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How can I construct, using Pauli, Hadamard and $T$ gate, a universal transformation $U$ such that $U|0\rangle$ has a less than $\frac{\pi}{4}$ complementary angle with $|0\rangle$?

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One way that you might do it is as follows. Consider the unitary sequence $V=HTHT$. Because it's unitary, we can write it in the form $$ V=e^{i\gamma}e^{-i\theta\vec{n}\cdot\vec{\sigma}}=e^{i\gamma}(\cos\theta I-i\sin\theta \ \vec{n}\cdot\vec{\sigma}) $$ where $\vec{n}\cdot\vec{n}=1$. If you work through the details, you'll find that $$ \cos\theta=\cos^2\frac{\pi}{8} $$ if memory serves.

This means that I can write $$ V^k=e^{i\gamma k}(\cos(k\theta) I-i\sin(k\theta) \ \vec{n}\cdot\vec{\sigma}). $$ So long as you pick a value of $k$ such that $|\cos(k\theta)|$ is close enough to 0, you're sorted (if you let $k$ get large enough, you can get arbitrarily close to 0). I'd probably just do this numerically (although if you want to get sophisticated, you could use continued fractions on $\theta/(2\pi)$).

I'm assuming that your condition is supposed to translate into a mathematical statement of $$ |\langle 0|U|0\rangle|\leq\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}. $$ If this is indeed the case, then $k=2$ may be fine, although $k=3$ is even safer as $$ \cos2\theta\approx 0.457107, \cos3\theta\approx-0.0732233. $$ Note that I've skipped an important step here because I've only been looking at $\cos(k\theta)$, not at $$ |\langle 0|U|0\rangle|=\sqrt{\cos^2(k\theta)+n^2_z\sin^2(k\theta)}. $$ Still, I claim that $n_z<1/\sqrt{2}$ so that so long as I find a small enough value of $\cos(k\theta)$ it will work out fine.

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