Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks

Abstract

Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process.

arXiv comment21 pages, 3 figures

References in corpus (12)

• Quantum Computation and Decision Trees
• A Quantum Random Walk Search Algorithm
• Exploring Topological Phases With Quantum Walks
• On the relationship between continuous- and discrete-time quantum walk
• Photonic quantum walk in a single beam with twisted light
• Quantum walks and wavepacket dynamics on a lattice with twisted photons
• Connecting the discrete and continuous-time quantum walks
• Electric quantum walks with individual atoms
• Relationship Between Quantum Walk and Relativistic Quantum Mechanics
• Staggered Quantum Walks on Graphs
• Asymptotic behavior of quantum walks on the line
• Staggered Quantum Walks with Hamiltonians