A limit distribution for a quantum walk driven by a five-diagonal unitary matrix

In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The five-diagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.

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A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽

10.3390/sym13071134 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1134

Author(s):

Kenta Higuchi ◽

Takashi Komatsu ◽

Norio Konno ◽

Hisashi Morioka ◽

Etsuo Segawa

Keyword(s):

Unit Circle ◽

Stationary State ◽

Discrete Time ◽

Quantum Walk ◽

Time Limit ◽

The Other ◽

Unitary Matrix ◽

Initial State ◽

Time Quantum ◽

Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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A limit theorem for a 3-period time-dependent quantum walk

Quantum Information and Computation ◽

10.26421/qic15.1-2-4 ◽

2015 ◽

Vol 15 (1&2) ◽

pp. 50-60

Author(s):

F. Alberto Grunbaum ◽

Takuya Machida

Keyword(s):

Asymptotic Behavior ◽

Limit Theorem ◽

Time Evolution ◽

Discrete Time ◽

Limit Distribution ◽

Shift Operator ◽

Quantum Walk ◽

Time Dependent ◽

Long Time ◽

Position Shift

We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.

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A limit theorem for a splitting distribution of a quantum walk

International Journal of Quantum Information ◽

10.1142/s0219749918500235 ◽

2018 ◽

Vol 16 (03) ◽

pp. 1850023

Author(s):

Takuya Machida

Keyword(s):

Limit Theorem ◽

Probability Distribution ◽

Random Walks ◽

Discrete Time ◽

Limit Distribution ◽

Quantum Walk ◽

Quantum Walks ◽

Unitary Evolution ◽

Time Quantum ◽

Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

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EARLY TIME PROPERTIES OF TIME EVOLUTION IN TWO-DIMENSIONAL SUBSPACE AND CP VIOLATION PROBLEM

International Journal of Modern Physics A ◽

10.1142/s0217751x9300151x ◽

1993 ◽

Vol 08 (21) ◽

pp. 3721-3745 ◽

Cited By ~ 7

Author(s):

K. URBANOWSKI

Keyword(s):

Time Evolution ◽

Early Time ◽

Dimensional Subspace ◽

Effective Hamiltonian ◽

Two Dimensional ◽

Matrix Elements ◽

The Matrix ◽

Long Time ◽

Time Region ◽

Short Time

Approximate formulae are given for the effective Hamiltonian H||(t) governing the time evolution in a subspace ℋ|| of the state space ℋ. The properties of matrix elements of H||(t) and the eigenvalue problem for H||(t) are discussed in the case of two-dimensional ℋ||. The eigenvectors of H||(t) for the short time region are found to be different from those for the long time region. The decay law of particles described by the eigenvectors of H||(t) is shown to have the form of the exponential function multiplied by some time-independent factor, equal to 1 only in the case of the [Formula: see text]-invariant theory. Some general properties of the matrix elements of H||(t) are tested in the Lee model.

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Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Probability Theory and Related Fields ◽

10.1007/pl00008755 ◽

2000 ◽

Vol 118 (4) ◽

pp. 574-594 ◽

Cited By ~ 4

Author(s):

J. M. Swart

Keyword(s):

Limit Distribution ◽

Time Limit ◽

Interacting Diffusions ◽

Long Time

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Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

Quantum Information and Computation ◽

10.26421/qic15.5-6-3 ◽

2015 ◽

pp. 406-418

Author(s):

Takuya Machida

Keyword(s):

Limit Theorems ◽

Quantum Walk ◽

Time Limit ◽

The Other ◽

Convergence In Distribution ◽

Initial State ◽

Uniform Limit ◽

Limit Measure ◽

Long Time ◽

Discrete Uniform

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

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LONG‐TIME RESPONSE PREDICTED BY EXACT ELASTIC RAY THEORY

Geophysics ◽

10.1190/1.1439584 ◽

1965 ◽

Vol 30 (3) ◽

pp. 363-368 ◽

Cited By ~ 8

Author(s):

T. W. Spencer

Keyword(s):

Formal Solution ◽

Intermediate Stage ◽

Time Limit ◽

Time Interval ◽

Individual Response ◽

Axially Symmetric ◽

Long Time ◽

The Difference ◽

The Individual ◽

Significant Figures

The formal solution for an axially symmetric radiation field in a multilayered, elastic system can be expanded in an infinite series. Each term in the series is associated with a particular raypath. It is shown that in the long‐time limit the individual response functions produced by a step input in particle velocity are given by polynomials in odd powers of the time. For rays which suffer m reflections, the degree of the polynomials is 2m+1. The total response is obtained by summing all rays which contribute in a specified time interval. When the rays are selected indiscriminately, the difference between the magnitude of the partial sum at an intermediate stage of computation and the magnitude of the correct total sum may be greater than the number of significant figures carried by the computer. A prescription is stated for arranging the rays into groups. Each group response function varies linearly in the long‐time limit and goes to zero when convolved with a physically realizable source function.

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Anomalies in Strongly Coupled Harmonic Quantum Brownian Motion

Open Systems & Information Dynamics ◽

10.1142/s1230161213500029 ◽

2013 ◽

Vol 20 (01) ◽

pp. 1350002 ◽

Cited By ~ 2

Author(s):

F. Giraldi ◽

F. Petruccione

Keyword(s):

Time Evolution ◽

Power Law ◽

Weak Coupling ◽

Inverse Power ◽

Strong Coupling Regime ◽

Weak Coupling Regime ◽

Spectral Densities ◽

Inverse Power Law ◽

Critical Configurations ◽

Long Time

The exact dynamics of a quantum damped harmonic oscillator coupled to a reservoir of boson modes has been formally described in terms of the coupling function, both in weak and strong coupling regime. In this scenario, we provide a further description of the exact dynamics through integral transforms. We focus on a special class of spectral densities, sub-ohmic at low frequencies, and including integrable divergencies referred to as photonic band gaps. The Drude form of the spectral densities is recovered as upper limit. Starting from special distributions of coherent states as external reservoir, the exact time evolution, described through Fox H-functions, shows long time inverse power law decays, departing from the exponential-like relaxations obtained for the Drude model. Different from the weak coupling regime, in the sub-ohmic condition, undamped oscillations plus inverse power law relaxations appear in the long time evolution of the observables position and momentum. Under the same condition, the number of excitations shows trapping of the population of the excited levels and oscillations enveloped in inverse power law relaxations. Similarly to the weak coupling regime, critical configurations give arbitrarily slow relaxations useful for the control of the dynamics. If compared to the value obtained in weak coupling condition, for strong couplings the critical frequency is enhanced by a factor 4.

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