Quantum Machine Learning is about the right balanceParameterized quantum circuits let us embed classical data into quantum states, tune them with adjustable gates, and read out predictions through measurement. The real question is whether these simple building blocks can be scaled into architectures that deliver genuine quantum advantage.September 10, 2025

Dear Quantum Machine Learner,

Today's post deals with parameterized quantum circuits (PQCs) and the balance between expressiveness and trainability. Finding the right balance is a key challenge not only in the field of quantum machine learning, but in many areas. Including writing itself as I am still experiencing.

I keep trying to stick to my release schedule, but it's difficult to find the right balance. Every new feature I add takes time to get right. Right now, creating Blackboard-style images is slowing me down the most. I draw them in TikZ (for those interested in my workflow), and although the results are worthwhile, they are very time-consuming. At the same time, I am introducing many new keywords that I still need to explain properly. For example, in today's post, I happened to write a detailed explanation of the term is a Unitary OperatorAn Unitary operator is a reversible quantum transformation. (therefore ) that encodesEncoding the data value according to a encoding scheme or feature mapQuantum Feature Map is... ("phi"). Different choices of correspond to different ways of embedding data into a Quantum StateQuantum State is..., such as Angle EncodingAngle Encoding, Amplitude EncodingAmplitude Encoding, or more elaborate Quantum Feature MapsQuantum Feature Map is...). So means apply the unitary defined by encoding rule to input . The is not a variable like , but a tag to remind you how the data is being encoded..

At the moment, I feel less like I'm operating a simple car antenna and more like I'm managing a large antenna system. It's powerful, but more difficult to control. Still, it's a thousand times better than my old platform, which in this analogy was really the car antenna. Especially in terms of page speed. Hopefully, you've noticed that the new page loads almost instantly, while the old PyQML page loads and loads and loads...

So, without further ado, let's load today's post.

Put up your antennas

—Frank ZickertAuthor of PyQML