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.
Figure 1 Which antenna works best?
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 A **unitary operator** is a linear operator ( U ) on a complex vector space that satisfies ( U^\dagger U = UU^\dagger = I ), meaning it preserves inner products. In simpler terms, it preserves the **length** and **angle** between vectors—so it represents a **reversible, norm-preserving transformation**. In quantum mechanics, unitary operators describe the evolution of isolated systems because they conserve probability. Learn more about Unitary Operator (therefore ) that Encoding is the process of converting information from one form into another, usually so it can be stored, transmitted, or processed more efficiently. For example, text can be encoded into binary for computers to handle, or sounds into digital signals for transmission. The key idea is that encoding changes the representation, not the meaning, of the data. Learn more about Encoding the data value according to a A **Quantum Feature Map** is a process that converts classical input data into a **quantum state** using a parameterized quantum circuit. This transformation encodes data into the **high-dimensional Hilbert space** of a quantum system, allowing quantum algorithms to exploit complex correlations not easily captured classically. Essentially, it’s the quantum analogue of feature mapping in machine learning, used to make data more separable for classification or regression tasks. Learn more about Quantum Feature Map ("phi"). Different choices of correspond to different ways of embedding data into a A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin. Learn more about Quantum State, such as Angle encoding is a method of loading classical data into a quantum state by mapping data values to rotation angles of qubits (e.g., using quantum gates like Rx Gate, Ry Gate, or Rz Gate) Each feature of the data is represented as the angle of the quantum state vector’s rotation, which changes its probability amplitudes. This allows continuous classical values to be embedded in qQuantum states for use in quantum algorithms or quantum circuits. Learn more about Angle Encoding, Amplitude encoding represents classical data as the amplitudes of a quantum state's basis vectors. If you have a normalized data vector , it’s encoded into an -qubit state . This allows data values to be stored in only qubits, but preparing the quantum state can be computationally expensive. Learn more about Amplitude Encoding, or more elaborate A **Quantum Feature Map** is a process that converts classical input data into a **quantum state** using a parameterized quantum circuit. This transformation encodes data into the **high-dimensional Hilbert space** of a quantum system, allowing quantum algorithms to exploit complex correlations not easily captured classically. Essentially, it’s the quantum analogue of feature mapping in machine learning, used to make data more separable for classification or regression tasks. Learn more about Quantum Feature Map). 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. Learn more about .
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...
