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Very recently Ma and Wu \cite{wu2024generalization} obtained a generalization of Fielder's lemma and applied to find adjacency, Laplacian, and signless Laplacian spectra of $P_n-$ product of commuting graphs. In this paper, we give a generalization of Fielder's lemma applying which not only one gets generalized result in \cite{wu2024generalization} as a particular case, but also one can find several kind of spectra of $H$-product of graphs when $H$ is an arbitrary graph. Moreover, we compute adjacency spectrum of $H-$ product of commuting graphs and universal adjacency spectrum of $H-$ product of commuting regular graphs.

The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we determine the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases.

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple graph $H$ with adjacency matrix $A(H)$ and degree diagonal matrix $D(H)$, the universal adjacency matrix is $U(H)= \alpha A(H)+\beta D(H)+ \gamma I +\eta J$, where $\alpha (\neq 0), \beta, \gamma, \eta \in \mathbb{R}$, $I$ is the identity matrix and $J$ is the all-ones matrix of suitable order. One can study many graph-associated matrices, such as adjacency, Laplacian, signless Laplacian, Seidel etc. in a unified manner through the universal adjacency matrix of a graph. Here we study universal adjacency eigenvalues and eigenvectors of power graphs, proper power graphs and their complements on the group $\mathbb{Z}_n$, dihedral group ${D}_n$, and the generalized quaternion group ${Q}_n$. Spectral results of no kind for the complement of power graph on any group were obtained before. We determine the full spectrum in some particular cases. Moreover, several existing results can be obtained as very specific cases of some results of the paper.

For a three-level system monitored by an ancilla, we show that quantum Zeno effect can be employed to control quantum jump for error correction. Further, we show that we can realize cNOT gate, and effect dense coding and teleportation. We believe that this work paves the way to generalize the control of a qudit.

We construct a double-toric surface code by exploiting the planar tessellation using a rhombus-shaped tile. With n data qubits, we are able to encode at least n/3 logical qubits or quantum memories. By a suitable arrangement of the tiles, the code achieves larger distances, leading to significant error-correcting capability. We demonstrate the robustness of the logical qubits thus obtained in the presence of external noise. We believe that the optimality of the code presented here will pave the way for design of efficient scalable architectures.

We employ the stochastic path-integral formalism and action principle for continuous quantum measurements - the Chantasri-Dressel-Jordan (CDJ) action formalism [1, 2] - to understand the stages in which quantum Zeno effect helps control the states of a simple quantum system. The detailed dynamics of a driven two-level system subjected to repeated measurements unravels a myriad of phases, so to say. When the detection frequency is smaller than the Rabi frequency, the oscillations slow down, eventually coming to a halt at an interesting resonance when measurements are spaced exactly by the time of transition between the two states. On the other hand, in the limit of large number of repeated measurements, the dynamics organizes itself in a rather interesting way about two hyperbolic points in phase space whose stable and unstable directions are reversed. Thus, the phase space flow occurs from one hyperbolic point to another, in different ways organized around the separatrices. We believe that the systematic treatment presented here paves the way for a better and clearer understanding of quantum Zeno effect in the context of quantum error correction.

We present a systematic treatment of a 0-\texorpdfstring{$\pi$}{Lg} qubit in the presence of a time-dependent external flux. A gauge-invariant Lagrangian and the corresponding Hamiltonian is obtained. The effect of the flux noise on the qubit relaxation is obtained using perturbation theory. Under a time-dependent drive of sinusoidal form, the survival probability, and transition probabilities have been studied for different strengths and frequencies. The driven qubit is shown to possess coherent oscillations among two distinct states for a weak to moderate strength close to resonant frequencies of the unperturbed qubit. The parameters can be chosen to prepare the system in its ground state. This feature paves the way to prolong the lifetime by combining ideas from weak measurement and quantum Zeno effect. We believe that this is an important variation of a topologically protected qubit which is tunable.

Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical methods, scalable algorithms and re-design of current state-of-the art numerical solvers are required for scaling to these machines with minimal overheads. One such approach for partial differential equations based solvers involves computation of spatial derivatives with possibly delayed or asynchronous data using high-order asynchrony-tolerant (AT) schemes to facilitate mitigation of communication and synchronization bottlenecks without affecting the numerical accuracy. In the present study, an effective methodology of implementing temporal discretization using a multi-stage Runge-Kutta method with AT schemes is presented. Together these schemes are used to perform asynchronous simulations of canonical reacting flow problems, demonstrated in one-dimension including auto-ignition of a premixture, premixed flame propagation and non-premixed autoignition. Simulation results show that the AT schemes incur very small numerical errors in all key quantities of interest including stiff intermediate species despite delayed data at processing element (PE) boundaries. For simulations of supersonic flows, the degraded numerical accuracy of well-known shock-resolving WENO (weighted essentially non-oscillatory) schemes when used with relaxed synchronization is also discussed. To overcome this loss of accuracy, high-order AT-WENO schemes are derived and tested on linear and non-linear equations. Finally the novel AT-WENO schemes are demonstrated in the propagation of a detonation wave with delays at PE boundaries.

The optical conductivity in a Kondo lattice system is presented in terms of the memory function formalism. I use Kondo-lattice Hamiltonian for explicit calculations. I compute the frequency dependent imaginary part of the memory function ($M^{\p\p}(\om)$), and the real part of the memory function $M^{\p}(\om)$ by using the Kramers-Kronig transformation. Optical conductivity is computed using the generalized Drude formula. I find that high frequency tail of the optical conductivity scales as $\sigma(\om) \sim \frac{1}{\om}$ instead of the Drude $\frac{1}{\om^2}$ law. Such a behaviour is seen in strange metals. My work points out that it may be the magnetic scattering mechanisms that are important for the anomalous behaviour of strange metals., Comment: 9

A colossal insulator-to-metal transition in high-spin pyrite phase of $MnS_{2}$ has been experimentally observed \cite{colomns2}. There are two possibilities behind this colossal insulator-to-metal transition: (1) migration of $Mn$ electrons to unoccupied $S^{2-}_{2}$ antibonding states under pressure which leads to conducting ligand states and hence metallic transition, and (2) possibility of band crossing transition. We have analyzed this experimental obervation theoretically using a toy statistical model and found that the transition is due to the migration of electrons from the transition metal ions to the ligand sites (i.e. the possibility (1)). The calculated resistivity compares well with the experimental data within the fitting parameters of the model., Comment: 5

I extend the calculations represented in \cite{konav} regarding the resistivity in Kondo lattice materials from $3d$ syatem to $2d$ systems. In the present work I consider a 2d system, and memory function is computed. However, results found in 2d case are different from 3d system . I find that in $2d$ in low temperature regime($ k_{B}T\ll \mu_d$) resistivity shows power law($\frac{1}{T}$) behaviour and in the high temeprature regime($ k_{B}T\gg\mu_d$) resistivity varies linearly with temperature. In $3d$ these behaviours are as $\frac{1}{T}$ and as $T^{\frac{3}{2}}$ respectively., Comment: 9

Direct numerical simulations (DNS) are an indispensable tool for understanding the fundamental physics of turbulent flows. Because of their steep increase in computational cost with Reynolds number ($R_{\lambda}$), well-resolved DNS are realizable only on massively parallel supercomputers, even at moderate $R_{\lambda}$. However, at extreme scales, the communications and synchronizations between processing elements (PEs) involved in current approaches become exceedingly expensive and are expected to be a major bottleneck to scalability. In order to overcome this challenge, we developed algorithms using the so-called Asynchrony-Tolerant (AT) schemes that relax communication and synchronization constraints at a mathematical level, to perform DNS of decaying and solenoidally forced compressible turbulence. Asynchrony is introduced using two approaches, one that avoids synchronizations and the other that avoids communications. These result in periodic and random delays, respectively, at PE boundaries. We show that both asynchronous algorithms accurately resolve the large-scale and small-scale motions of turbulence, including instantaneous and intermittent fields. We also show that in asynchronous simulations the communication time is a relatively smaller fraction of the total computation time, especially at large processor count, compared to standard synchronous simulations. As a consequence, we observe improved parallel scalability up to $262144$ processors for both asynchronous algorithms.

We theoretically analyse D.C. resistivity($\rho$) in the Kondo-lattice model using the powerful memory function approach. The complete temperature evolution of $\rho$ is investigated using the W\"{o}lfle-G\"{o}tze expansion of the memory function. The resistivity in this model originates due to spin-flip magnetic scattering of conduction $s$-electron off the quasi-localized $d$ or $f$ electron spins. We find the famous resistivity upturn at lower temperature regime ($k_B T<<\mu_d$), where $\mu_d$ is the effective chemical potential of $d$-electrons. In the high temperature regime $(\mu_d<

A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and stability. The resulting mathematical framework reduces to a minimization problem subjected to equality and inequality constraints. We show that the framework can provide analytical results for optimal schemes and their numerical performance including, for example, the type of errors that appear for spectrally optimal schemes. By coupling the problem in this unified framework, one can effectively decouple the requirements for order of accuracy and spectral resolution, for example. Alternatively, we show how the framework exposes the tradeoffs between e.g. accuracy and stability and how this can be used to construct explicit schemes that remain stable with very large time steps. We also show how spectrally optimal schemes only bias odd-order derivatives to remain stable, at the expense of accuracy, while leaving even-order derivatives with symmetric coefficients. Schemes constructed within this framework are tested for diverse model problems with an emphasis on reproducing the physics accurately.

We calculate resistivity in the paramagnetic phase just above the curie temperature in a $2d$ ferromagnetic metal. The required dynamical susceptibility in the formalism of resistivity is calculated within the Random Phase Approximation(RPA). The mechanism of resistivity is magnetic scattering in which $s$-band electrons are scattered off the magnetic spin fluctuations of d-band electrons. We use the $s$-$d$ Hamiltonian formalism. We find that near the quantum critical point the resistivity in $2d$ scales as $T^{\frac{4}{3}}$, whereas in $3d$ it scales as $T^{\frac{5}{3}}$. In contrast to it, resistivity due to phonon scattering is given by $T^5$ in low temperature limit as is well known. Our RPA result agrees with the Self-Consistence Renormalisation(SCR) theory result., Comment: 11 pages

An introduction to the Zwanzig-Mori-G\"{o}tze-W\"{o}lfle memory function formalism (or generalized Drude formalism) is presented. This formalism is used extensively in analyzing the experimentally obtained optical conductivity of strongly correlated systems like cuprates and Iron based superconductors etc. For a broader perspective both the generalised Langevin equation approach and the projection operator approach for the memory function formalism are given. The G\"{o}tze-W\"{o}lfle perturbative expansion of memory function is presented and its application to the computation of the dynamical conductivity of metals is also reviewd. This review of the formalism contains all the mathematical details for pedagogical purposes., Comment: 17 pages; 2 figures and 2 tables