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Quantum Information Processing
Published in: Quantum Information Processing 3/2021

01-03-2021

Quantum walks defined by digraphs and generalized Hermitian adjacency matrices

Authors: Sho Kubota, Etsuo Segawa, Tetsuji Taniguchi

Published in: Quantum Information Processing | Issue 3/2021

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Abstract

We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix and exhibit explicit formulas of supports of their square. Also, we provide tables on the identification of digraphs by their eigenvalues.
previous article Convergence theorems on multi-dimensional homogeneous quantum walks
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Footnotes
1
This dynamics of the quantum walk treated on a half plain of the two-dimensional lattice [6] can be regarded as a quantum walk on a digraph, which is not symmetric, because a quantum walker moves to x and y directions, alternatively.
 
2
Quantum walks we will define are a generalization of the Grover walks. As we will see later, walkers in our quantum walk are on \(G^{\pm }\). In the case of Grover walks, the connectedness of graphs is usually assumed, so we need assume that \(G^{\pm }\) is connected.
 
Literature
1.
2.
Biamonte, M., Faccin, M.: Complex networks from classical to quantum. Commun. Phys. 2(1), 1–10 (2019)CrossRef
3.
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, Berlin (2011)MATH
4.
Chagas, B., Portugal, R.: Discrete-Time Quantum Walks on Oriented Graphs. arXiv:2001.04814v2, (2020)
5.
Emms, D., Hancock, E.R., Severini, S., Wilson, R.C.: A matrix representation of graphs and its spectrum as a graph invariant. Electr. J. Combin. 13(1), (2006)
6.
Endo, T., Konno, N., Obuse, H., Segawa, E.: Sensitivity of quantum walks to boundary of two-dimensional lattices: approaches from the CGMV method and topological phases. J. Phys. A: Math. Theor. 50, 455302 (2017)ADSCrossRef
7.
Flamini, F., Sciarrino, F., Spagnolo, N.: Photonic quantum information processing: a review. Rep. Prog. Phys. 82(1), 016001 (2018)ADSCrossRef
8.
Guo, K.: Quantum walks on strongly regular graphs. Master’s thesis, University of Waterloo (2010)
9.
10.
Godsil, C., Lato, S.: Perfect State Transfer on Oriented Graphs. arXiv:2002.04666, (2020)
11.
12.
13.
Gräfe, M., Szameit, A.: Integrated photonic quantum walks. J. Phys. B: Atom. Mol. Opt. Phys. 53(7), 073001 (2020)ADSCrossRef
14.
Higuchi, Y., Konno, N., Sato, I., Segawa, E.: A note on the discrete-time evolutions of quantum walk on a graph. J. Math. Ind. 5(2013B–3), 103–109 (2013)MathSciNetMATH
15.
Higuchi, Y., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267, 4197–4235 (2014)MathSciNetCrossRef
16.
Higuchi, Y., Konno, N., Sato, I., Segawa, E.: Periodicity of the discrete-time quantum walk on a finite graph. Interdiscip. Inf. Sci. 23, 75–86 (2017)MathSciNet
17.
Hoyer, S., Meyer, D.A.: Faster transport with a directed quantum walk. Phys. Rev. A 79, 024307 (2009)ADSCrossRef
18.
Kubota, S.: Strongly regular graphs with the same parameters as the symplectic graph. Sib. Elektron. Mat. Izv. 13, 1314–1338 (2016)MathSciNetMATH
19.
Konno, N., Sato, I.: On the relation between quantum walks and zeta functions. Quant. Inf. Process. 11, 341–349 (2012)MathSciNetCrossRef
20.
Konno, N., Sato, I., Segawa, E.: Phase measurement of quantum walks: application to structure theorem of the positive support of the Grover Walk. Electr. J. Combin. 26, 26 (2019)MathSciNetCrossRef
21.
Kubota, S., Segawa, S., Taniguchi, T., Yoshie, Y.: Periodicity of Grover walks on generalized Bethe trees. Linear Algebra Appl. 554, 371–391 (2018)MathSciNetCrossRef
22.
Lato, S.: Quantum Walks on Oriented Graphs. Masters Thesis (2019)
23.
Liu, J., Li, X.: Hermitian-adjacency matrices and Hermitian energies of mixed graphs. Linear Algebra Appl. 466, 182–207 (2015)MathSciNetCrossRef
24.
25.
Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2013)MATH
26.
27.
28.
Ren, P., Aleksic, T., Emms, D., Wilson, R.C., Hancock, E.R.: Quantum walks, Ihara zeta functions and cospectrality in regular graphs. Quant. Inf. Process. 10, 405–417 (2011)MathSciNetCrossRef
29.
Severini, S.: The underlying digraph of a coined quantum random walk. Preprint arXiv:quant-ph/0210055, (2002)
30.
31.
Yoshie, Y.: A characterization of the graphs to induce periodic Grover walk. Yokohama Math. J. 63, 9–23 (2017)MathSciNetMATH
32.
33.
Zhan, H.: Discrete Quantum Walks on Graphs and Digraphs, Ph.D. thesis (2018)
Metadata
Title
Quantum walks defined by digraphs and generalized Hermitian adjacency matrices
Authors
Sho Kubota
Etsuo Segawa
Tetsuji Taniguchi
Publication date
01-03-2021
Publisher
Springer US
Published in
Quantum Information Processing / Issue 3/2021
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-021-03033-z

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