arXiv subjects

Bibhas Adhikari

Publications and source records attributed to Bibhas Adhikari.

At least 37 records · Page 2Linked to original sources

arXiv2023

Modeling interdisciplinary interactions among Physics, Mathematics & Computer Science

Interdisciplinarity has over the recent years have gained tremendous importance and has become one of the key ways of doing cutting edge research. In this paper we attempt to model the citation flow across three different fields -- Physics (PHY), Mathematics (MA) and Computer Science (CS). For instance, is there a specific pattern in which these fields cite one another? We carry out experiments on a dataset comprising more than 1.2 million articles taken from these three fields. We quantify the citation interactions among these three fields through temporal bucket signatures. We present numerical models based on variants of the recently proposed relay-linking framework to explain the citation dynamics across the three disciplines. These models make a modest attempt to unfold the underlying principles of how citation links could have been formed across the three fields over time.

cs.DL↗

arXiv2023

A characterization of orthogonal permutative matrices of order 4

Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and rational orthogonal permutative matrices of order $4.$ We show that any such matrix can always be expressed as a linear combination of up to four permutation matrices. Finally we determine several matrix spaces generated by linearly independent permutation matrices such that any orthogonal matrix in these spaces is always permutative or direct sum of orthogonal permutative matrices up to permutation of its rows and columns.

math.CO↗

arXiv2022

One-connection rule for structural equation models

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph $G=(V, D,B)$ is parameterized by a rational function with parameters for each vertex and edge in $G$. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when $D$, the directed part of the mixed graph $G$, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from $D$ through some small operations. This closed form expression then allows us to show that if $G$ is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

math.ST↗

arXiv2022

Localization of two-dimensional quantum walks defined by generalized Grover coins

Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.

quant-ph↗

arXiv2022

Limit theorems and localization of three state quantum walks on a line defined by generalized Grover coins

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on one dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position, for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.

quant-ph↗

arXiv2021

Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

In this paper, we propose a unified approach for solving structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, $\star$-even and $\star$-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases. The obtained results are supported with numerical examples.

math.NA↗

arXiv2020

Perfect state transfer on hypercubes and its implementation using superconducting qubits

We propose a protocol for perfect state transfer between any pair of vertices in a hypercube. Given a pair of distinct vertices in the hypercube we determine a sub-hypercube that contains the pair of vertices as antipodal vertices. Then a switching process is introduced for determining the sub-hypercube of a memory enhanced hypercube that facilitates perfect state transfer between the desired pair of vertices. Furthermore, we propose a physical architecture for the pretty good state transfer implementation of our switching protocol with fidelity arbitrary close to unity, using superconducting transmon qubits with tunable couplings. The switching is realised by the control over the effective coupling between the qubits resulting from the effect of ancilla qubit couplers for the graph edges. We also report an error bound on the fidelity of state transfer due to faulty implementation of our protocol.

quant-ph↗

arXiv2020

Preserving spectral properties of structured matrices under structured perturbations

This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues while keeping the Jordan chains unchanged. Moreover, a no spillover structured perturbation of a structured matrix is obtained whose rank is equal to the number of eigenvalues (including multiplicities) which are modified, and in addition, preserves the rest of the eigenvalues and the corresponding Jordan chains which need not be known. The specific structured matrices considered in this paper form Jordan and Lie algebra corresponding to an orthosymmetric scalar product.

math.NA↗

arXiv2020

A model for the spread of an epidemic from local to global: A case study of COVID-19 in India

In this paper we propose an epidemiological model for the spread of COVID-19. The dynamics of the spread is based on four fundamental categories of people in a population: Tested and infected, Non-Tested but infected, Tested but not infected, and non-Tested and not infected. The model is based on two levels of dynamics of spread in the population: at local level and at the global level. The local level growth is described with data and parameters which include testing statistics for COVID-19, preventive measures such as nationwide lockdown, and the migration of people across neighboring locations. In the context of India, the local locations are considered as districts and migration or traffic flow across districts are defined by normalized edge weight of the metapopulation network of districts which are infected with COVID-19. Based on this local growth, state level predictions for number of people tested with COVID-19 positive are made. Further, considering the local locations as states, prediction is made for the country level. The values of the model parameters are determined using grid search and minimizing an error function while training the model with real data. The predictions are made based on the present statistics of testing, and certain linear and log-linear growth of testing at state and country level. Finally, it is shown that the spread can be contained if number of testing can be increased linearly or log-linearly by certain factors along with the preventive measures in near future. This is also necessary to prevent the sharp growth in the count of infected and to get rid of the second wave of pandemic.

physics.soc-ph↗

arXiv2020

Why lockdown : On the spread of SARS-CoV-2 in India, a network approach

We analyze the time series data of number of districts or cities in India that are affected by COVID-19 from March 01, 2020 to April 17, 2020. We study the data in the framework of time series network data. The networks are defined by using the geodesic distances of the districts or cities specified by the latitude and longitude coordinates. We particularly restrict our analysis to all but districts in the north-eastern part of India. Unlike recent studies on the projection of the number of people infected with SARS-CoV-2 in the near future, in this note, the emphasis is on understanding the dynamics of the spread of the virus across the districts of India. We perform spectral and structural analysis of the model networks by considering several measures, notably the spectral radius, the algebraic connectivity, the average clustering coefficient, the average path length and the structure of the communities. Furthermore, we study the overall expansion properties given by the number of districts or cities before and after lockdown. These studies show that lockdown has a significant impact on the spread of SARS-CoV-2 in districts or cities over long distances. However, this impact is only observed after approximately two weeks of lockdown. We speculate that this happened due to the insufficient number of tests for COVID-19 before the lockdown which could not stop the movement of people infected with the virus but not detected, over long distances.

cs.SI↗

arXiv2020

On the eigenvalue region of permutative doubly stochastic matrices

This paper is devoted to the study of eigenvalue region of the doubly stochastic matrices which are also permutative, that is, each row of such a matrix is a permutation of any other row. We call these matrices as permutative doubly stochastic (PDS) matrices. A method is proposed to obtain symbolic representation of all PDS matrices of order $n$ by finding equivalence classes of permutationally similar symbolic PDS matrices. This is a hard problem in general as it boils down to finding all Latin squares of order $n.$ However, explicit symbolic representation of matrices in these classes are determined in this paper when $n=2, 3, 4.$ It is shown that eigenvalue regions are same for doubly stochastic matrices and PDS matrices when $n=2, 3.$ It is also established that this is no longer true for $n=4,$ and two line segments are determined which belong to the eigenvalue region of doubly stochastic matrices but not in the eigenvalue region of PDS matrices. Thus a conjecture is developed for the boundary of the eigenvalue region of PDS matrices of order $4.$ Finally, inclusion theorems for eigenvalue region of PDS matrices are proved when $n\geq 2.$

math.CO↗

arXiv2020

Periodicity of lively quantum walks on cycles with generalized Grover coin

In this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as a linear sum of permutation matrices, which is a generalization of the Grover matrix. First we provide a complete characterization of orthogonal matrices of order $3\times 3$ which are linear sum of permutation matrices. Consequently, we determine several groups of complex, real and rational orthogonal matrices. We establish that an orthogonal matrix of order $3\times 3$ is a linear sum of permutation matrices if and only if it is permutative. Finally we determine period of lively quantum walk on cycles when the coin operator belongs to the group of orthogonal (real) linear sum of permutation matrices.

quant-ph↗

arXiv2020

Updating structured matrix pencils with no spillover effect on unmeasured spectral data and deflating pair

This paper is devoted to the study of perturbations of a matrix pencil, structured or unstructured, such that a perturbed pencil will reproduce a given deflating pair while maintaining the invariance of the complementary deflating pair. If the latter is unknown, it is referred to as no spillover updating. The specific structures considered in this paper include symmetric, Hermitian, $\star$-even, $\star$-odd and $\star$-skew-Hamiltonian/Hamiltonian pencils. This study is motivated by the well-known Finite Element Model Updating Problem in structural dynamics, where the given deflating pair represents a set of given eigenpairs and the complementary deflating pair represents the remaining larger set of eigenpairs. Analytical expressions of structure preserving no spillover updating are determined for deflating pairs of structured matrix pencils. Besides, parametric representations of all possible unstructured perturbations are obtained when the complementary deflating pair of a given unstructured pencil is known. In addition, parametric expressions are obtained for structured updating with certain desirable structures which relate to existing results on structure preservation of a symmetric positive definite or semi definite matrix pencil.

math.NA↗

arXiv2019

Corona product of signed graphs and its application to signed network modelling

The notion of corona of two graphs was introduced by Frucht and Harary in 1970. In this paper we generalize their definition of corona product of two graphs and introduce corona product of two signed graphs by utilizing the framework of marked graphs, which was introduced by Beineke and Harary in 1978. We study the structural and spectral properties of corona product of signed graphs. Further we define signed corona graphs by considering corona product of a fixed small signed graph with itself iteratively, and we call the small graph as the seed graph for the corresponding corona graphs. Signed corona graphs can be employed as a signed network generative model for large growing signed networks. We study structural properties of corona graphs that include statistics of signed links, all types of signed triads and degree distribution. Besides we analyze algebraic conflict of signed corona graphs generated by specially structured seed graphs. Finally we show that a suitable choice of a seed graph can produce corona graphs which preserve properties of real signed networks.

math.CO↗

arXiv2019

The rise and rise of interdisciplinary research: Understanding the interaction dynamics of three major fields -- Physics, Mathematics & Computer Science

The distinction between sciences is becoming increasingly more artificial -- an approach from one area can be easily applied to the other. More exciting research nowadays is happening perhaps at the interfaces of disciplines like Physics, Mathematics and Computer Science. How do these interfaces emerge and interact? For instance, is there a specific pattern in which these fields cite each other? In this article, we investigate a collection of more than 1.2 million papers from three different scientific disciplines -- Physics, Mathematics, and Computer Science. We show how over a timescale the citation patterns from the core science fields (Physics, Mathematics) to the applied and fast-growing field of Computer Science have drastically increased. Further, we observe how certain subfields in these disciplines are shrinking while others are becoming tremendously popular. For instance, an intriguing observation is that citations from Mathematics to the subfield of machine learning in Computer Science in recent times are exponentially increasing.

cs.DL↗

arXiv2019

Condition for zero and non-zero discord in graph Laplacian quantum states

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph-theoretic conditions for zero and non-zero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and $X$-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.

quant-ph↗

arXiv2018

Proposal for Dimensionality Testing in Quantum Private Query

Recently, dimensionality testing of a quantum state has received extensive attention (Ac{í}n et al. Phys. Rev. Letts. 2006, Scarani et al. Phys. Rev. Letts. 2006). Security proofs of existing quantum information processing protocols rely on the assumption about the dimension of quantum states in which logical bits are encoded. However, removing such assumption may cause security loophole. In the present paper, we show that this is indeed the case. We choose two players' quantum private query protocol by Yang et al. (Quant. Inf. Process. 2014) as an example and show how one player can gain an unfair advantage by changing the dimension of subsystem of a shared quantum system. To resist such attack we propose dimensionality testing in a different way. Our proposal is based on CHSH like game. As we exploit CHSH like game, it can be used to test if the states are product states for which the protocol becomes completely vulnerable.

quant-ph↗

arXiv2018

Construction of cospectral graphs

Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory specially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a clustered graph.

math.CO↗