Analysis

This question tests students' ability to derive the general formula of a geometric sequence from the sum of its first $n$ terms and to calculate the sum of the squares of the first $n$ terms of a geometric sequence based on its first term and common ratio.

Solution

Solution: $a_1 = S_1 = 1$,

For $n \geqslant 2$, $a_n = S_n - S_{n-1} = 2^{n-1}$,

Thus, the first term of the geometric sequence is $1$, and the common ratio $q$ is $2$,
Then $a_n = 2^{n-1}$
Then $a_n^2 = 4^{n-1}$, which is a geometric sequence with the first term as $1$ and the common ratio as $4$,
Therefore, $a_1^2 + a_2^2 + \cdots + a_n^2 = \frac{1-4^n}{1-4} = \frac{1}{3}\left(4^n - 1\right)$,
Hence, the correct answer is $\boxed{\text{D}}$.