# Why primitive rotation is $53.13^\circ$ in the quantum Turing machine used by Vitanyi for Quantum Kolmogrov Complexity?

Right now I am going through Quantum Kolmogorov Complexity Based on Classical Descriptions by Vitanyi.

In the introduction, the author assumed the primitive rotation $\theta = 53.13^\circ$ to have countably many quantum Turing machine.

Is this choice of angle arbitrary?

• N.B. $53.15^\circ \approx \arcsin (4/5)$. Restricting to rational amplitudes with a denominator of 5 is an old convention from the early days of quantum computing, dating back at least to Bernstein and Vazirani, for those who are looking for a special case to make comparisons between quantum complexity and counting complexity. – Niel de Beaudrap Sep 15 '16 at 7:09
• I will check their paper. – Omar Shehab Sep 15 '16 at 7:47
• @Niel: I am fairly sure that Bernstein and Vazirani were the first ones to use arcsin(4/5). And if it's not already clear from Niel's comment, $4/5$ is chosen because $3^2 + 4^2 = 5^2$. – Peter Shor Nov 19 '16 at 22:37