The Controlled NotYou've been misled about the CNOT gate. Open almost any textbook and you'll read: "If the control qubit is is a basis state., flip the target." But CNOT is not a cause-and-effect gate. Let's take a look at what it is instead.October 02, 2025

Dear Quantum Machine Learner,

In recent posts, I have added slide decks to my blog posts. And the feedback has been overwhelmingly positive. The slides serve one purpose: they give you a quick overview of the topic. Yet, they also do something unexpected. They help me clarify my own texts.

This clarity does not come naturally. Creating a slide deck forces me to narrow the focus of each slide. This is valuable because when too many ideas are packed into a slide, the main message can easily get lost. At the same time, working under such constraints forces me to think more deeply about the core message: What is really important here?

When you have unlimited space, you can circle around a topic, look at it from different angles, and ultimately convey the message. However, when you only have space for a single short line, you have to commit: that's the crux of the matter. And surprisingly, this process often changes my own understanding of the topic.

A good example of this is the controlled NOT operator. Until recently, I considered it to be the quantum equivalent of an if-then-else. This is how it is described in most textbooks, and for years I had subscribed to this view and come to terms with the problems and difficulties it caused. However, when I thought about it under tight constraints, I realized that this view is misleading. The if-then-else structure promotes classical thinking, and that is precisely what holds us back from understanding quantum algorithms.

So, strictly speaking, I haven't discovered anything new of course, but I have rediscovered a more useful perspective. And it is precisely this change that is the focus of today's post.

So, here is today's post is companion. A PDF file summarizing the key points.

Enjoy the post with my rediscovered insights on What the CNOT Operator Really is

—Frank ZickertAuthor of PyQML