scholarly journals A new type of quantum walks based on decomposing quantum states

2021 ◽  
Vol 21 (7&8) ◽  
pp. 541-556
Author(s):  
Chusei Kiumi
Keyword(s):  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
Quantum States ◽  
The Other ◽  
Imaginary Component ◽  
Probability Measures ◽  
Real Component ◽  
New Type ◽  
Weak Limit Theorem

In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.

scholarly journals Limit theorems for quantum walks with memory

2010 ◽  
Vol 10 (11&12) ◽  
pp. 1004-1017
Author(s):  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Limit Theorems ◽  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
One Dimension ◽  
Counting Method ◽  
Return Probability ◽  
Weak Limit Theorem ◽  
Path Counting ◽  
With Memory

Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

scholarly journals WEAK LIMITS FOR QUANTUM WALKS ON THE HALF-LINE

2013 ◽  
Vol 11 (06) ◽  
pp. 1350054 ◽  
Author(s):  
CHAOBIN LIU ◽  
NELSON PETULANTE
Keyword(s):  
Limit Theorem ◽  
General Condition ◽  
Transition Probabilities ◽  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
Half Line ◽  
Weak Limits ◽  
Terminal Behavior ◽  
Weak Limit Theorem

For a discrete two-state quantum walk (QW) on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half-line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.

scholarly journals Weak limit theorem for a nonlinear quantum walk

2018 ◽  
Vol 17 (9) ◽  
Author(s):  
Masaya Maeda ◽  
Hironobu Sasaki ◽  
Etsuo Segawa ◽  
Akito Suzuki ◽  
Kanako Suzuki
Keyword(s):  
Limit Theorem ◽  
Quantum Walk ◽  
Weak Limit ◽  
Nonlinear Quantum ◽  
Weak Limit Theorem

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

2006 ◽  
Vol 09 (02) ◽  
pp. 287-297 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Probability Theory ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Homogeneous Tree ◽  
One Dimensional ◽  
Time Quantum ◽  
New Type ◽  
The One

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

scholarly journals A limit law of the return probability for a quantum walk on a hexagonal lattice

2015 ◽  
Vol 13 (07) ◽  
pp. 1550054 ◽  
Author(s):  
Takuya Machida
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Hexagonal Lattice ◽  
Quantum Walk ◽  
Quantum Walks ◽  
The Other ◽  
View Point ◽  
Initial State ◽  
Random Walker ◽  
Return Probability

A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random walker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we take care of a discrete-time three-state quantum walk on a hexagonal lattice from the view point of mathematics. The mathematical result shows a limit of the return probability when the walker starts off at the origin. The result of the limit tells us about a possibility of localization at the position and a dependence of localization on the initial state.

One-dimensional three-state quantum walks: Weak limits and localization

2016 ◽  
Vol 19 (04) ◽  
pp. 1650025 ◽  
Author(s):  
Chul Ki Ko ◽  
Etsuo Segawa ◽  
Hyun Jae Yoo
Keyword(s):  
Limit Distribution ◽  
Evolution Operator ◽  
Adjoint Operator ◽  
Quantum Walk ◽  
Vacuum Expectation ◽  
Weak Limit ◽  
Quantum Walks ◽  
Continuous Part ◽  
One Dimensional ◽  
Weak Limits

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

scholarly journals AN ALGEBRAIC STRUCTURE FOR ONE-DIMENSIONAL QUANTUM WALKS AND A NEW PROOF OF THE WEAK LIMIT THEOREM

2013 ◽  
Vol 16 (02) ◽  
pp. 1350018
Author(s):  
TATSUYA TATE
Keyword(s):  
Limit Theorem ◽  
Characteristic Function ◽  
Algebraic Structure ◽  
Chebyshev Polynomials ◽  
Transition Probability ◽  
Weak Limit ◽  
Quantum Walks ◽  
One Dimensional ◽  
Natural Computation ◽  
Weak Limit Theorem

An algebraic structure for one-dimensional quantum walks is introduced. This structure characterizes, in some sense, one-dimensional quantum walks. A natural computation using this algebraic structure leads us to obtain an effective formula for the characteristic function of the transition probability. Then, the weak limit theorem for the transition probability of quantum walks is deduced by using simple properties of the Chebyshev polynomials.

scholarly journals Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

2016 ◽  
Vol 22 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Shimpei ENDO ◽  
Takako ENDO ◽  
Norio KONNO ◽  
Etsuo SEGAWA ◽  
Masato TAKEI
Keyword(s):  
Limit Theorem ◽  
Quantum Walk ◽  
Weak Limit ◽  
Two Phase ◽  
Weak Limit Theorem

Optimal class-specific witnesses for three-qubit entanglement from Greenberger-Horne-Zeilinger symmetry

2013 ◽  
Vol 13 (3&4) ◽  
pp. 210-220
Author(s):  
Christopher Eltschka ◽  
Jens Siewert
Keyword(s):  
Ghz State ◽  
Quantum States ◽  
Point Of View ◽  
The Other ◽  
Mixed States ◽  
Other Hand ◽  
New Type ◽  
The One ◽  
Qubit States ◽  
Symmetric States

Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.

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