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Consider a qubit
$$\left| \psi \right> = (\alpha_1 + i\alpha_2 ) \left|0\right> + (\beta_1 + i\beta_2 )\left|1\right>$$
Now if i pass this through a series of quantum gates or any typical quantum circuit can I know the values of $\alpha$'s &$\beta$'s. before measuring it or without measuring it?
The single qubit state is
$$|\psi \rangle = \alpha |0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \hspace{2 cm} \alpha, \beta \in \mathbb{C}$$
and if you know what $\alpha$ and $\beta$ is ahead of time then of course you can keep track of all the matrix vector multiplications as applying a quantum gate is nothing but applying a unitary matrix, $U \in \mathbb{C}^2 \times \mathbb{C}^2 \ \ \ \textrm{s.t} \ \ \ U^\dagger U = I $, to the vector $|\psi \rangle$. So from the information of the resulting vector from the matrix-vector multiplcation, you can determine the probability of observing $|0\rangle$ and $|1\rangle$. This is what it is being done if you use something like statevector_simulator fro Qiskit.
For a single qubit, this is not a problem... but when you start getting into the 50 qubits range... storing $2^{50}$ complex values is quite daunting already and we are not talking about the cost of matrix-vector multiplication yet. With all that being said, here I am assuming a general setting. There are cases where this exact simulation can be done efficiently classically.
