Core Quantum RoutinesThere are a few routines that build the core of almost any quantum algorithm, inclduding quantum machine learning. Today, we start with the Quantum Fourier Transform.September 25, 2025

Dear Quantum Machine Learner,

The Quantum Fourier Transform is the most popular routine in quantum computing. You see it in textbooks, tutorials, and lecture slides. It is introduced as the workhorse behind Shor's algorithm and the quantum phase estimation.

It is the core routine that provides exponential speedup in comparison to the classil Fourier transform. So, of course, every student wants to understand how it works.

Surprisingly, the lecturers, books, and online tutorials seem to want to keep this knowledge secret. They pack their knowledge in walls of symbols to make sure really nobody can understand.

For a long time, I also had difficulty understanding how the Quantum Fourier Transform works, or what it actually does. After initial attempts, I quickly moved this routine to the bottom of my list. I thought, OK, this is really advanced stuff. I'll do it once I have a really deep understanding.

But when the time came, I was surprised to find that the Quantum Fourier Transform is pretty straightforward. I kept digging to see what I'd obviously missed. Because there was no way this holy grail of quantum computing could be that simple.

Yet, it is. Conceptually, the Quantum Fourier Transform is much simpler than other core routines, such as the Grover iterate. Let me therefore share with you a first look at what the Quantum Fourier Transform does.

Once again, I prepared a PDF file that summarizes the key points and gives you an introduction without many words.

—Frank ZickertAuthor of PyQML