hermiemonterde - Research

A quantum walk in a park (PIMS Emergent Seminar, Oct '26).mp4

For a gentle overview of continuous quantum walks on graphs, here is my talk called A quantum walk in a park held at the PIMS Emergent Research Seminar last October 2026.

Research Interests

I am generally interested in the interaction between graph theory and linear algebra, and their applications. My particular interests include:

Algebraic graph theory: graph spectra, graphs with well-structured eigenbases, extremal problems on graph eigenvalues, association schemes,

Combinatorial matrix analysis: inverse eigenvalue problems for graphs, matrix decomposition, combinatorial matrices

Quantum information and computation: continuous-time quantum walks, perfect state transfer, vertex sedentariness, quantum search on marked vertices

Current projects

Continuous quantum walks

A continuous quantum walk on a graph describes the propagation of quantum states in a spin network. Here, a spin network is modelled by an undirected graph whose vertices and edges represent the qubits and their interactions in a network, respectively. Perfect state transfer (PST) refers to the accurate transmission of quantum states, while fractional revival (FR) represents quantum entanglement amongst some qubits in the network. These two phenomenon allow quantum algorithms to outperform classical counterparts. I am working to unravel the role of the structural and algebraic properties of graphs in the occurrence of PST, FR and other types of transfer in spin networks, including uniform mixing, pretty good state transfer, peak state transfer and vertex sedentariness.

Graph spectra and eigenvectors

The eigenvalues and eigenvectors of real symmetric matrices associated with a graph (e.g. adjacency, Laplacian and signless Laplacian matrix) reveal many of a graphs structural properties. At present, I am working to establish a connection between quantum walks and some graph invariants, such as the bounding matrix and the average mixing matrix of a graph. I also work on extremal problems on certain linear combinations of certain eigenvalues of the adjacency matrix of a graph. Moreover, I am interested on establishing the relationship between the nullity of the adjacency matrix of graph and its structure, as well as developing the theory of graphs with structured Laplacian eigenbases.

Inverse eigenvalue problem for graphs (IEPG)

The IEPG determines the possible spectra of real symmetric matrices M whose pattern of nonzero off-diagonal entries is described by the edges of a given graph G. That is, the (u,v) entry of M is nonzero if and only there is an edge between u and v in G. Right now, I am working on some IEPG parameters on multigraphs, as well as importing techniques from IEPG to further understand continuous quantum walks. I am also interested in the hollow version of IEPG for graphs with prescribed spectral property, achievable multiplicity partitions, minimum number of distinct eigenvalues, strong properties, and inverse eigenvector problems.

Association Schemes

An association scheme is a set of binary relations on a set S that partition S x S and satisfy strict regularity and symmetry conditions. Right now, we are working on near-factorizations in association schemes, which are factorizations of the matrix λ(J−I) into a product of 0-1 matrices S and T that belong to the association scheme, where λ>0 is an integer, J is the all-ones matrix and I is the identity matrix. I am also currently working on quantum walks on Cayley graphs and graphs in associations schemes, where I exploit the underlying structure of the scheme to produce desirable types of quantum state transfer in highly regular and sometimes highly symmetric graphs.

Published Papers

H. Monterde. Local ϵ-uniform mixing in continuous quantum walks, to appear in the Journal of Algebraic Combinatorics (2026).
K. Meagher, H. Monterde, Sedentary quantum walks on bipartite graphs, to appear in the Canadian Journal of Mathematics (2026).
S. Ojha, H. Monterde, H. Pal. Laplacian state transfer in graphs with involutions, Discrete Mathematics, 350(3): 115493, 2027. [arXiv]
H. Monterde, New results on vertex sedentariness, Discrete Mathematics, 349(4): 114959 (2026). [arXiv]
S. Kirkland, H. Monterde, Quantum walks on join graphs, Discrete Mathematics, 349(3): 114832 (2026). [arXiv]
H. Monterde, H. Pal, S. Kirkland, Laplacian quantum walks on blow-up graphs, Linear Algebra and its Applications (2025). [arXiv]
D. McLaren, H. Monterde, S. Plosker, Weakly Hadamard diagonalizable graphs and quantum state transfer, Linear and Multilinear Algebra, 73(17): 3763–3790 (2025). [arXiv]
C. Godsil, S. Kirkland, H. Monterde, Perfect state transfer between real pure states, SIAM Journal on Matrix Analysis and Applications, 46(3): 2093-2115 (2025). [arXiv]
S. Kim, H. Monterde, B. Ahmadi, A. Chan, S. Kirkland, S. Plosker, A generalization of quantum pair state transfer, Quantum Information Processing 23: 369 (2024). [arXiv]
B. Bhattacharjya, H. Monterde, H. Pal, Quantum walks on blow-up graphs, Journal of Physics A: Mathematical and Theoretical, 57(33): 335303 (2024). [arXiv]
S. Kirkland, H. Monterde, S. Plosker, Quantum state transfer between twins in weighted graphs, Journal of Algebraic Combinatorics 58: 623–649 (2023). [arXiv]
H. Monterde, Fractional revival between twin vertices, Linear Algebra and its Applications 676: 25-43 (2023). [arXiv]
H. Monterde, Sedentariness in quantum walks, Quantum Information Processing 22: 273 (2023). [arXiv]
H. Monterde, Strong cospectrality and twin vertices in weighted graphs, The Electronic Journal of Linear Algebra 38: 494-518 (2022). [arXiv]
H. Monterde, A. Paras, On the sum of strictly k-zero matrices, Science Diliman 29(1): 37-52 (2017).

Submitted and in preparation

K. Meagher, H. Monterde, Tung T. Nguyen, Sedentariness in Cayley graphs, submitted (2026).
J. Cooper, H. de Vera, H. Monterde, S.A. Talebpour, Xuzhu Wang, Quantum walks and graph operations, submitted (2026).
A. Herman, A. Lacaze-Masmonteil, K. Meagher, H. Monterde, Near-factorizations in association schemes, submitted (2026).
S. Kirkland, H. Monterde, S. Plosker, Perfect (s,r)-state transfer, submitted (2026).
H. Monterde, H. Pal. More graphs with pair state transfer, submitted (2026).
H. Kumar, L. Liu, H. Monterde, S. Pragada, M. Tait. Maximum spectral sum of graphs, submitted (2026).
L. de Lima, R. Del-Vecchio, H. Monterde, H. Teixeira. Structured eigenbases and pair state transfer on threshold graphs, submitted (2026).
C. Godsil, S. Kirkland, S. Mohapatra, H. Monterde, H. Pal, Quantum walks on finite and bounded infinite graphs, submitted (2026).

Theses

Quantum pure state transfer, PhD Dissertation, UManitoba, 2025
Quantum state transfer between twins in graphs, MSc Thesis, UManitoba, 2021
On the sum of strictly k-zero matrices, MSc Thesis, UP Diliman, 2015

Graduate supervision

Quantum walks on graphs, PIMS First-Year Interest Groups

PIMS FYIG graduate student collaborators

Quantum Search Working Group, BIRS Algebraic Graph Theory 2025

Collaborators

Bahman Ahmadi - Bikash Bhattacharjya - Ludovick Bouthat - Ada Chan - Joy Cooper - Leonardo de Lima - Homer de Vera - Renata del Vecchio - Shaun Fallat - Chris Godsil - Tracy Hall - Allen Herman - Mark Kempton - Sooyeong Kim - Stephen Kirkland - Hitesh Kumar - Alice Lacaze-Masmonteil - Gabor Lippner - Lele Liu - Kate Lorenzen - Darian McLaren - Karen Meagher - Sarojini Mohapatra - Samir Mondal - Shahla Nasserasr - Tung T. Nguyen - Swornalata Ojha - Hiranmoy Pal - Agnes Paras - Johnna Parenteau - Sarah Plosker - Shivaram Pragada - Carolyn Reinhart - Bryan Shader - Mike Tait - Aysa Tajeri - Christino Tamon - Alireza Talebpour - Heber Teixeira - Ruth Wang - Tom Wong - Xiaohong Zhang

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