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In the paper by Dawson and Nielsen where they develop an algorithm for the Solovay-Kitaev Theorem, they analyze the lenght of the output noting how, for an approximation of degree $n$, the lenght of the output sequence is $5l_{(n-1)}$ where $l_{(n-1)}$ is the lenght of the sequence for the approximation of degree $n-1$. They conclude that the algorithm produces a sequence asymptotically of lenght $O(5^n)$.
We can then work on the recurrence for the approximation error $\epsilon_n$, and by isolating $n$ we arrive at this formula: $$n = \left \lceil {\frac{\ln\left(\frac{\ln(1/\epsilon \cdot c_{appr}^2)}{\ln(1/\epsilon_0 \cdot c_{appr}^2)}\right)}{\ln\left(\frac{3}{2}\right)}}\right \rceil $$
Then they affirm how substituting the right hand side of this formula into $O(5^n)$ allows us to rewrite it in terms of $\epsilon$ to get the result stated in the theorem of page 4 of the paper: $O\left(\ln\left(\frac{1}{\epsilon}\right)^{3.97}\right)$, but I don't get wich passages we need to perform to arrive at such conclusion. Can anyone help me?
Thank you.

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They use the rule of conversion of logarithms: $$ \log_b x = \frac{\log_c x}{\log_c b}.$$

For clarity define the following quantity: $$ A:= \frac{\ln(1/\epsilon \cdot c_{appr}^2)}{\ln(1/\epsilon_0 \cdot c_{appr}^2)}. $$ Now we can write $$ n = \left \lceil {\frac{\ln \left( \frac{\ln(1/\epsilon \cdot c_{appr}^2)}{\ln(1/\epsilon_0 \cdot c_{appr}^2)} \right)}{\ln(3/2)}}\right \rceil = \left \lceil {\frac{\ln(A)}{\ln(3/2)}}\right \rceil. $$

We use the log conversion rule to rewrite $\ln(A)$: $$ \ln(A) = \frac{\log_5 A}{\log_5 e}. $$ So we get $$ n = \left \lceil {\frac{\frac{\log_5 A}{\log_5 e}}{\ln(3/2)}}\right \rceil. $$ Now, substitute the above equation into $O(5^n)$ and ignore the ceiling operation with the Big $O$: $$ 5^n = 5^{\frac{\frac{\log_5 A}{\log_5 e}}{\ln(3/2)}} = \left ( 5^{\log_5 A} \right)^{\frac{\frac{1}{\log_5 e}}{\ln(3/2)}} = (A)^{\frac{\frac{1}{\log_5 e}}{\ln(3/2)}}. $$ Next, we use the log conversion again to rewrite $\log_5 e$ as follows: $$ \log_5 e = \frac{\ln e}{\ln 5} = \frac{1}{\ln 5}. $$ Hence, we can write $$ 5^n = (A)^{\frac{\ln 5}{\ln(3/2)}}. $$ So, we have $$ 5^n = (A)^{\frac{\ln 5}{\ln(3/2)}} = \left( \frac{\ln(1/\epsilon \cdot c_{appr}^2)}{\ln(1/\epsilon_0 \cdot c_{appr}^2)} \right)^{\frac{\ln 5}{\ln(3/2)}}. $$

Note that $\ln(1/\epsilon_0 \cdot c_{appr}^2)$ is constant in $\epsilon$, so it will disappear when we push it through Big $O$. Also, we can absorb the constant $c^2_{appr}$ into $\epsilon$. Therefore, we get the final result: $$ l_{\epsilon} = O\left( \ln(1/\epsilon ) ^{\frac{\ln 5}{\ln(3/2)}} \right). $$ We also have $\frac{\ln 5}{\ln(3/2)} \approx 3.97$. So we get the result in the theorem: $$ l_{\epsilon} = O\left( \ln(1/\epsilon ) ^{3.97} \right). $$

We can do the same thing for $O(3^n)$ to get : $$ t_{\epsilon} = O\left( \ln(1/\epsilon ) ^{\frac{\ln 3}{\ln(3/2)}} \right). $$ The above can also be written as presented in the theorem: $$ t_{\epsilon}= O\left( \ln(1/\epsilon ) ^{2.71} \right). $$

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