The $\gamma_c$-Peak: Covariant Recovery on Four Organic Qubit Platforms

Abstract

We characterize where, in the noise parameter space of the uniform dephasing--depolarizing channel $\mathcal N_\gamma^\delta=\mathcal E_\delta\!\circ\!\mathcal D_\gamma$, a deterministic, \emph{nonlinear, target-informed} denoising heuristic yields its largest fidelity gain. The procedure pulls the off-diagonal magnitudes of the noisy state toward those of a known target, with efficiency set by a SWAP-test-purified catalyst; it is not a quantum channel, and target access is an explicit classical resource, so the results describe a benchmark procedure, not blind error correction. Our main tool is the covariant purification map $\mathcal P_\mathrm{cov}(\rho)=(\rho+\rho^2)/(1+\mathrm{Tr}\,\rho^2)$, an exact closed form for one SWAP-test purification round (a rederivation of symmetrization purification: Barenco \emph{et al.}, Cirac--Ekert--Macchiavello) that reduces the catalyst to a scalar eigenvalue iteration. With it we derive the $d\to\infty$ fidelity-gain peak location on Haar-random pure states (Theorem~3): $\gamma_{\rm peak}(d)\to\gamma^\star(r,\delta)$, with $\gamma^\star(2,0.1)=0.4725$ and limiting magnitude $0.2262$. Bootstrap-quantified sweeps to $d=256$ are consistent with both limits. The peak is resource- and protocol-dependent: a catalyst-only reference moves it from $\approx0.50$ to $\approx0.34$, and one purification round instead of two to $\approx0.39$. Bell and uniform $d=4$ states admit unique-peak theorems for the $r=0$, $\delta=0$ protocol member. All results are reproducible from the open-source \texttt{organic-qc-bench} package with seed~42.