Energy-Scaled Zero-Noise Extrapolation for Gottesman-Kitaev-Preskill Code

TL;DR

This work tackles the finite-energy limitation of Gottesman-Kitaev-Preskill (GKP) codes in bosonic qubits by introducing Energy-Scaled Zero-Noise Extrapolation (ES-ZNE), which uses the mean photon number $\bar{n}$ as a tunable noise knob and extrapolates measurements to the infinite-energy limit using a power-law model. Through numerical simulations of a GKP qubit under a pure-loss channel with Petz recovery, ES-ZNE mitigates finite-energy errors and reveals the intrinsic performance of the ideal GKP code, including a threshold at $x_{\text{crit}} \approx 0.4055$ beyond which correction deteriorates. The approach is extended to two qubits with independent noise, showing reliable extrapolations for entangled and random states and quantifying computational gains via energy-cutoff requirements. ES-ZNE offers a practical, software-based route to enhance near-term bosonic quantum processors by trading sampling overhead for reduced physical resource needs like high squeezing. Overall, the method isolates finite-energy artifacts from fundamental code performance and aligns with theoretical capacity bounds for pure-loss channels.

Abstract

The performance of Gottesman-Kitaev-Preskill (GKP) codes, an approach to hardware-efficient quantum error correction, is limited by the finite squeezing capabilities of current experimental platforms. To circumvent this hardware demand, we introduce Energy-Scaled Zero-Noise Extrapolation (ES-ZNE), a quantum error mitigation protocol that uses the mean photon number of the GKP code as a tunable effective noise parameter. The protocol measures logical observables at a series of accessible finite energies and extrapolates the results to the ideal, infinite-energy limit using an ansatz based on the code's asymptotic error scaling. Through simulating a GKP qubit under a pure-loss channel, we demonstrate that ES-ZNE successfully mitigates finite-energy errors, recovering the ideal expectation values (within numerical uncertainty) in the shallow-noise regime. Furthermore, by computationally removing artifacts arising from the finite-energy encoding, our method characterizes the intrinsic performance of the ideal GKP code, revealing a sharp error threshold beyond which the code's corrective power diminishes. These results establish ES-ZNE as a practical, software-based strategy for enhancing the performance of near-term bosonic quantum processors, trading sampling overhead for demanding physical resources like high squeezing.

Figures (7)

• Figure 1: Visualization of the extrapolated results and pipeline.Extrapolation: Power-law extrapolation of expectation value $\langle X \rangle (n_{\rm max})$, with zoomed-in Wigner functions of the GKP lattice at $n_{\rm max} = 1, 4, 30$. State Evolution ($n_{\rm max}=4$): illustration of the GKP code state evolution. From left to right: (i) The maximally mixed state of the codespace, $\tfrac{1}{2}P_L$, showing the characteristic grid structure. (ii) The encoded logical $|+\rangle_L$ state, exhibiting interference fringes indicative of quantum superposition. (iii) The state after a pure-loss channel with transmissivity $\eta=0.82$, showing contraction towards the origin and decoherence. (iv) The state after the application of the Petz recovery map, where the grid structure and interference fringes are partially restored. Simulation Pipeline: A schematic of the operational workflow, detailing the transitions from a logical input state ($\rho_{\rm in}$) to a physical encoded state ($\rho_{\rm enc}$), through the noisy channel ($\rho_{\rm noisy}$) and recovery ($\rho_{\rm rec}$), to the final decoded logical state ($\rho_L$) upon which observables are measured. Repeat the pipeline by tuning $n_{\rm max}$, and extrapolate to $n_{\rm max} \rightarrow \infty$.
• Figure 2: Baseline performance of finite-energy GKP codes. The conditional expectation value $\langle X \rangle$ is plotted against loss depth $x = -\ln(\eta)$ (log scale) for varying mean photon numbers $\bar{n}$ (color bar). The data reveals two distinct regimes separated by a threshold near $x \approx 0.45$. In the low-noise regime, increasing energy (lighter colors) monotonically improves coherence. Conversely, in the high-noise regime, the trend reverses, and higher-energy states degrade more rapidly as the channel capacity is exceeded.
• Figure 3: Single-Qubit Zero-Noise Extrapolation. (a): Conditional expectation values $\langle X \rangle_{\text{cond}}$ and power-law extrapolations for three representative loss depths. Shaded regions indicate the uncertainty of the extrapolated limit $L$. For the protected regime ($x=0.2, 0.4$), we observe monotonic improvement with energy. In contrast, the post-threshold regime ($x=0.556$) exhibits performance degradation as energy increases. (b): Residual diagnostic $|y(n) - L|$ on a log-log scale. The datasets for $x=0.2$ and $0.4$ display linear behavior, confirming the validity of the power-law ansatz, whereas the $x=0.556$ data deviates significantly from this scaling.
• Figure 4: Extrapolation across regimes. Extrapolated infinite-energy limit of $\langle X \rangle$ against loss depth $x = -\ln(\eta)$, compared with measurements from finite-energy simulations ($n_{\rm max}=4, 10$). The extrapolated measurement achieves higher expectation with high certainty before the threshold, but performs worse than finite-energy measurements beyond the threshold with higher uncertainty.
• Figure 5: Two-Qubit Zero-Noise Extrapolation. (a): Power-law extrapolation of the conditional expectation value $\langle XX \rangle_{\text{cond}}$ for the maximally entangled state $|\Phi^+\rangle_L$. Data is shown for independent pure-loss channels at depths $x \approx 0.2$ (blue) and $x \approx 0.4$ (purple). Shaded regions indicate the uncertainty of the extrapolated limit $L$. The low-loss case successfully recovers the ideal correlation ($L \approx 1$), while the higher-loss case reveals residual decoherence ($L \approx 0.82$) consistent with the single-qubit threshold behavior. (b): Residual diagnostic $|y(n) - L|$ on a log-log scale. The linear behavior confirms that the power-law ansatz accurately models finite-energy errors in the two-qubit setting.