Approximate recovery and implementation error

For adaptive programmes, include the classical history register in each complete channel. A controlled operation is blockwise on that classical register; a uniform bound over its possible histories bounds the controlled step. No postselected normalization is used.

If

then $d_{\diamond }({\mathrm{Ad}}_{\tilde{U}},{\mathrm{Ad}}_U)\le \eta$. Indeed, expanding $\tilde{U}X{\tilde{U}}^{†}-UX{U}^{†}$ into two terms bounds its trace norm by $2\eta {\Vert X\Vert }_1$, also after tensoring a reference.

For the streaming tree, a uniform elementary-gate error ${\delta }_g$ and destructive single-qubit-readout error ${\delta }_m$ give

An error in the supplied encoding/access channel can be added in the same half-diamond metric.

Approximate-QEC and universal-recovery results motivate possible choices of $\mathcal{D}$ BenyApprox,Cotler. No state-specific recovery guarantee is silently promoted here to a worst-case diamond bound.