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I want to express the square root of NOT as a time-dependent unitary matrix such that each $n$ units of time, the square root of NOT is produced.

More precisely, I want to find a $U(t_0,t_1)$ such that $U(t_0,t_1) = \sqrt{\text{NOT}}$, if $t_1-t_0=n$ for some $n$.

One possible solution is to express $\sqrt{\text{NOT}}$ as a product of rotation matrices, and then, parametrize the angles in a clever way to depend on the time. But I do not know how to express $\sqrt{\text{NOT}}$ as a product of rotation matrices.

Any help?

Sanchayan Dutta
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Alejandro Díaz-Caro
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1 Answers1

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$$ \sqrt{NOT} = e^{(\frac{i \pi}{4} I_2 - \frac{i \pi}{4} \sigma_x)}\\ U(t) = e^{\frac{t-t_0}{t_1 - t_0} (\frac{i \pi}{4} I_2 - \frac{i \pi}{4} \sigma_x)} $$

AHusain
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