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I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read $$ ||PA |0\rangle |\psi \rangle - |0\rangle U_r |\psi \rangle||=\mathcal{O}(\delta) \tag{15}$$ and

$$ ||\text{Tr}_\text{anc}(PA |0\rangle |\psi \rangle) - U_r |\psi\rangle \langle\psi| U_r ||=\mathcal{O}(\delta) \tag{16}$$

and I would like to understand how to rigorously argue that they are true which I struggle with because I am not very seasoned when it comes to handling complexity and Landau-notation.

Some context: The authors present a new technique that allows them to approximate $U_r |\psi \rangle$ by executing a circuit that acts like

$$PA |0\rangle |\psi \rangle = |0\rangle \left(\frac{3}{s}\tilde{U}-\frac{4}{s^3} \tilde{U}\tilde{U}^{\dagger}\tilde{U} \right)|\psi \rangle \tag{1}$$

and the two equations (15),(16) hold under the assumption that

$$ |s-2|=\mathcal{O}(\delta) \tag{2}$$ $$ ||\tilde{U}-U_r||=\mathcal{O}(\delta) \tag{3}$$

where I suppose the norm $||\bullet||$ is the spectral norm of the operators.

What I know how to do: I am able to show that (3) implies that $$|\tilde{U}\tilde{U}^\dagger-\boldsymbol{1}||=\mathcal{O}(\delta) \tag{4}$$ where I basically use that errors in unitaries only increase linearly which in turn I was able to prove from basic properties of the spectral norm.

What I struggle with: I want to apply (4) to (1) and replace $\tilde{U}\tilde{U}^{\dagger}$ by the identity plus some error and go from there but I do not know how I can do arithmetic with the Landau symbol.

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