Fig. 3.

Subroutine costs for each component of implementing the block-encoding. The labels C$\{n\}$ correspond to the costs enumerated in protocol SI Appendix, section IV.D. CR is the reflection cost which given the additional register and augmented $\lambda$ is $n_T+2n_{\eta }+6n_n+n_M+16$ where $n_T=10+log(\lambda /ϵ)$, with $ϵ$ being the target precision of time evolution, and $n_{\eta }=\lceil log(\eta )\rceil$.