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IET Quantum Communication

Continuous‐Time Quantum Walk on the Homogeneous Tree

Qi Han, Rong Zhang, Lijie Gou, Shuai Wang

Published Jan 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT In this paper, we explore continuous‐time quantum walk on homogeneous tree. By applying the stratification method, we treat vertices in the same stratum of a homogeneous tree as a single vertex and obtain the probability amplitudes for different strata in finite homogeneous tree parts. In particular, when (where denotes the number of strata), we derive the probability amplitudes for each stratum of the homogeneous tree, considering the homogeneous tree's transition from a finite to infinite structure.

DOI
10.1049/qtc2.70031

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ABSTRACT In this paper, we explore continuous‐time quantum walk on homogeneous tree. By applying the stratification method, we treat vertices in the same stratum of a homogeneous tree as a single vertex and obtain the probability amplitudes for different strata in finite homogeneous tree parts. In particular, when (where denotes the number of strata), we derive the probability amplitudes for each stratum of the homogeneous tree, considering the homogeneous tree's transition from a finite to infinite structure.
In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.
In this work, we consider the application of continuous time quantum walking(CTQW) to the Maximum Clique(MC) Problem. Performing CTQW on graphs will generate distinct periodic probability amplitude for different vertices. We will show that the intensity of the probability amplitude at frequency indeed implies the clique structure of some special kinds of graph. And recursive algorithms with time complexity O(N5)O(N^5)O(N5) in classical computers for finding the maximum clique are proposed. We have experimented on random graphs where each edge exists with probabilities 0.3, 0.5 and 0.7. Although counter examples are not found for random graphs, whether these algorithms are universal is not known to us.
The continuous-time quantum walk on the underlying graphs of association schemes has been studied, via the algebraic combinatorics structures of association schemes, namely semi-simple modules of their Bose–Mesner and Terwilliger algebras. It is shown that the Terwilliger algebra stratifies the graph into a ( d + 1) disjoint union of strata which is different from the stratification based on distance, except for distance regular graphs. In underlying graphs of association schemes, the probability amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes, such that the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the same, therefore there are at most ( d + 1) different observing probabilities. The importance of association scheme in continuous-time quantum walk is shown by some worked out examples such as arbitrary finite group association schemes followed by symmetric S n , Dihedral D 2 m and cyclic groups. At the end it is shown that the highest irreducible representations of Terwilliger algebras pave the way to use the spectral distributions method of Jafarizadeh and Salimi (2005 Preprint quant-ph/0510174) in studying quantum walk on some rather important graphs called distance regular graphs.
In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.