Quantum Graph Theory Research Articles

Application of quantum graph theory to metamaterial design: Negative refraction of acoustic waveguide modes

We leverage quantum graph theory to quickly and accurately characterize acoustic metamaterials comprising networks of interconnected pipes. Anisotropic bond lengths are incorporated in the model that correspond to space-coiled acoustic structures to exhibit dispersion spectra reminiscent of hyperbolic metamaterials. We construct two metasurfaces with embedded graph structure and, motivated by the graph theory, infer and fine-tune their dispersive properties to engineer nonresonant negative refraction of acoustic surface waves at their interface. Agreement between the graph model, full-wave simulations, and experiments bolsters quantum graph theory as a new paradigm for metamaterial design. Published by the American Physical Society 2024

Geometric spectral theory of quantum graphs

Geometric spectral theory of quantum graphs

Quantum graph wave external triggering: Energy transfer and damping.

The propagation of wave trains resulting from a local external trigger inside a network described by a metric graph is analyzed using quantum graph theory. The external trigger is a finite-time perturbation imposed at one vertex of the graph, leading to a consecutive wave train into the network, supposedly at rest before the applied external perturbation. A complete analytical solution for the induced wave train is found having a specific spectrum as well as mode's amplitudes. Furthermore the precise condition by which the external trigger can transfer a maximal energy to any specific natural mode of the quantum graph is derived. Finally, the wave damping associated with boundary-layer dissipation is computed within a multiple time-scale asymptotic analysis. Exponential damping rates are explicitly found related to their corresponding mode's eigenvalue. Each mode energy is then obtained, as well as their exponential damping rate. The relevance of these results to the physics of waves within networks are discussed.

Sturm-Liouville problems and global bounds by small control sets and applications to quantum graphs

We develop a Logvinenko–Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on L2-norms of eigenfunctions – and linear combinations thereof – in terms of their L2- and W1,2-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs.

RECOVERING THE SHAPE OF A QUANTUM CATERPILLAR TREE BY TWO SPECTRA

existence of co-spectral (iso-spectral) graphs is a well-known problem of the classical graph theory. However, co-spectral graphs exist in the theory of quantum graphs also. In other words, the spectrum of the Sturm-Liouville problem on a metric graph does not determine alone the shape of the graph. Сo-spectral trees also exist if the number of vertices exceeds eight. We consider two Sturm-Liouville spectral problems on an equilateral metric caterpillar tree with real L2 (0,l) potentials on the edges. In the first (Neumann) problem we impose standard conditions at all vertices: Neumann boundary conditions at the pendant vertices and continuity and Kirchhoff’s conditions at the interior vertices. The second (Dirichlet) problem differs from the first in that in the second problem we set the Dirichlet condition at the root (one of the pendant vertices of the stalk of the caterpillar tree, i.e. the central path of it). Using the asymptotics of the eigenvalues of these two spectra we find the determinant of the normalized Laplacian of the tree and the determinant of the prime submatrix of the normalized laplacian obtained by deleting the row and the column corresponding to the root. Expanding the fraction of these determinants into continued fraction we receive full information on the shape of the tree. In general case this continued fraction is branched. We prove that in the case of a caterpillar tree the continued fraction does not branch and the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree. A concrete example is shown. The known pair of co-spectral trees with minimal number (eight) of vertices belongs to the class of caterpillar trees. Keywords: metric graph, tree, pendant vertex, interior vertex, edge, caterpillar tree, Sturm-Liouville equation, potential, eigenvalues, spectrum, Dirichlet boundary condition, Neumann boundary condition, root, continued fraction, adjacency matrix, prime submatrix, normalized Laplacian

Boosted Binary Quantum Classifier via Graphical Kernel.

In terms of the logical structure of data in machine learning (ML), we apply a novel graphical encoding method in quantum computing to build the mapping between feature space of sample data and two-level nested graph state that presents a kind of multi-partite entanglement state. By implementing swap-test circuit on the graphical training states, a binary quantum classifier to large-scale test states is effectively realized in this paper. In addition, for the error classification caused by noise, we further explored the subsequent processing scheme by adjusting the weights so that a strong classifier is formed and its accuracy is greatly boosted. In this paper, the proposed boosting algorithm demonstrates superiority in certain aspects as demonstrated via experimental investigation. This work further enriches the theoretical foundation of quantum graph theory and quantum machine learning, which may be exploited to assist the classification of massive-data networks by entangling subgraphs.

Upper Bound for the Diameter of a Tree in the Quantum Graph Theory

Upper Bound for the Diameter of a Tree in the Quantum Graph Theory

Some examples of quantum graphs

We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space $$M_2$$ . Secondly, we apply the theory of 2-cocycle deformations to Cayley graphs of abelian groups. This defines a twisting procedure that produces a quantum graph, which is quantum isomorphic to the original one. For instance, we define the anticommutative hypercube graphs. Thirdly, we construct an example of a quantum graph, which is not quantum isomorphic to any classical graph.

A mathematical modelling to detect sickle cell anemia using Quantum graph theory and Aquila optimization classifier

Recently genetic disorders are the most common reason for human fatality. Sickle Cell anemia is a monogenic disorder caused by A-to-T point mutations in the β-globin gene which produces abnormal hemoglobin S (Hgb S) that polymerizes at the state of deoxygenation thus resulting in the physical deformation or erythrocytes sickling. This shortens the expectancy of human life. Thus, the early diagnosis and identification of sickle cell will aid the people in recognizing signs and to take treatments. The manual identification is a time consuming one and might outcome in the misclassification of count as there is millions of red blood cells in one spell. So as to overcome this, data mining approaches like Quantum graph theory model and classifier is effective in detecting sickle cell anemia with high precision rate. The proposed work aims at presenting a mathematical modeling using Quantum graph theory to extract elasticity properties and to distinguish them as normal cells and sickle cell anemia (SCA) in red blood cells. Initially, input DNA sequence is taken and the elasticity property features are extracted by using Quantum graph theory model at which the formation of spanning tree is made followed by graph construction and Hemoglobin quantization. After which, the extracted properties are optimized using Aquila optimization and classified using cascaded Long Short-Term memory (LSTM) to attain the classified outcome of sickle cell and normal cells. Finally, the performance assessment is made and the outcomes attained in terms of accuracy, precision, sensitivity, specificity, and AUC are compared with existing classifier to validate the proposed system effectiveness.

The electron transmission properties in a non-planar system of two chained rings

The electron transmission properties in a model of two chained orthogonal quantum rings with input and output wires were investigated by using the quantum graph theory and quantum waveguide theory. The model was obtained for three-dimensional space. It also was shown that changing of the orientation of second ring in the respect to the field is a way to control the electron transmission and reflection.

Edge States of Schrödinger Equations on Graphene with Zigzag Boundaries

In this paper, we compare the spectra of bulk and edge Hamiltonians on graphene from the point of view of the theory of quantum graphs. Since the bulk Hamiltonians with even potentials on the graphene has been studied by Kuchment and Post, our main targets here are the edge Hamiltonians on the graphene with boundaries. In this paper, we obtain their spectral properties under the Dirichlet boundary condition on the zigzag edges and draw pictures of the dispersion relation. Then, we obtain the existence of embedded eigenvalues for edge states in spectral bands.

Multi-secret Sharing Model based on Hermite Interpolation Polynomial and Quantum Graph State

With the rise of the Internet plus blockchain, quantum secure communication has become a hot topic of interdisciplinary research. Secret sharing is the key technology to realize blockchain. In this paper, the Hermite interpolation polynomial combined with the idea of homomorphic cryptography is used to construct a multi-secret sharing model based on quantum graph theory. This paper relies on the theory of quantum theory to have mature theoretical and experimental foundations in quantum preparation, storage transmission, and measurement feasibility and overcomes the existing program feasibility problems and potential security problems with classical cryptography. This model is outsourced, verifiable, and extensible and has a high information rate. It can effectively resist external interception, replay, entanglement, tampering attacks and internal forgery, impersonation, deception, and collusion attacks. This paper has carried on a detailed analysis of the model.

Elastodynamics on graphs—wave propagation on networks of plates

We consider the wave dynamics on networks of plates coupled along 1D joints. This set-up can be mapped onto an extension of wave graph systems studied in, for example, quantum graph theory. In the elastic case, different mode-types (flexural, longitudinal and shear waves) propagate in each plate and do so at different wave speeds. The flexural (or bending) modes are described in terms of fourth order equations introducing an always evanescent wave component into the system. Waves encounter plate intersections and can be transmitted, reflected or mode converted. The intersection or vertex scattering matrices mix different waves which can be propagating (open) and evanescent (closed). The local scattering matrices and the global transfer operator are no longer unitary; the consequences of this non-unitarity on secular equations and the Weyl law will be discussed. The findings are of relevance to describing complex engineering structures such as networks of beams and plates.

A compositional approach to quantum functions

We introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyze dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.

An alpha particle model for Carbon-12

We introduce a new model for the Carbon-12 nucleus and compute its lowest energy levels. Our model is inspired by previous work on the rigid body approximation in the B=12 sector of the Skyrme model. We go beyond this approximation and treat the nucleus as a deformable body, finding several new states. A restricted set of deformations is considered, leading to a configuration space C which has a graph-like structure. We use ideas from quantum graph theory in order to make sense of quantum mechanics on C even though it is not a manifold. This is a new approach to Skyrmion quantisation and the method presented in this paper could be applied to a variety of other problems.

Physical approach to quantum networks with massive particles

This paper treats a quantum network from a physical approach, explicitly finds the physical eigenstates and compares them to the quantum-graph description. The basic building block of a quantum network is an X-shaped potential well made by crossing two quantum wires, and we consider a massive particle in such an X well. The system is analyzed using a variational method based on an expansion into modes with fast convergence and it provides a very clear intuition for the physics of the problem. The particle is found to have a ground state that is exponentially localized to the center of the X well, and the other symmetric solutions are formed so to be orthogonal to the ground state. This is in contrast to the predictions of the conventionally used so-called Kirchoff boundary conditions in quantum graph theory that predict a different sequence of symmetric solutions that cannot be physically realized. Numerical methods have previously been the only source of information on the ground-state wave function and our results provide a different perspective with strong analytical insights. The ground-state wave function has the shape of a solitonic solution to the non-linear Schr{\"o}dinger equation, enabling an analytical prediction of the wave number. When combining multiple X wells into a network or grid, each site supports a solitonic localized state. The solitons only couple to each other and are able to jump from one site to another as if they were trapped in a discrete lattice.

Vibrational quantum graphs and their application to the quantum dynamics of CH5.

The first application of quantum graphs to the vibrational quantum dynamics of molecules is reported. The quantum-graph model is applied to the quasistructural molecular ion CH5+, whose nuclear dynamics challenges the traditional understanding of chemical structures and molecular spectra. The vertices of the quantum graph represent versions of the equilibrium structure with distinct atom numbering, while the edges refer to collective nuclear motions transforming the versions of the equilibrium structure into one another. These definitions allow the mapping of the complex vibrational quantum dynamics of CH5+ onto the motion of a particle confined in a quantum graph. The quantum-graph model provides a simple understanding of the low-energy vibrational quantum dynamics of CH5+ and is able to reproduce the low-lying vibrational energy levels of CH5+ (and CD5+) with remarkable accuracy.

Performing quantum computing experiments in the cloud

Quantum computing technology has reached a second renaissance in the past five years. Increased interest from both the private and public sector combined with extraordinary theoretical and experimental progress has solidified this technology as a major advancement in the 21st century. As anticipated by many, the first realisation of quantum computing technology would occur over the cloud, with users logging onto dedicated hardware over the classical internet. Recently IBM has released the {\em Quantum Experience} which allows users to access a five qubit quantum processor. In this paper we take advantage of this online availability of actual quantum hardware and present four quantum information experiments that have never been demonstrated before. We utilise the IBM chip to realise protocols in Quantum Error Correction, Quantum Arithmetic, Quantum graph theory and Fault-tolerant quantum computation, by accessing the device remotely through the cloud. While the results are subject to significant noise, the correct results are returned from the chip. This demonstrates the power of experimental groups opening up their technology to a wider audience and will hopefully allow for the next stage development in quantum information technology.

Laplacians on quantum hypergraphs

AbstractWe introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Degeneracies in the length spectra of metric graphs

The spectral theory of quantum graphs is related via an exact trace formula to the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e., finding how many different lengths exist for periodic orbits with a given period and the average number of periodic orbits with the same length) is necessary for the systematic study of spectral fluctuations using the trace formula. This is a combinatorial problem which we solve exactly for complete (fully connected) graphs with arbitrary number of vertices.