Logical error and leakage add under code-preserving gate approximation

If $G=0$, both products are the empty product $I$ and the sum of errors is $0$, so the full-space error is $0$; [F2] then gives zero logical block error and zero leakage.

For $G\ge 1$, insert and subtract the mixed products to obtain the exact telescoping identity $V_G\cdots V_1-W_G\cdots W_1={\sum }_{j=1}^G{V}_G\cdots V_{j+1}(V_j-W_j)W_{j-1}\cdots W_1\mathrm{.}$ It follows by expanding the sum: consecutive terms cancel, leaving only $V_G\cdots V_1-W_G\cdots W_1$. Every outer product is unitary, so its operator norm is one and each summand has norm at most ${\delta }_j$. The triangle inequality proves the displayed sum bound.

The product $W_{\mathrm{t}\mathrm{o}\mathrm{t}}$ preserves the code because each factor does. Apply [F2] with $\delta =\mathrm{\parallel }{V}_{\mathrm{t}\mathrm{o}\mathrm{t}}-W_{\mathrm{t}\mathrm{o}\mathrm{t}}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$; step 1.1 bounds this by ${\sum }_j{\delta }_j$ using step 1.2 when $G\ge 1$ and step 1.1 when $G=0$. This gives the claimed logical error and leakage bounds, including on arbitrary ambient states before projection.