frame. For odd-period quantum walks, 0-energy edge states and π-energy edge states coexist equally on the boundary, but for even-period quantum walks, these two kinds of edge states may exist alone or together. The relations between the number of edge state and high winding number are discussed by considering coin parameter as a step function of position. The validity of the bulk-edge correspondence is confirmed. We also reveal that the occupation probability of the walker at the boundary exhibits dynamical oscillation due to the coexistence of 0-energy and π-energy edge states.

Circle Hough Transform (CHT) method applied to coin identifier and recognition based on computer vision and android developer

The coin recognition system is a very important role in currency processing systems which is aims to minimize human miscalculation and time to calculate coin values. The coin identifier and recognition to calculate the value of Indonesian coins based on android developer presented in this paper. The two coin parameters are used for coin recognition which are radius and colour of the coins. The Circle Hough Transform (CHT) method with gray scaling and erosion process as pre-processing image used to calculate radius parameters, based on android developer and computer vision. After the radius value is obtained then the average RGB value of the coins is detected to determine the colour parameters. By calculating the diameter and determining the colour, the coins can be recognized and the total value of the coins in the input image can be known. The experimental results show that the system has been obtained indicate that the system successfully recognized coins based on determined algorithm. That is mean the application using computer vision can recognition coin and count the value of coins.

Quantum walker as a probe for its coin parameter

In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\theta$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols, and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space $H_w(\theta)$ increases with $\theta$ and with time which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of {\em bounded} and {\em unbounded} DTQWs, and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare $H_w(\theta)$ to the full QFI $H_f(\theta)$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\theta$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.

Experimental realization of a momentum-space quantum walk

We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential applications include atom interferometry and engineering asymmetric walks.

Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped Dirac equation

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\hat{C}(\theta_1)$ and $\hat{C}(\theta_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters $\theta_1$ and $\theta_2$ in the coin operations we show that the standard deviation [$\sigma_{\theta_1, \theta_2} (t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations $\theta_1$ or $\theta_2$, $\sigma_{\theta_1, \theta_2}(t) = \min \{\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) \}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using $\theta_1$ and $\theta_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta=0$, here we show the recovery of the massless Dirac equation with non-zero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.

Robust quantum state engineering through coherent localization in biased-coin quantum walks

We address the performance of a coin-biased quantum walk as a generator for non-classical position states of the walker. We exploit a phenomenon of coherent localization in the position space - resulting from the choice of small values of the coin parameter and assisted by post-selection - to engineer large-size coherent superpositions of position states of the walker. The protocol that we design appears to be remarkably robust against both the actual value taken by the coin parameter and strong dephasing-like noise acting on the spatial degree of freedom. We finally illustrate a possible linear-optics implementation of our proposal, suitable for both bulk and integrated-optics platforms.

Quantum state engineering using one-dimensional discrete-time quantum walks

Quantum state preparation in high-dimensional systems is an essential requirement for many quantum-technology applications. The engineering of an arbitrary quantum state is, however, typically strongly dependent on the experimental platform chosen for implementation, and a general framework is still missing. Here we show that coined quantum walks on a line, which represent a framework general enough to encompass a variety of different platforms, can be used for quantum state engineering of arbitrary superpositions of the walker's sites. We achieve this goal by identifying a set of conditions that fully characterize the reachable states in the space comprising walker and coin, and providing a method to efficiently compute the corresponding set of coin parameters. We assess the feasibility of our proposal by identifying a linear optics experiment based on photonic orbital angular momentum technology.

Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.

Ansatz for Majorana modes for a quantum walk in a finite wire

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 $E=0,\pi$. 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 modes are provided.

One-Dimensional Three-State Quantum Walk with Single-Point Phase Defects

In this paper, we study a three-state quantum walk with a phase defect at a designated position. The coin operator is a parametrization of the eigenvectors of the Grover matrix. We numerically investigate the properties of the proposed model via the position probability distribution, the position standard deviation, and the time-averaged probability at the designated position. It is shown that the localization effect can be governed by the phase defect’s position and strength, coin parameter and initial state.

Persistence of unvisited sites in quantum walks on a line

We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk there is no connection between the behaviour of persistence and the scaling of variance. In particular, we find that for a two-state quantum walks persistence follows an inverse power-law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk the scaling of persistence is given by two contributions. The first is the inverse power-law. The second contribution to the asymptotic behaviour of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behaviour.

Effect of Quantum Coins on Two-Particle Quantum Walks**Supported by the National Natural Science Foundation of China under Grant Nos 11104128, 61205119 and 41206084, the Natural Science Foundation of Jiangxi-Provincial Office of Education under Grant No GJJ13485, and the Doctor Start-up Foundation of Nanchang Hangkong University under Grant No EA201008229.

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.