Logical and Process Thresholds for Long-Lifetime Quantum Computation via Surface-Code Quantum Error Correction

Abstract

Surface--code threshold theorems traditionally assume an abstract local noise model, typically an i.i.d.\ or weakly correlated Pauli channel on a flat hardware lattice, and establish exponential suppression of logical error below a critical physical error rate~\cite{dennis2002topological,aharonov2008fault,aliferis2006quantum,fowler2012surface,terhal2015quantum}. In this setting the surface code functions primarily as a quantum memory architecture, with fault--tolerant computation obtained by supplementing the code with transversal gates and magic--state distillation under the same phenomenological noise assumptions~\cite{nielsen2010quantum,watrous2018theory}. Such analyses, however, do not natively accommodate curved or slowly drifting chronogeometries, structured vacuum or spectral noise, or process--level equivalence between distinct QEC schedules. This work develops a triadic--Fourfold fault--tolerance framework for surface--code quantum computation in the IQR/PQS setting of~\cite{Borom:2025chrono}. The starting point is a full fourfold inter--modal correlator $\hat G_\Lambda$ and its ideal counterpart $\hat G^0_\Lambda$, defined on a QEC diagnostic realized reference configuration and a QEC--realized weave equipped with a Good--clock $C=(\hat T,U_t)$. The associated noise kernel $\Delta\hat G$ is required to lie in a Triadic--Fourfold Noise Class $\mathcal N_{\mathrm{tri\text{-}4F}}$ with correlational, vacuum, and spectral sectors satisfying suitable clustering and time--scale separation conditions~\cite{breuer2002theory,riera2012thermal}. From this data the analysis extracts triad--split local error rates $(p_{\mathrm{corr}},p_{\mathrm{vac}},p_{\mathrm{spec}})$, triad--resolved leakage parameters $\varepsilon_{\mathrm{ps}}^{(a)}$, and process differentials $\delta_a$ as explicit functionals of $\Delta\hat G$. The main Triadic--Fourfold Threshold Theorem shows that the \emph{standard} surface--code family on the emergent Twofold register still exhibits an exponential logical threshold controlled by the correlational error rate $p_{\mathrm{corr}}$, while vacuum and spectral contributions appear only through small, explicitly quantified correction terms $f_{\mathrm{vac}},f_{\mathrm{spec}}$. At the same time, a triadic process threshold bound ensures that, provided the global process differential $\delta_{\mathrm{tot}}$ is sufficiently small, all QEC schedules implementing a given logical circuit are holosymmetrically equivalent in the dual--history se

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