Topological Graphs and the Quantum Transition to encode information topologically
By Tanush
This research advances graph theory and its applications by introducing a novel framework in which edges are represented not as static scalar weights, but as vector-valued functions. Through this approach, graphs are encoded as webs of B-splines, enabling edges to capture topology, geometry, and node significance in ways that linear definitions cannot. The spline structure allows connections to self-adjust while preserving smoothness and local control, creating a bridge between discrete combinatorial structures and continuous geometric representation.
Topological information is incorporated through Hodge Laplacian, which provide spectral invari- ants of the original graph. These invariants can then be expressed within the spline web and further compressed into classical geometric invariants—such as line integrals, curvature, and torsion—using the Frenet–Serret framework. This interplay between discrete topology, spline geometry, and in-variant compression highlights the adaptability of the model, ability to switch between invariants,and allows for high expression to be encoded into the edges.This research extends our own spline-based graph framework into quantum mechanics by introducing the Quantum Topographical Spline Basis (QTSB). QTSB redefines spline recursion in terms of quantum normalization and phase, creating a Hilbert-space formulation that adapts B-splines to quantum principles. This basis provides a mathematically rigorous way to encode qubit information with compactness and expressiveness, while remaining consistent with the rules of Quantum Mechanics. By replacing Cox-De Boor’s Partition of Unity with our new Partition of Squared unity and adding an additional parameterizing term of phi to the Basis vector N, I am able to introduce imaginary numbers and splines that are able to not only add together but also have constructive and destructive properties which is extremely useful. It also manages to encode several nodes together in one state, and relate topological information with them.
This research has the potential to influence multiple areas of science and technology by providing a unified framework that links graph theory, geometry, and quantum computation. On the classical side, a spline-based graph representation offers new tools for analyzing networks in fields such as transportation, communication, and data science, where richer edge structures can capture geometric and topological properties that traditional graphs overlook. Such capabilities could lead to more efficient routing algorithms, improved models of physical systems, and deeper insights into the structural properties of large-scale networks.In the quantum domain, the Quantum Topographical Spline Basis (QTSB) introduces a novel quantum basis that can completely revolutionize how we see quantum. Because my QTSB is not normalized in each pair but in each shifting pair, if offers a unique opportunity to use spline advantages and apply them to a quantum-system that meets all the requirements but rather than local normalized is adjacency normalized(a term I invented). This allows you to essentially be able to use in quantum mechanics. In addition, because degree is different in the quantum sense, it essentially allows to recur over superposition. Similar to a Bell State, it allows you to superpose over a superposed state, but unlike the bell curve, it has no limit on the degree you place except the limits faced computationally or for the problem at hand. This also encodes many nodes together and also forms the basis of quantum graphs represented as spline webs.