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I am reading an article on representation of digital images using qubits. But I am not able to understand the notations of the article. Can somebody help me: $|0\rangle$: It means the first basis vector in the respective vector space. But what does $ |0\rangle^{3q+ 2n}$ mean?
A single qubit in the $0$ state is often written as $|0\rangle$.
If we have two independent qubits in the zero state we can write them as $|0\rangle\otimes|0\rangle$. This is often also written as $|0\rangle|0\rangle = |00\rangle =|0\rangle^{\otimes2}=|0\rangle^2$. This extends naturally to more qubits.
Coming back to your question: $|0\rangle^{\otimes 3q+2}$ (or $|0\rangle^{3q+2}$), it are just $3q+2$ qubits, all in the zero state $|0\rangle$.
Also, think about the dimensions these basis vectors are creating. $|0\rangle$ is a $2\times 1$ vector, which is in the standard basis looks like: $$\begin{bmatrix} 1\\0 \end{bmatrix}$$Then, continuing $$|0\rangle^{\otimes2} = |00\rangle=|0\rangle|0\rangle = |0\rangle \otimes |0\rangle = \begin{bmatrix} 1\\0 \end{bmatrix} \otimes \begin{bmatrix} 1\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix}$$
They all mean the same thing.
