On the Scope of Compression-Error Bounds in Tensor-Network Error Mitigation: A propagated-residual counterexample, an exact observable-level decomposition, and two sufficient repairs

Abstract

Tensor-network error mitigation (TEM) applies a matrix-product-operator approximation to the inverse of a characterized global noise map in classical post-processing. This work retains the exact, untruncated TEM inversion identity under invertible layerwise noise and analyzes what a sequence of local MPO-compression errors implies for the final truncated mitigation map and for selected observables. We identify several scope issues in the compression-error analysis of Appendix G of Filippov et al., including the printed Frobenius identity, the observable-vector norm, the treatment of local compression residuals, and PTM orientation. We define the local compression residual and derive an exact propagated-residual identity showing that each residual must be transported through all subsequent TEM updates. A two-qubit CPTP example demonstrates that later inverse-noise layers can amplify an early compression residual, so an unweighted sum of local residual norms is not a valid general bound on the final TEM map error. Two sufficient certification routes are provided: a global propagated-norm bound and an exact observable-relative decomposition. We also show that stability under successive increases of bond dimension is a useful practical diagnostic but not, by itself, a general error certificate without an independent tail bound. These results do not invalidate exact TEM or the reported numerical benchmarks; they delimit the scope of compression-error certification and provide mathematically sufficient alternatives.