4D (Part 3): The Topology of Pastries

(Continued from 4D (Part 2) Hole in the Wall)

I love pastries–just as much as I hate math.  Anyone who knows me, even in passing, would agree that this is the truth.  However, these exigent emotions of epic inverse proportion has just been summarily breached, thanks in no small part by three cool dudes who figured out how to make mathematics as tasty as donuts.

Back in 2016, three guys won the Nobel prize for their startling discovery of something important in the field of quantum physics.  Their names are David Thouless, Duncan Haldane and Michael Kosterlitz [1] and they are, without a doubt, the top Topologists in their field.  Before I go any further, please allow me to define the term topology as it is utilized in quantum physics.

Topology n. for mathematics:  The study of those properties of geometric forms that remain invariant under certain transformations, such as bending or stretching.

I’m of the camp of thought that less is sometimes not as elegant as people make them out to be.  Less words that are used to describe something may seem elegant, but if it does not make much sense to the reader, then I’d rather see more simple words strung together so the rest of us slow folks can catch up.

Simply speaking, topology is the modern version of geometry, the study of all different sorts of spaces.  Back in the ancient days when I first learned geometry, a line was different from a circle, and a circle was different from an oval.  Topology, however doesn’t care about all that because the geometry of a shape really DOES NOT depends on your unique vantage point, or the position that you happen to be observing that shape.

For example, if you look at a 2D circle on its face, it certainly looks like a circle, but if you turn it until all you can see is its edge, it suddenly turns into a line.  Twist it a bit and look at it at an angle, and it becomes an oval.  Pull on the corners and the circle becomes a triangle.  Pull four corners out and it becomes a square, but no matter what we do, it does not matter, topographically.  It would still be indistinguishable, as a geometric form, because it has not changed at all.

What distinguishes different kinds of geometry from each other is, therefore, the kinds of transformations that are allowed before it is really considered changed.  In this case, twisting and bending this 2D circle does not change its geometry, no matter how or where you view it.

Only tearing it apart or punching a hole through it will change its geometry.  Case in point.  Here we have the oft-used and abused elephant picture.

Both images depict an elephant, but the vantage point is different.  Changing camera angles does not change the geometry of the elephant.  The only way for this geometry to change is for us to either physically hack the elephant into various pieces or shoot a hole through the elephant (which would of course, kill it, so please don’t do that).

To translate this into quantum physics topology, materials are described as mathematical objects with set numbers of holes. [2]  For example, a donut, a pretzel and a cinnamon bun may be different in many ways to the rest of us pastry lovers, but to a physics topologist, the only difference would be the number of holes each pastry has–ergo, their topological invariant.

In other words, you can twist and bend a donut all you want, but it will only have one hole.  It also holds true for the cinnamon bun and the pretzel.  The only way to change the donut so that it is no longer a donut is to take a bite.

To translate this pastry example into quantum physics (as applicable to quantum computers), we would have to compare the normal computer conductor to that of a superconductor, in which case, it would be the equivalent of a donut transforming into a bun.

A computer conductor is a set of wires that allow electricity, light, heat, sound, or other forms of energy to pass through from various devices to other devices.  These wires transmit tiny bursts of electricity that either pass through or don’t pass through based on whether something is switched on or off.  This on/off energy transmission is mathematically represented by zeroes and ones (0/1) in combinations called bytes.

Since we live in 2018 and not the dark ages, most people understand this very rudimentary level of computer science, so I won’t belabor the point.  Quantum computing is, however, on a vastly different and far superior level.

Imagine being able to add as many as twenty zeroes and ones in a single output.  These are qubits.

Classical computers (what I’m using right now to create this posting) encode information as on or off bits denoted at the 0-position or 1-position.  Quantum computers, on the other hand, use qubits superpositions of both at once, which means qubits have access into another dimension, allowing it to tap into that strange magical ability called entanglement.

Entanglement is another ball of wax, which I will touch upon before I can tie in my personal experiences on this, so hang on tight.

(to be continued…)

…

[1]  https://www.theguardian.com/science/2016/oct/04/david-thouless-duncan-haldane-and-michael-kosterlitz-win-nobel-prize-in-physics

[2] http://mathworld.wolfram.com/Topology.html