Combinatorial Topology And Quantum Computing
Combinatorial topology is a type of math that helps us study the shapes of objects. Imagine you have a big pile of toys, like blocks or Legos, and you want to know how many ways you can combine them to make different shapes.
Simply put, Combinatorial Topology helps us count how many different shapes we can make with the same number of blocks by looking at their properties, like how many corners or edges they have. For example, a square has four corners and four edges, while a triangle has three corners and three edges. It’s like figuring out how many different ways you can arrange the same set of toys to make different creations!
Combinatorial Topology is a hot topic in mathematics because it helps us understand and classify the shapes of objects in a precise and systematic way. It is used in many different math, science, and engineering areas, including computer graphics, robotics, and materials science.
By studying the shapes of objects using combinatorial topology, we can also discover interesting properties and relationships between them. For example, we can figure out how two shapes can be transformed into each other by stretching or bending without tearing them apart, which is important in understanding the structure of materials and biological molecules.
There are many different formulas used in combinatorial topology, depending on what properties of objects we want to study. However, Euler’s formula is one of the most fundamental formulas in combinatorial topology.
Euler’s formula relates the number of vertices (corners), edges, and faces of a three-dimensional object. It says that for any polyhedron (a solid object with flat faces), the number of vertices minus the number of edges plus the number of faces is always equal to 2:
vertices – edges + faces = 2
This formula tells us that the number of vertices, edges, and faces of a polyhedron are not independent of each other but are related in a precise way. It also has many applications in geometry, topology, and other areas of mathematics. Here is a simplified version of the same Euler’s formula:
V − E + F = 2
Where V is the number of vertices, E is the number of edges, and F is the number of faces of a three-dimensional object (specifically, a polyhedron).
One example of a more advanced and complex version of Euler’s formula is the generalized Gauss-Bonnet theorem, which relates the topology (the study of the properties of space that are preserved under continuous transformations) of a surface to its geometry (the study of the properties of space that are not affected by continuous transformations).
The generalized Gauss-Bonnet theorem states that for any compact (bounded and without boundary) two-dimensional surface S with a smooth (continuous and differentiable) metric (a way of measuring distances and angles), the total Gaussian curvature K of the surface is related to its Euler characteristic χ by the formula:
\int_{S} K . dA = 2\pi \chi
Where dA is an infinitesimal area element on the surface, and the integral is taken over the entire surface. The Gaussian curvature K measures the intrinsic curvature of the surface at each point, and the Euler characteristic χ is a topological invariant that characterizes the shape of the surface.
This formula is more advanced and complex than Euler’s formula because it relates the curvature of a surface to its topology, which is a deeper and more abstract concept. It has many applications in differential geometry, topology, and physics, including the study of black holes and the geometry of the universe.
Alright, I rambled extensively about combinatorial topology to get to the point of sharing with you that the mathematics of Combinatorial Topology has applications in the field of quantum computing, too, especially in the study of topological quantum computing (TQC). In other words, Topological quantum computing (TQC) solves the fragility of the qubits at the hardware level by using topological invariants of quantum systems.
Topological quantum computing is a type of quantum computing that uses topological properties of materials to store and manipulate quantum information. It is based on the idea that certain physical systems, such as materials with topological order, can support protected quantum states that are robust against local perturbations and noise. TQC is a way of storing information using special types of quantum states that are like knots or said to encode in “knotted.” These knots help keep the information safe so it doesn’t disappear. Scientists have proven that this works well in certain materials called quantum Hall liquids. They have used this method to make quantum memory more stable. This field is a mix of math, physics, and computer science.
Combinatorial topology is used in TQC to understand the topological properties of materials and to design new materials with specific topological properties. For example, it has been used to study the properties of topological phases of matter, such as topological insulators and topological superconductors, which are potential candidates for building topological quantum computers. Does it ring a bell?! If you have been following my posts, your answer should be: Yes! Or check out my previous post on Breakthrough with Exotic Sub-Atomic Matter.
It has also been used to study the topological properties of quantum error-correcting codes, which are used to protect quantum information from errors and decoherence in quantum computing. By understanding the topological properties of these codes, researchers can design more efficient and robust quantum error-correcting codes for use in quantum computing.
I know I have oversimplified the combinatorial topology in this post. For those seeking more advanced information on this subject, I would recommend the following book:
Alexandrov, P.S. (1998). Combinatorial Topology Volume 2. Dover Publications Inc. (Original work published 1965).
Hamed Nazari
Hamed is an innovative and results-driven Chief Scientist with expertise in Quantum Science, Engineering, and AI. He has worked for leading tech companies in Silicon Valley and served as an Adjunct Professor at UC Berkeley and UCLA.