Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

• URL: http://arxiv.org/abs/2309.15194v1
• Date: Tue, 26 Sep 2023 18:53:35 GMT
• Title: Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins
• Authors: Rohit Sarma Sarkar, Bibhas Adhikari
• Abstract summary: We study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: In this paper we study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups, which are graphs with both directed and undirected edges. We consider the walks with coins that are one-parameter continuous deformation of the Grover matrix and can be written as linear combinations of certain permutation matrices. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix. Finally, we investigate the localization property of the walks through numerical simulations and observe that the walks localize for a wide range of coins for different sizes of the graphs.

Related papers

• Irrational quantum walks [0.0]
  We develop theory to study any quantum walk generated by an integral Hamiltonian. We use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix.
  arXiv  Detail & Related papers  (2022-08-18T17:21:47Z)
• Quantitative approach to Grover's quantum walk on graphs [62.997667081978825]
  We study Grover's search algorithm focusing on continuous-time quantum walk on graphs. Instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph endowed Laplacians.
  arXiv  Detail & Related papers  (2022-07-04T19:33:06Z)
• Discrete Quantum Walks on the Symmetric Group [0.0]
  In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cayley graphs generated by the symmetric group ($sym$) with appropriate generating sets.
  arXiv  Detail & Related papers  (2022-03-28T23:48:08Z)
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution [52.77024349608834]
  Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. We present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians.
  arXiv  Detail & Related papers  (2021-08-06T19:38:01Z)
• Localization of two-dimensional quantum walks defined by generalized Grover coins [0.17205106391379021]
  We study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices. We show that the proposed walks localize at its initial position for initial coin states when the coin belongs to classes which contain the Grover matrix.
  arXiv  Detail & Related papers  (2021-02-28T13:54:29Z)
• Random Walks: A Review of Algorithms and Applications [37.226218097358284]
  In computer science, classical random walks and quantum walks can be used to calculate the proximity between nodes and extract the topology in the network. Various random walk related models can be applied in different fields, which is of great significance to downstream tasks such as link prediction, recommendation, computer vision, semi-supervised learning, and network embedding.
  arXiv  Detail & Related papers  (2020-08-09T03:41:56Z)
• A Matrix Chernoff Bound for Markov Chains and Its Application to Co-occurrence Matrices [30.49132891333353]
  We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Our proof is based on the matrix expander (regular undirected graph) Chernoff bound [Garg et al. STOC '18] and scalar Chernoff-Hoeffding bounds for Markov chains. Our matrix Chernoff bound for Markov chains can be applied to analyze the behavior of co-occurrence statistics for sequential data.
  arXiv  Detail & Related papers  (2020-08-06T05:44:54Z)
• Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks [0.0]
  We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. We show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks.
  arXiv  Detail & Related papers  (2020-08-03T22:15:01Z)
• Space-Time Correspondence as a Contrastive Random Walk [47.40711876423659]
  We cast correspondence as prediction of links in a space-time graph constructed from video. We learn a representation in which pairwise similarity defines transition probability of a random walk. We demonstrate that a technique we call edge dropout, as well as self-supervised adaptation at test-time, further improve transfer for object-centric correspondence.
  arXiv  Detail & Related papers  (2020-06-25T17:56:05Z)
• Continuous-time quantum walks in the presence of a quadratic perturbation [55.41644538483948]
  We address the properties of continuous-time quantum walks with Hamiltonians of the form $mathcalH= L + lambda L2$. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry.
  arXiv  Detail & Related papers  (2020-05-13T14:53:36Z)
• Periodicity of lively quantum walks on cycles with generalized Grover coin [0.17205106391379021]
  We extend the study of three state lively quantum walks on cycles by considering the coin operator as a linear sum of permutation matrices. We establish that an orthogonal matrix of order $3times 3$ is a linear sum of permutation matrices if and only if it is permutative.
  arXiv  Detail & Related papers  (2020-03-29T06:32:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.