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Fix $\epsilon > 0$, and suppose $U$ and $S$ are $n$ qubit unitaries such that $\| U - S \| \leq \epsilon$ (operator norm). Furthermore, let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ be the probability of obtaining the output string $y \in \{0,1\}^n$ given the input string $x \in \{0,1\}^n$ w.r.t. the unitary $U$. ($P_S(y \mid x)$ is the same thing, but w.r.t. the unitary $S$.)

I am trying to prove that $U$ and $S$ are close in the weak multiplicative sense, i.e., that $$ |P_U(y \mid x) - P_S(y \mid x)| \leq \epsilon P_U(y \mid x). $$ While I think this has to be true simply because $U$ and $S$ are effectively the same operation, I am struggling to get the inequalities to work out. Any help would be appreciated.

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    $\begingroup$ This is not possible. Consider $U=I$ and $|x\rangle$ and $|y\rangle$ orthonormal. Then the RHS is zero, but the LHS may be positive (depending on $S$). $\endgroup$ Commented Jan 11, 2023 at 23:48
  • $\begingroup$ True, thank you! $\endgroup$ Commented Jan 11, 2023 at 23:52

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