Quantum Graphs Research Articles

A note on Ambarzumian’s theorem for quantum graphs

Based on the main result presented in Bifulco and Kerner (Proc Am Math Soc 152:295–306, 2024), we derive Ambarzumian–type theorems for Schrödinger operators defined on quantum graphs.

Gap distributions of Fourier quasicrystals with integer weights via Lee–Yang polynomials

Recent work of Kurasov and Sarnak provides a method for constructing one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a special class of multivariate polynomials called Lee–Yang polynomials. In particular, they provided a non-periodic FQ with unit coefficients and uniformly discrete support, answering an open question posed by Meyer. Their method was later shown to generate all one-dimensional Fourier quasicrystals with \mathbb{N} -valued coefficients ( \mathbb{N} -FQ). In this paper, we characterize which Lee–Yang polynomials give rise to non-periodic \mathbb{N} -FQs with unit coefficients and uniformly discrete support, and show that this property is generic among Lee–Yang polynomials. We also show that the infinite sequence of gaps between consecutive atoms of any \mathbb{N} -FQ has a well-defined distribution, which, under mild conditions, is absolutely continuous. This generalizes previously known results for the spectra of quantum graphs to arbitrary \N -FQs. Two extreme examples are presented: first, a sequence of \mathbb{N} -FQs whose gap distributions converge to a Poisson distribution. Second, a sequence of random Lee–Yang polynomials that results in random \mathbb{N} -FQs whose empirical gap distributions converge to that of a random unitary matrix (CUE).

On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use.

Norm-Resolvent Convergence for Neumann Laplacians on Manifold Thinning to Graphs

Norm-resolvent convergence with an order-sharp error estimate is established for Neumann Laplacians on thin domains in Rd, d≥2, converging to metric graphs in the limit of vanishing thickness parameter in the “resonant” case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to those of the δ′ type.

Inverse spectral problem for the Schrödinger operator on the square lattice

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schrödinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.

Closed and open superconducting microwave waveguide networks as a model for quantum graphs.

We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equationis effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.

A Comparison between Invariant and Equivariant Classical and Quantum Graph Neural Networks

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such as graph neural networks (GNNs), have been leveraged for various data analysis tasks in high-energy physics. One typical task is jet tagging, where jets are viewed as point clouds with distinct features and edge connections between their constituent particles. The increasing size and complexity of the LHC particle datasets, as well as the computational models used for their analysis, have greatly motivated the development of alternative fast and efficient computational paradigms such as quantum computation. In addition, to enhance the validity and robustness of deep networks, we can leverage the fundamental symmetries present in the data through the use of invariant inputs and equivariant layers. In this paper, we provide a fair and comprehensive comparison of classical graph neural networks (GNNs) and equivariant graph neural networks (EGNNs) and their quantum counterparts: quantum graph neural networks (QGNNs) and equivariant quantum graph neural networks (EQGNN). The four architectures were benchmarked on a binary classification task to classify the parton-level particle initiating the jet. Based on their area under the curve (AUC) scores, the quantum networks were found to outperform the classical networks. However, seeing the computational advantage of quantum networks in practice may have to wait for the further development of quantum technology and its associated application programming interfaces (APIs).

Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms

We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional C ∗ C^* -algebras B B equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on L 2 ( B ) L^2(B) , the quantum adjacency matrices, or as certain operator bimodules over B ′ B’ . We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such “non-tracial” quantum graph corresponds to a “tracial” quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from unital completely positive (UCP) maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.

Deep quantum graph dreaming: deciphering neural network insights into quantum experiments

Despite their promise to facilitate new scientific discoveries, the opaqueness of neural networks presents a challenge in interpreting the logic behind their findings. Here, we use a eXplainable-AI technique called inception or deep dreaming, which has been invented in machine learning for computer vision. We use this technique to explore what neural networks learn about quantum optics experiments. Our story begins by training deep neural networks on the properties of quantum systems. Once trained, we ‘invert’ the neural network—effectively asking how it imagines a quantum system with a specific property, and how it would continuously modify the quantum system to change a property. We find that the network can shift the initial distribution of properties of the quantum system, and we can conceptualize the learned strategies of the neural network. Interestingly, we find that, in the first layers, the neural network identifies simple properties, while in the deeper ones, it can identify complex quantum structures and even quantum entanglement. This is in reminiscence of long-understood properties known in computer vision, which we now identify in a complex natural science task. Our approach could be useful in a more interpretable way to develop new advanced AI-based scientific discovery techniques in quantum physics.

Financial fraud detection using quantum graph neural networks

Financial fraud detection using quantum graph neural networks

Time evolution and the Schrödinger equation on time dependent quantum graphs * *Dedicated to Michael Berry on his forthcoming anniversary.

The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.

Quantum Secure Multi-Party Summation with Graph State.

Quantum secure multi-party summation (QSMS) is a fundamental problem in quantum secure multi-party computation (QSMC), wherein multiple parties compute the sum of their data without revealing them. This paper proposes a novel QSMS protocol based on graph state, which offers enhanced security, usability, and flexibility compared to existing methods. The protocol leverages the structural advantages of graph state and employs random graph state structures and random encryption gate operations to provide stronger security. Additionally, the stabilizer of the graph state is utilized to detect eavesdroppers and channel noise without the need for decoy bits. The protocol allows for the arbitrary addition and deletion of participants, enabling greater flexibility. Experimental verification is conducted to demonstrate the security, effectiveness, and practicality of the proposed protocols. The correctness and security of the protocols are formally proven. The QSMS method based on graph state introduces new opportunities for QSMC. It highlights the potential of leveraging quantum graph state technology to securely and efficiently solve various multi-party computation problems.

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.

A Quantum Spatial Graph Convolutional Neural Network Model on Quantum Circuits.

This article proposes a quantum spatial graph convolutional neural network (QSGCN) model that is implementable on quantum circuits, providing a novel avenue to processing non-Euclidean type data based on the state-of-the-art parameterized quantum circuit (PQC) computing platforms. Four basic blocks are constructed to formulate the whole QSGCN model, including the quantum encoding, the quantum graph convolutional layer, the quantum graph pooling layer, and the network optimization. In particular, the trainability of the QSGCN model is analyzed through discussions on the barren plateau phenomenon. Simulation results from various types of graph data are presented to demonstrate the learning, generalization, and robustness capabilities of the proposed quantum neural network (QNN) model.

Quantum graph theoretic encryption and dynamic key evolution in secure quantum communication networks

The dawn of the quantum era marks a pivotal moment in the evolution of digital communications, bringing forth both unparalleled potential and unprecedented challenges. This paper delves into the realm of Quantum Graph Theoretic Encryption and Dynamic Key Evolution, a frontier in the quest for impenetrable communication networks in the quantum computing age. Quantum computing, with its ability to perform complex calculations at speeds unattainable by traditional computers, offers a beacon of hope and a source of peril. The very principles that endow quantum computers with exceptional power also render traditional cryptographic methods vulnerable. This dichotomy underscores the urgency for innovative encryption techniques that can withstand the onslaught of quantum computational capabilities. When merged with the principles of quantum mechanics, graph theory opens a new dimension of cryptographic possibilities. However, the dynamic nature of digital communications necessitates more than just robust encryption; it requires agility. This is where the concept of Dynamic Key Evolution comes into play. This paper is structured to guide the reader through this novel integration of quantum mechanics, graph theory, and dynamic cryptography. We begin by laying a foundational understanding of quantum computing, followed by an exploration of graph theory in the context of encryption. We then introduce our proposed method of Quantum Graph Theoretic Encryption, highlighting its advantages and potential applications. Subsequently, we delve into the concept of Dynamic Key Evolution, illustrating its significance in maintaining the integrity of quantum encryption in a rapidly changing digital landscape.

Characterization of Directed Graphs Representing C*-Algebra of Complex Matrices

Quantum mechanics is a study that plays a major role in determining the biological intelligence of Artificial Intelligence (AI). Point particle systems in quantum mechanics can be explained using C*-Algebra which is called CAR-algebra. There is a special case in the CAR-algebra which is isomorphic to the C*-algebra of complex matrices. On the other hand, C*-algebras of direct sum of complex matrix spaces is isomorphic to C*-algebra constructed by orthogonal projection and partial isometries operators via the Cuntz-Krieger relations of a directed graph. This article will provide a basis for the relationship between quantum mechanics and graphs through a discussion of the characterization of graphs that can represent C*-algebra of complex matrices. It is found that C*-algebra complex matrices n × n is a directed graph without cycles with n – 1 arrows, a single source, and has n path from the source.

Reducibility of n Subcentrally Symmetric Quantum Graphs

Reducibility of n Subcentrally Symmetric Quantum Graphs

Bogoliubov-de Gennes equation on graphs: A model for tree-branched Majorana wire network

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

Quantum Algorithms for Charged Particle Track Reconstruction in the LUXE Experiment

The LUXE experiment is a new experiment in planning in Hamburg, which will study quantum electrodynamics at the strong-field frontier. LUXE intends to measure the positron production rate in this unprecedented regime using, among others, a silicon tracking detector. The large number of expected positrons traversing the sensitive detector layers results in an extremely challenging combinatorial problem, which can become computationally expensive for classical computers. This paper investigates the potential future use of gate-based quantum computers for pattern recognition in track reconstruction. Approaches based on a quadratic unconstrained binary optimisation and a quantum graph neural network are investigated in classical simulations of quantum devices and compared with a classical track reconstruction algorithm. In addition, a proof-of-principle study is performed using quantum hardware.

Closed form expressions for the Green’s function of a quantum graph—a scattering approach

In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.