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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.