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microsoft / QuantumKatas

Tutorials and programming exercises for learning Q# and quantum computing
MIT License
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Systems of Equations not properly explained #893

Closed hiibolt closed 11 months ago

hiibolt commented 12 months ago

Hello!

On line 55 of the Multi-Qubit Systems workbook, it is not properly explained how the constants are found. To my limited knowledge of math, is it not true that at least some of the constants needs to be known?

If so, how were they determined here?

tcNickolas commented 11 months ago

The solution to this system of equations is not unique, you can multiply the $\alpha$ and $\beta$ given as the solutions by any complex number with absolute value 1 and $\gamma$ and $\delta$ - by its adjoint and you'll get a valid solution as well. In practice, these solutions differ from the given one only by the global phases of the qubit states, which is not observable, so they are ignored.

To solve the system, you can fix $\alpha$ to be a non-negative real number, and, taking into account the normalization requirement for both qubit states ($|\alpha|^2 + |\beta|^2 = 1$ and $|\gamma|^2 + |\delta|^2 = 1$), find out that the absolute value of each of those numbers is $\frac1{\sqrt2}$; after this, you only need to adjust the relative phases of each number.