Hybrid bound for imperfect QSP phase rotations

Write the ordered product as $A_n{A}_{n-1}\cdots A_1$, where each factor is one phase or signal gate, and write the implemented factors as ${\tilde{A}}_j$. Insert and subtract products with the first $k$ factors implemented and all remaining factors ideal; this gives the exact telescoping identity ${\tilde{A}}_n\cdots {\tilde{A}}_1-A_n\cdots A_1={\sum }_{k=1}^n{\tilde{A}}_n\cdots {\tilde{A}}_{k+1}({\tilde{A}}_k-A_k)A_{k-1}\cdots A_1\mathrm{.}$

Every prefix and suffix in each summand is a product of unitaries, hence has operator norm $1$ by [F3]. For any operators $S,T$ and unit vector $v$, [F5] gives $\mathrm{\parallel }(S+T)v\mathrm{\parallel }\le \mathrm{\parallel }Sv\mathrm{\parallel }+\mathrm{\parallel }Tv\mathrm{\parallel }\le (\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}+\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}})\mathrm{\parallel }v\mathrm{\parallel }$; taking the supremum gives the operator-norm triangle inequality. Together with submultiplicativity from [F3], this bounds the norm of the telescoping sum by the sum of individual factor errors. Grouping phase and signal factors gives ${\sum }_j{\zeta }_j+{\sum }_k{\eta }_k$, and setting all ${\eta }_k=0$ gives the phase-only assertion. The resource coordinates remain separate by [F4].