scholarly journals The role of coherence on two-particle quantum walks

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Li-Hua Lu ◽  
Shan Zhu ◽  
You-Quan Li
Keyword(s):  
Correlation Function ◽  
Average Distance ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Mixed States ◽  
Initial State ◽  
Photon Correlation ◽  
Two Photon ◽  
Degree Of Coherence

AbstractWe investigate the dynamical properties of the two-boson quantum walk in systems with different degrees of coherence, and where the effect of the coherence on the two-boson quantum walk can be naturally introduced. A general analytical expression for the two-boson correlation function, for both pure states and mixed states, is given.We propose a possible two-photon quantum-walk scheme with a mixed initial state, and find that the twophoton correlation function and the average distance between two photons can be influenced by the initial photon distribution, the relative phase, or the degree of coherence. The propagation features of our numerical results can be explained by our analytical two-photon correlation function.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

2021 ◽  
pp. 2250001
Author(s):  
Ce Wang
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Initial State ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Higher Dimensional ◽  
Gauss Type ◽  
Open Quantum Random Walks ◽  
Limit Probability ◽  
Creation Operators

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

scholarly journals Realization of the probability laws in the quantum central limit theorems by a quantum walk

2013 ◽  
Vol 13 (5&6) ◽  
pp. 430-438
Author(s):  
Takuya Machida
Keyword(s):  
Probability Theory ◽  
Limit Theorem ◽  
Central Limit ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Central Limit Theorems ◽  
Initial State

Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.

Entanglement for quantum walks on the line

2011 ◽  
Vol 11 (9&10) ◽  
pp. 855-866
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Initial State ◽  
Von Neumann ◽  
Time Quantum ◽  
The One ◽  
Path Counting

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

scholarly journals Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 504
Author(s):  
Ce Wang ◽  
Caishi Wang
Keyword(s):  
Discrete Time ◽  
Quantum Information Processing ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Integer Lattice ◽  
Initial State ◽  
Higher Dimensional ◽  
Time Quantum ◽  
Gauss Type ◽  
Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

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.

scholarly journals Photon spin and the shape of the two-photon correlation function

1997 ◽  
Vol 391 (3-4) ◽  
pp. 469-473 ◽  
Author(s):  
Claus Slotta ◽  
Ulrich Heinz
Keyword(s):  
Correlation Function ◽  
Photon Correlation ◽  
Two Photon

scholarly journals QUANTUM WALKS ON GENERAL GRAPHS

2006 ◽  
Vol 04 (05) ◽  
pp. 791-805 ◽  
Author(s):  
VIV KENDON
Keyword(s):  
Continuous Time ◽  
Prior Information ◽  
Quantum Computer ◽  
Quantum Walk ◽  
Quantum Walks ◽  
General Graph ◽  
Resource Requirements ◽  
General Graphs

Quantum walks, both discrete (coined) and continuous time, on a general graph of N vertices with undirected edges are reviewed in some detail. The resource requirements for implementing a quantum walk as a program on a quantum computer are compared and found to be very similar for both discrete and continuous time walks. The role of the oracle, and how it changes if more prior information about the graph is available, is also discussed.

scholarly journals Limit theorems for the discrete-time quantum walk on a graph with joined half lines

2012 ◽  
Vol 12 (3&4) ◽  
pp. 314-333
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Half Line ◽  
Initial State ◽  
Time Quantum ◽  
Weak Convergence Theorem ◽  
The One

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

scholarly journals Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 85
Author(s):  
Luca Razzoli ◽  
Matteo G. A. Paris ◽  
Paolo Bordone
Keyword(s):  
Transport Properties ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Transport Processes ◽  
Initial State ◽  
Transport Efficiency ◽  
Harvesting Systems ◽  
Time Quantum ◽  
Modeling Transport

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.

scholarly journals Implementing graph-theoretic quantum algorithms on a silicon photonic quantum walk processor

2021 ◽  
Vol 7 (9) ◽  
pp. eabb8375
Author(s):  
Xiaogang Qiang ◽  
Yizhi Wang ◽  
Shichuan Xue ◽  
Renyou Ge ◽  
Lifeng Chen ◽  
...  
Keyword(s):  
Large Scale ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Graph Theoretic ◽  
Two Photon ◽  
Full Control ◽  
Silicon Photonic ◽  
Vertex Graph ◽  
Particle Exchange

Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglement-driven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled two-photon quantum walks on any five-vertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates single-particle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to large-scale, programmable quantum walk processors for classically intractable applications.

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