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I have this qubit state:
$$ H \left[ \frac{1}{\sqrt{2}} |0\rangle + \left( \sqrt{\frac{2}{7}}+\frac{1}{\sqrt{7}}i \right) |1\rangle \right] $$
How to solve this given Hadamard gate on qubit?
Hadamard matrix should be multiplied with vector 2x1 (1 and 1) but what to do with numbers in front of it?
You can think about this in two ways:
Quantum operators are linear, meaning any operator $\hatΩ$ obeys the following relation: $$\hatΩ \left[ α|ψ⟩ + β|φ⟩ \right] = α \left(\hatΩ|ψ⟩\right) + β \left(\hatΩ|φ⟩\right) $$ In other words, you can just pull the constants out in front, calculate out the results of $H|0⟩$ and $H|1⟩$ as usual, then re-distribute your constants.
Generally $α$ and $β$ can be any complex number at all, but in your case $α=\frac{1}{\sqrt{2}}$ and $β=\left(\sqrt\frac{2}{7} + \frac{i}{\sqrt{7}}\right)$.
Matrix operations are designed to make these linear operations easier to think about. The vector you are trying to apply $H$ to can be written as: $$ |Ψ⟩ = \left[ \begin{array}{c} α \\ β \end{array} \right] $$ Meanwhile, $H$ can be written as: $$ H = \frac{1}{\sqrt{2}} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] $$ The final result is: $$ H|Ψ⟩ = \frac{1}{\sqrt{2}} \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{c} α \\ β \end{array} \right] = \frac{1}{\sqrt{2}} \left[ \begin{array}{c} α+β \\ α-β \end{array} \right]$$ So, the final answer is: $$ H|Ψ⟩ = \frac{1}{\sqrt{2}} \left[ (α+β) |0⟩ + (α-β) |1⟩ \right] $$
