# d-Path Laplacians and Quantum Transport on Graphs

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Quantum Transport Controlled by d-Path Laplacians

#### 3.1. Quantum Transport on a Ring

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 3.2. Quantum Transport in Complete and Star Graphs

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. d-Path Laplacians versus Fractional Graph Laplacian

**Theorem**

**3.**

**Lemma**

**4.**

## 5. Conclusions and Future Outlook

## Funding

## Conflicts of Interest

## References

- Grigor’yan, A. Heat kernels on manifolds, graphs and fractals. In European Congress of Mathematics; Birkhäuser: Basel, Switzerland, 2001; pp. 393–406. [Google Scholar]
- Kondor, R.I.; Lafferty, J. Diffusion kernels on graphs and other discrete structures. In Proceedings of the 19th International Conference on Machine Learning, Sydney, Australia, 8–12 July 2002; Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 2002; pp. 315–322. [Google Scholar]
- Bai, X.; Hancock, E.R. Heat kernels, manifolds and graph embedding. In Proceedings of the Joint IAPR International Workshops on Statistical Techniques in Pattern Recognition (SPR) and Structural and Syntactic Pattern Recognition (SSPR), Lisbon, Portugal, 18–20 August 2004; Springer: Berlin, Germany, 2004; pp. 198–206. [Google Scholar]
- Olfati-Saber, R.; Fax, J.A.; Murray, R.M. Consensus and cooperation in networked multi-agent systems. Proc. IEEE.
**2007**, 95, 215–233. [Google Scholar] [CrossRef] [Green Version] - Mesbahi, M.; Egerstedt, M. Graph Theoretic Methods in Multiagent Networks; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Suau, P.; Hancock, E.R.; Escolano, F. Graph characteristics from the Schrödinger operator. In Proceedings of the International Workshop on Graph-Based Representations in Pattern Recognition, Vienna, Austria, 15–17 May 2013; Springer: Berlin, Germany, 2013; pp. 172–181. [Google Scholar]
- Escolano, F.; Hancock, E.R.; Lozano, M.A. Skeletal Graphs from Schrödinger Magnitude and Phase. International Workshop on Graph-Based Representations in Pattern Recognition, Vienna, Austria, 13–15 May 2015; Springer: Cham, Switzerland, 2015; pp. 335–344. [Google Scholar]
- Emms, D.; Wilson, R.C.; Hancock, E.R. Graph matching using the interference of continuous-time quantum walks. Pattern Recog.
**2009**, 42, 985–1002. [Google Scholar] [CrossRef] - Emms, D.; Wilson, R.; Hancock, E. Graph embedding using a quasi-quantum analogue of the hitting times of continuous time quantum walks. Quantum Inform. Comput.
**2009**, 9, 231–254. [Google Scholar] - Berkolaiko, G.; Kuchment, P. Introduction to Quantum Graphs; American Mathematical Society: Providence, RI, USA, 2013. [Google Scholar]
- Merris, R. Laplacian matrices of graphs: A survey. Lin. Algebra Appl.
**1994**, 197, 143–176. [Google Scholar] [CrossRef] [Green Version] - Mohar, B. The Laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications; Alavi, Y., Chartrand, G., Oellermann, Schwenk, A.J., Wiley, O.R., Eds.; Springer: Cham, Switzerland, 1991; Volume 2, pp. 871–898. [Google Scholar]
- Grone, R.; Merris, R.; Sunder, V.S. The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl.
**1990**, 11, 218–238. [Google Scholar] [CrossRef] - Grone, R.; Merris, R. The Laplacian spectrum of a graph II. SIAM J. Matrix Anal. Appl.
**1994**, 7, 221–229. [Google Scholar] [CrossRef] - Senft, D.C.; Ehrlich, G. Long jumps in surface diffusion: One-dimensional migration of isolated adatoms. Phys. Rev. Lett.
**1995**, 74, 294–297. [Google Scholar] [CrossRef] - Linderoth, T.R.; Horch, S.; Lægsgaard, E.; Stensgaard, I.; Besenbacher, F. Surface diffusion of Pt on Pt(110): Arrhenius behavior of long jumps. Phys. Rev. Lett.
**1997**, 78, 4978–4981. [Google Scholar] [CrossRef] - Schunack, M.; Linderoth, T.R.; Rosei, F.; Lægsgaard, E.; Stensgaard, I.; Besenbacher, F. Long jumps in the surface diffusion of large molecules. Phys. Rev. Lett.
**2002**, 88, 156102. [Google Scholar] [CrossRef] [Green Version] - Yu, C.; Guan, J.; Chen, K.; Bae, S.C.; Granick, S. Single-molecule observation of long jumps in polymer adsorption. ACS Nano
**2013**, 7, 9735–9742. [Google Scholar] [CrossRef] [Green Version] - Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys.
**2008**, 129, 11B603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Caruso, F.; Chin, A.W.; Datta, A.; Huelga, S.F.; Plenio, M.B. Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport. Chem. Phys.
**2009**, 131, 09B612. [Google Scholar] [CrossRef] [Green Version] - Côté, R.; Russell, A.; Eyler, E.E.; Gould, P.L. Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys.
**2006**, 8, 156. [Google Scholar] [CrossRef] - Ates, C.; Eisfeld, A.; Rost, J.M. Motion of Rydberg atoms induced by resonant dipole—Dipole interactions. New J. Phys.
**2008**, 10, 045030. [Google Scholar] [CrossRef] - Cáceres, M.O.; Nizama, M. The quantum Levy walk. J. Phys. A Math. Theor.
**2010**, 43, 455306. [Google Scholar] [CrossRef] - Paparo, G.D.; Martin-Delgado, M.A. Google in a quantum network. Sci. Rep.
**2012**, 2, 444. [Google Scholar] [CrossRef] [Green Version] - Estrada, E. Path Laplacian matrices: Introduction and application to the analysis of consensus in networks. Lin. Algebra Appl.
**2012**, 436, 3373–3391. [Google Scholar] [CrossRef] [Green Version] - Riascos, A.P.; Mateos, J.L. Long-range navigation on complex networks using Lévy random walks. Phys. Rev. E
**2012**, 86, 056110. [Google Scholar] [CrossRef] [Green Version] - Estrada, E.; Hameed, E.; Hatano, N.; Langer, M. Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case. Lin. Algebra Appl.
**2017**, 523, 307–334. [Google Scholar] [CrossRef] [Green Version] - Estrada, E.; Hameed, E.; Langer, M.; Puchalska, A. Path Laplacian operators and superdiffusive processes on graphs. II. Two-dimensional lattice. Lin. Algebra Appl.
**2018**, 555, 373–397. [Google Scholar] [CrossRef] [Green Version] - Estrada, E.; Delvenne, J.C.; Hatano, N.; Mateos, J.L.; Metzler, R.; Riascos, A.P.; Schaub, M.T. Random multi-hopper model: Super-fast random walks on graphs. J. Complex Net.
**2018**, 6, 382–403. [Google Scholar] [CrossRef] [Green Version] - Riascos, A.P.; Mateos, J.L. Fractional quantum mechanics on networks: Long-range dynamics and quantum transport. Phys. Rev. E
**2015**, 92, 052814. [Google Scholar] [CrossRef] [PubMed] - Nandkishore, R.; Huse, D.A. Many-body localization and thermalization in quantum statistical mechanics. Ann. Rev. Cond. Matt. Phys.
**2015**, 6, 15. [Google Scholar] [CrossRef] [Green Version] - Altman, E.; Vosk, R. Universal dynamics and renormalization in many body localized systems. Ann. Rev. Cond. Matt. Phys.
**2015**, 6, 383. [Google Scholar] [CrossRef] [Green Version] - Flouris, K.; Jimenez, M.M.; Debus, J.D.; Herrmann, H.J. Confining massless Dirac particles in two-dimensional curved space. Phys. Rev. B
**2018**, 98, 155419. [Google Scholar] [CrossRef] [Green Version] - Gutman, D.B.; Protopopov, I.V.; Burin, A.L.; Gornyi, I.V.; Santos, R.A.; Mirlin, A.D. Energy transport in the Anderson insulator. Phys. Rev. B
**2016**, 93, 245427. [Google Scholar] [CrossRef] [Green Version] - Tikhonov, K.S.; Mirlin, A.D. Many-body localization transition with power-law interactions: Statistics of eigenstates. Phys. Rev. B
**2018**, 97, 214205. [Google Scholar] [CrossRef] [Green Version] - Hauke, P.; Heyl, M. Many-body localization and quantum ergodicity in disordered long-range Ising models. Phys. Rev. B
**2015**, 92, 134204. [Google Scholar] [CrossRef] [Green Version] - Nag, S.; Garg, A. Many-body localization in the presence of long-range interactions and long-range hopping. Phys. Rev. B
**2019**, 99, 224203. [Google Scholar] [CrossRef] [Green Version] - Stanković, M.S.; Vidanović, M.V.; Tricković, S.B. Summation of Some Trigonometric and Schlömilch Series. J. Comput. Anal. Appl.
**2003**, 5, 313–331. [Google Scholar] - Higham, N.J. Functions of Matrices: Theory and Computation; SIAM: Philadelphia, PA, USA, 2008. [Google Scholar]
- Arias, J.H.; Gómez-Gardeñes, J.; Meloni, S.; Estrada, E. Epidemics on plants: Modeling long-range dispersal on spatially embedded networks. J. Theor. Biol.
**2018**, 453, 1–3. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Estrada, E.; Gambuzza, L.V.; Frasca, M. Long-range interactions and network synchronization. SIAM J. Appl. Dyn. Syst.
**2018**, 17, 672–693. [Google Scholar] [CrossRef] - Powell, B.J. An introduction to effective low-energy Hamiltonians in condensed matter physics and chemistry. arXiv
**2009**, arXiv:0906.1640. [Google Scholar] - De Mendoza, I.H.; Pachón, L.A.; Gómez-Gardeñes, J.; Zueco, D. Synchronization in a semiclassical Kuramoto model. Phys. Rev. E
**2014**, 90, 052904. [Google Scholar] [CrossRef] [PubMed] [Green Version] - DeVille, L. Synchronization and stability for quantum Kuramoto. J. Stat. Phys.
**2019**, 174, 160–187. [Google Scholar] [CrossRef] [Green Version] - Fleischmann, R.; Geisel, T.; Ketzmerick, R.; Petschel, G. Quantum diffusion, fractal spectra, and chaos in semiconductor microstructures. Physica D
**1995**, 86, 171–181. [Google Scholar] [CrossRef] - Agliari, E.; Blumen, A.; Mülken, O. Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. A Math. Theor.
**2008**, 41, 445301. [Google Scholar] [CrossRef]

**Figure 1.**Illustration of the time evolution of the states of the nodes in a linear chain of five nodes using a classical (

**a**) and quantum (

**b**) transport dynamics with random initial condition at the nodes.

**Figure 2.**Probability of return for a given vertex of a ring graph with $n=100,001$ nodes using combinatorial Laplacian $\mathcal{L}$ (red dotted line) and using ${\mathcal{L}}_{s=2}$ (blue continuous line). The broken black line indicates the behavior $\overline{\pi}\propto {t}^{-1}.$

**Figure 3.**Plot of the transition probability ${\pi}_{50,j}$ for a quantum particle located at $t=0$ at position 51 of a ring having 101 nodes. (

**a**) quantum transport controlled by the Schrödinger equation with d-path Laplacian ${\mathcal{L}}_{s=2}$; (

**b**) quantum transport controlled by the Schrödinger equation without LRI ${\mathcal{L}}_{s\to \infty}$. The values of ${\pi}_{50,j}$ are given in logarithmic scale for better visualization. The colobar corresponds to the transition probability in logarithmic scale.

**Figure 4.**Probability at every node of a ring of 101 nodes using the Schrödinger equation without LRI (

**a**–

**d**) and with Mellin transform using $s=2$ (

**e**–

**h**). The probability at t = 0 is equal to one at the node 51 and zero elsewhere. The plots correspond to times $t=1$ (

**a**,

**e**), 10 (

**b**,

**f**), 100 (

**c**,

**g**), 1000 (

**g**,

**h**).

**Figure 5.**(

**a**) plot of the average return probability versus time in a complete graph ${K}_{100};$ (

**b**) transition probabilities between the node $v=50$ and the rest of the nodes in the complete graph ${K}_{100}.$ Only the zoomed region between nodes 40 and 60 is shown for the sake of visibility. The colobar in (

**b**) corresponds to the return probability.

**Figure 6.**Plot of the transition probability ${\pi}_{50,j}$ for a quantum particle located at $t=0$ at position $v=50$ of a barbell graph consisting of two cliques of 33 nodes separated by a path of 33 nodes (

**a**) without LRI (

**b**) and with ${\mathcal{L}}_{s=2}$. The central region consisting of nodes 34–66 is formed by the path. The colobars correspond to the transition probability.

**Figure 7.**Plot of the transition probability ${\pi}_{1,j}$ for a quantum particle located at $t=0$ at position $v=1$ of the tree illustrated in (

**a**) without LRI (

**b**), with ${\mathcal{L}}_{s=2}$ (

**c**) and with ${L}^{\gamma =0.25}$ (

**d**). The colobar corresponds to the transition probability.

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Estrada, E.
*d*-Path Laplacians and Quantum Transport on Graphs. *Mathematics* **2020**, *8*, 527.
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**AMA Style**

Estrada E.
*d*-Path Laplacians and Quantum Transport on Graphs. *Mathematics*. 2020; 8(4):527.
https://doi.org/10.3390/math8040527

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Estrada, Ernesto.
2020. "*d*-Path Laplacians and Quantum Transport on Graphs" *Mathematics* 8, no. 4: 527.
https://doi.org/10.3390/math8040527