Clarify normalization notation in amplitude encoding lesson

[matti-sarjala]

URL to the relevant course

https://learning.quantum.ibm.com/course/quantum-machine-learning/data-encoding

Learning course: “Data encoding”

Section:
“Amplitude encoding”

Problem:
The normalization symbol \alpha appears to be used in two different ways. Earlier in the section, \alpha denotes the norm of the data vector, because the state is written with 1/\alpha. Later in a “Check your understanding” answer, \alpha is used as the inverse normalization factor itself.

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• new content request
• typo
• code bug
• out-of-date content
• broken link
• other

Describe the fix or the content request.

In the Learning lesson Data encoding, the amplitude encoding section appears to use the normalization symbol \alpha in two different ways.

Earlier in the section, the state is written as

|\psi^{(j)}_x\rangle =
\frac{1}{\alpha}
\sum_{i=1}^N x^{(j)}_i |i\rangle

with

\sum_{i=1}^N |x^{(j)}_i|^2 = |\alpha|^2.

This makes \alpha the norm of the data vector.

Later, in the “Check your understanding” answer for the vector (9,8,6,2,9,2), the state is written as

|\psi\rangle =
\alpha(9|000\rangle+8|001\rangle+\cdots)

and then

\alpha = \frac{1}{\sqrt{270}}.

Here, \alpha is used as the inverse normalization factor instead.

This is mathematically understandable, but it may be confusing pedagogically because the same symbol switches meaning within the lesson.

One possible minimal fix would be to use different notation, for example:

• use A = \sqrt{\sum_i |x_i|^2} for the vector norm in the explanatory section
• use 1/A as the normalization factor in the examples

or alternatively keep \alpha consistently as either the norm or the inverse normalization factor throughout the section.

This is a documentation/learning clarity issue only. No code or API behavior is affected.

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