What is the "additive error" of Swap Test?

Question

I'm learning the Swap Test, a quantum circuit to calculate the inner product of two quantum states $|\langle \phi|\psi\rangle|^2 $:

For the error analysis of this quantum circuit, according to Swap Test:

"This allows one to, for example, estimate the squared inner product between the two states, $|\langle \phi|\psi\rangle|^2 $, to $\epsilon$ additive error by taking the average over $O(\frac {1}{\varepsilon ^{2}})$ runs of the swap test."

What's the "additive error $\epsilon$ " from a statistical point of view? Is it MSE, RMSE, or Variance? Can RMSE serve as an additive error of the Swap Test?

Answer 1

Additive error $\epsilon$ means that, if the true value of the expectation value we are attempting to measure is $\langle \hat{O} \rangle_{\textrm{True}}$, then our estimate $\langle \hat{O} \rangle_{\textrm{Estimated}}$ falls within the range $\langle \hat{O} \rangle_{\textrm{Estimated}} \in [\langle \hat{O} \rangle_{\textrm{True}} - \epsilon, \langle \hat{O} \rangle_{\textrm{True}} + \epsilon]$. Conversely, a multiplicative error $\epsilon$ implies $\langle \hat{O} \rangle_{\textrm{Estimated}} \in [\langle \hat{O} \rangle_{\textrm{True}} (1 - \epsilon), \langle \hat{O} \rangle_{\textrm{True}} (1 + \epsilon)]$. See, e.g., this Theoretical Computer Science question for a more in-depth discussion of the differences between the two.