## Jeremy Côté

Bits, ink, particles, and words.

# The Curse of Dimensionality

If there’s one field of mathematics that everyone encounters in their daily life, I would argue that it’s combinatorics (with perhaps geometry being the other one). The rules of combinatorics cast a shadow over our lives. They affect how we make decisions and form the scaffolding for how options in our lives are displayed to us.

In this essay, I want to explore the idea which is known as the curse of dimensionality.

# Physics On A Cube

One of my favourite mathematical pieces of writing is Flatland, by Edwin Abbott Abbott (the book is in the public domain, so you can download it from Wikipedia). Published over a century ago, it’s a story1 involving residents (Flatlanders) who live in a two-dimensional world. Without giving too much of the story away (because you should seriously read it!), the inhabitants find themselves shocked when a strange shape dips into their world. That other “shape” is a sphere, which we know lives in a three-dimensional space. This confuses the residents to no end, and only a brave soul dares to push their mind further to explore the possibility of there being another dimension available.

1. Which, of course, is also a product of its time.

# A Game of Loops

When I hear the word “quantum”, I think of all the misconceptions and crazy ideas people associate with it in a lot of popular media. Physicists are great (and terrible) at coming up with names, and the word “quantum” is such an example of a word with a lot of baggage attached. Pair it with the word “computer”, however, and the misconceptions skyrocket, sometimes turning into full-blown hype. The reality (at the time of this writing) is much more modest: quantum computing presents an opportunity for thinking of computation differently, and the subsequent years will see how this plays out when theory meets experiment and engineering.

There’s a ton to talk about when it comes to quantum computing, but in this essay, I want to share with you something called a quantum error correcting code. It does what it says on the tin, and corrects errors that can accumulate during a computation. There are many such proposals, but one of the most popular is called the surface code, whose name will make sense as we dive into the details. The surface code is a proposal for how we can build a quantum computer that is robust to errors, but is only one step in the process. This essay is devoted to the surface code, how it works, and the challenges it faces when it comes to implementation.