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These topological qubits would be more stable,leading to more accurate computation and a better quantum computer.Topology was originally studied as a branch of purely abstract mathematics.Thanks to the pioneering work of Thouless, Haldane, and Kosterlitz,we now know it can be used to understand the riddles of nature and to revolutionize the future of technologies.

These qubits can solve problems exponentially faster than classical digital computers.The problem is that this data is so delicate that interaction with the environment can destroy it.But in some exotic topological phases,the subatomic particles can become protected.In other words, the qubits formed by them can't be changed by small or local disturbances.

The electronics of the future could be built to use this perfectly efficient electron highway.The topological properties of subatomic particles could also transform quantum computing.Quantum computers take advantage of the fact that subatomic particles can be in different states at the same time to store information in something called qubits.

Because the electrons are stuck in these loops,they're not conducting electricity.But at the edge of the material,the orbits become open, connected,and they all point in the same direction.So electrons can jump from one orbit to the next and travel all the way around the edge.This means that the material conducts electricity around the edge but not in the middle Here's where topology comes in.

That's because these properties lend surprising stability and remarkable characteristics to some exotic phases of matter in the delicate quantum world.One example is called a topological insulator.Imagine a film of electrons.If a strong enough magnetic field passes through them,each electron will start traveling in a circle,which is called a closed orbit.

That's because particles,like electrons and photons,are subject to the strange laws of quantum physics,which involve a great deal of uncertainty that we don't see at the scale of coffee cups.But the Nobel Laureates discovered that topological properties do exist at the quantum level.And that discovery may revolutionize materials science,electronic engineering, and computer science.

You could reshape a donut into a coffee cup and it would still have just one.That topological property is stable. On the other hand, a pretzel has three holes.There are no smooth incremental changes that will turn a donut into a pretzel.You'd have to tear two new holes.For a long time, it wasn't clear whether topology was useful for describing the behaviors of subatomic particles.

This conductivity isn't affected by small changes in the material,like impurities or imperfections.That's just like how the hole in the coffee cup isn't changed by stretching it out.The edge of such a topological insulator has perfect electron transport:no electrons travel backward, no energy is lost as heat,and the number of conducting pathways can even be controlled.

But what does that mean?First of all, topology is a branch of mathematics that focuses on fundamental properties of objects.Topological properties don't change when an object is gradually stretched or bent.The object has to be torn or attached in new places.A donut and a coffee cup look the same to a topologist because they both have one hole.

What if electricity could travel forever without being diminished?What if a computer could run exponentially faster with perfect accuracy?What technology could those abilities build?We may be able to find out thanks to the work of the three scientists who won the Nobel Prize in Physics in 2016.David Thouless,Duncan Haldane,and Michael Kosterlitz won the award for discovering that even microscopic matter at the smallest scale can exhibit macroscopic propertiesn and phases that are topological.

When force, a vector, is combined with another vector that provides orientation,we have the physical quantity called stress,which is an example of a second order tensor.These tensors are also used outside of video games for all sorts of purposes,including scientific simulations,car designs,and brain imaging.Scalars, vectors, and the tensor family present us with a relatively simple way of making sense of complex ideas and interactions, and as such, they're a prime example of the elegance, beauty, and fundamental usefulness of mathematics.

Various types of tensors contain different amounts of information.Does that mean there's something that can convey more information than vectors?Absolutely.Say you're designing a video game,and you want to realistically model how water behaves.Even if you have forces acting in the same direction with the same magnitude,depending on how they're oriented,you might see waves or whirls.

And just as different words in two languages can convey the same idea,different representations from two bases can describe the same vector.The vector is the essence of what's being communicated,regardless of the language used to describe it.It turns out that scalars also share this coordinate invariance property.In fact, all quantities with this property are members of a group called tensors.

If we look at the two arrays,they're clearly not the same,but an array alone doesn't completely describe a vector.Each needs a basis to give it context,and when we properly assign them,we see that they are in fact describing the same vector.You can think of elements in the array as individual letters.Just as a sequence of letters only becomes a word in the context of a particular language,an array acquires meaning as a vector when assigned a coordinate basis.

When your friend thinks about where he has to move,it can be written mathematically as 2x + 3y,or, like this, which is called an array.Since you're facing the other way,your coordinate basis points in opposite directions,which we can call x prime and y prime,and your movement can be written like this,or with this array.

No matter which way you face,or what coordinate system you place over the camp ground,the vector doesn't change.Let's use the familiar Cartesian coordinate system with its x and y axes.We call these two directions our coordinate basis because they're used to describe everything we graph.Let's say the tent starts at the origin and ends up over here at point B.The straight arrow connecting the two points is the vector from the origin to B.

What does that mean?Let's say you and a friend are moving your tent.You stand on opposite sides so you're facing in opposite directions.Your friend moves two steps to the right and three steps forward while you move two steps to the left and three steps back.But even though it seems like you're moving differently,you both end up moving the same distance in the same direction following the same vector.

Vector quantities also have a magnitude plus an extra piece of information,direction.To navigate to your bench,you need to know how far away it is and in what direction,not just the distance,but the displacement.What makes vectors special and useful in all sorts of fields is that they don't change based on perspective but remain invariant to the coordinate system.

Physicists,air traffic controllers,and video game creators all have at least one thing in common:vectors.What exactly are they,and why do they matter?To answer,we first need to understand scalars.A scalar is a quantity with magnitude.It tells us how much of something there is.The distance between you and a bench,and the volume and temperature of the beverage in your cup are all described by scalars.