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31 lines
1 KiB
Markdown
31 lines
1 KiB
Markdown
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Un [[gate quantistico]] che permette di effettuare una rotazione su asse arbitrario.
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Usando la [[formula di Eulero]], esso corrisponde a:
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$$
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\def \varX {{\color{coral} \theta}}
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\def \varY {{\color{cornflowerblue} \phi}}
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\def \varZ {{\color{yellowgreen} \lambda}}
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\def \varI {{\color{hotpink} i}}
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\Huge
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\mathbf{U}(\varX, \varY, \varZ) = \begin{bmatrix}
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\cos \left( \frac{\varX}{2} \right) &
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- e^{\varI \varZ} \sin \left( \frac{\varX}{2} \right) \\
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e^{\varI \varY} \sin \left( \frac{\varX}{2} \right) &
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e^{\varI \varY + \varI \varZ} \sin \left( \frac{\varX}{2} \right)
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\end{bmatrix}
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$$
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Espanso, sarebbe:
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$$
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\def \varX {{\color{coral} \theta}}
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\def \varY {{\color{cornflowerblue} \phi}}
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\def \varZ {{\color{yellowgreen} \lambda}}
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\def \varI {{\color{hotpink} i}}
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\mathbf{U}(\varX, \varY, \varZ) = \begin{bmatrix}
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\cos \frac{\varX}{2} &
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- (\cos \varZ + \varI \sin \varZ) \cdot \sin \frac{\varX}{2} \\
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(\cos \varY + \varI \sin \varY) \cdot \sin \frac{\varX}{2} &
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(cos (\varY + \varZ) + \varI \sin (\varY + \varZ)) \cdot \sin \frac{\varX}{2}
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\end{bmatrix}
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$$
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