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Lecture notes in computer science

A Continuous-Time Quantum Walk Kernel for Unattributed Graphs

Luca Rossi, Andrea Torsello A, Edwin R. Hancock

Published Jan 1, 2013• 19 citations
Publisher
DOI
10.1007/978-3-642-38221-5_11

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In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection.

Physical Review E•Feb 23, 2015•46 citations
Continuous time quantum walks built on graph Laplacians produce non monotonic graph propagation features through quantum interference during evolution, which classical diffusion cannot achieve. Such walks deliver a physically motivated propagation scheme for temporal graph signal modeling. We propose Quantum Walk Temporal Architecture (QWTA), which replaces classical graph propagation with a CTQW-type parameterized spectral propagator. QWTA embeds effective observation intervals into the spectral phase modulation of the propagation kernel. QWTA-Base preserves exact CTQW spectral evolution and serves as a physically faithful propagation reference. QWTA-GR further introduces phase soft clipping and gated residual fusion to stabilize propagation. The results show that explicit phase encoding of irregular time intervals, when combined with suitable stabilization, provides a physically motivated and competitive graph propagation design for temporal graph forecasting with missing historical observations.
In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection.