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In the example on the Qiskit page (https://qiskit.org/textbook/ch-applications/hhl_tutorial.html#A.-Some-mathematical-background) section 3 for the HHL algorithm the author is trying to solve the system of equations written as $A \vec{x}=\vec{b}$, where: $$ A=\left(\begin{array}{cc} 1 & -1 / 3 \\ -1 / 3 & 1 \end{array}\right), \quad \vec{b}=\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \quad \text { and } \quad \vec{x}=\left(\begin{array}{c} x_{1} \\ x_{2} \end{array}\right). $$

The final state of the system implementing the algorithm is: $$ \frac{\frac{3}{2 \sqrt{2}}\left|u_{1}\right\rangle+\frac{3}{4 \sqrt{2}}\left|u_{2}\right\rangle}{\sqrt{45 / 32}}=\frac{|x\rangle}{\|x\|}. $$ Given that: $$ |x\rangle={\frac{3}{2 \sqrt{2}}\left|u_{1}\right\rangle+\frac{3}{4 \sqrt{2}}\left|u_{2}\right\rangle} $$

where: $$ \left|u_{1}\right\rangle=\left(\begin{array}{c} 1 \\ -1 \end{array}\right) \quad \text { and } \quad\left|u_{2}\right\rangle=\left(\begin{array}{l} 1 \\ 1 \end{array}\right). $$

How do i determine the solution to the system which is $\left[\begin{array}{ll}1.125 & 0.375\end{array}\right]$?

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