Energy-Scaled Zero Noise Extrapolation Extends the Operating Regime of GKP Codes

The Gottesman-Kitaev-Preskill (GKP) code is a leading candidate for error correction in bosonic systems, but its performance is limited by photon loss and finite energy (mean photon number, n). These imperfections introduce errors that limit practical deployment. We propose and demonstrate Energy-Scaled Zero Noise Extrapolation (ES-ZNE), an error mitigation protocol that systematically removes finite-energy errors. Unlike conventional ZNE, which scales external noise, our protocol leverages the code's intrinsic mean photon number n as a controllable noise-scaling parameter. We simulate the encode-noise-recover process under a pure-loss channel, applying a near-optimal Petz recovery map for single- and two-qubit observables. By fitting expectation values to a power-law model in n and extrapolating to the infinite-energy limit, we achieve substantial error reduction. At moderate photon loss (18%), ES-ZNE recovers ideal expectation values to within numerical uncertainty. Even at significant loss (33%), ~96% of the ideal value is restored. This work provides a demonstration of ZNE adapted to intrinsic bosonic code parameters, extending the useful operating regime of near-term GKP processors.