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We derive a counting formula for the eigenvalues of Schr\"odinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
We investigate quantum circuits for graph representation learning, and propose equivariant quantum graph circuits (EQGCs), as a class of parameterized quantum circuits with strong relational inductive bias for learning over graph-structured…
The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction…
Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. These consist of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose…
We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges,…
Quantum graph states are critical resources for various quantum algorithms, and also determine essential interconnections in distributed quantum computing. There are two schemes for generating graph states probabilistic scheme and…
We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are…
We review the quantum spectral curve (QSC) formalism for anomalous dimensions of planar ${\cal\ N}=4$ SYM, including its $\gamma$-deformation. Leaving aside its derivation, we concentrate on formulation of the "final product" in its most…
The quantum hidden subgroup approach is an actively studied approach to solve combinatorial problems in quantum complexity theory. With the success of the Shor's algorithm, it was hoped that similar approach may be useful to solve the other…
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined…
Using a quantum processor to embed and process classical data enables the generation of correlations between variables that are inefficient to represent through classical computation. A fundamental question is whether these correlations…
We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an…
We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number…
Graph Neural Networks (GNNs) are powerful machine learning models that excel at analyzing structured data represented as graphs, demonstrating remarkable performance in applications like social network analysis and recommendation systems.…
We derive a set of spectral statistics whose power spectrum is characterized, in the case of chaotic quantum systems, by colored noise $1/f^{\gamma}$, where the integer parameter $\gamma$ critically depends on the specific energy-level…
We introduce n-particle quantum graphs with singular two-particle interactions in such a way that eigenfunctions can be given in the form of a Bethe ansatz. We show that this leads to a secular equation characterising eigenvalues of the…
We consider the challenging problem of statistical inference for exponential-family random graph models based on a single observation of a random graph with complex dependence. To facilitate statistical inference, we consider random graphs…
We present a non-perturbative analysis of the power-spectrum of energy level fluctuations in fully chaotic quantum structures. Focussing on systems with broken time-reversal symmetry, we employ a finite-$N$ random matrix theory to derive an…
We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and…