Estimating the performance boundary of Gottesman-Kitaev-Preskill codes and number-phase codes

Abstract

Bosonic quantum error-correcting codes encode logical information in a harmonic oscillator, with the Gottesman-Kitaev-Preskill (GKP) and number-phase (NP) codes representing two fundamentally different encoding paradigms. Although both have been extensively studied, it remains unclear under what physical noise conditions (including photon loss and dephasing) one encoding intrinsically outperforms the other. Here we estimate a quantitative performance boundary between GKP and NP codes under general photon loss-dephasing noise. By optimizing code parameters within each encoding family, we identify the noise regimes in which each code exhibits a fundamental advantage. In particular, we find that the crossover occurs when the dephasing strength is approximately two orders of magnitude smaller than the loss strength, revealing a sharp separation between operational regimes. Beyond this specific comparison, our work establishes a practical and extensible methodology for benchmarking bosonic codes and optimizing their parameters, providing concrete guidance for the experimental selection and deployment of bosonic encodings in realistic noise environments.

Figures (5)

• Figure 1: Overview of the bosonic lattice codes and the optimization workflow. The yellow panel shows the Wigner function of the logical state $|0_L\rangle$ for a hexagonal GKP code ($\Delta=0.2$) and the diamond NP code ($f=0.5,s=2,r=0,n=4$), together with the set of variational parameters. The green panel illustrates the GPU-accelerated evaluation of the near-optimal fidelity used as the performance metric. The blue panel summarizes the core mechanism of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), showing how code parameters are iteratively updated to maximize performance. The green and blue rounded-corner arrows indicate the iterative feedback loop between fidelity evaluation and parameter optimization.
• Figure 2: Optimal parameters of GKP codes versus loss and dephasing: (a) $\alpha$, (b) $\beta\mathrm{-real}$, and (c) $\Delta$. The shape parameters $\alpha$ and $\beta\mathrm{-real}$ are compared with the hexagonal lattice values ($\alpha \approx 1.347$ and $\beta\mathrm{-real} \approx 0.6734$). For most noise conditions, the optimized parameters remain close to the hexagonal lattice; systematic deviations appear when both loss and dephasing are strong. (d)-(f) show the Wigner functions of $|0_L\rangle$ for the hexagonal lattice and two deviated configurations. The corresponding noise strengths and code parameters are: (d) $\{\gamma t=0.05, \kappa t=10^{-4}, \alpha=1.35, \beta=0.672+1.16 \mathrm{i}, \Delta=0.18\}$; (e) $\{\gamma t=0.15, \kappa t=7.2 \times10^{-3}, \alpha=1.29, \beta=0.443+1.22 \mathrm{i}, \Delta=0.35\}$; (f) $\{\gamma t=0.15, \kappa t=6.6 \times10^{-3}, \alpha=1.29, \beta=0.875+1.22 \mathrm{i}, \Delta=0.34\}$.
• Figure 3: Optimal parameters of NP codes versus loss and dephasing: (a) $f$, (b) $s$, (c) Gaussian-state squeezing parameter $r$, (d) Gaussian-state mean photon number $n$, and (e) logical-state mean photon number $s \times n$. The parameters $f$ and $s$ determine the code-lattice geometry in the code space. (f)-(i) show the Wigner functions of $|0_L\rangle$ for four noise regimes: (f) lower left, (g) lower right, (h) top left, and (i) top right. The corresponding noise strengths and code parameters are: (f) $\{\gamma t=0.01, \kappa t=10^{-4}, f=0.500, s=5, r=0.0527, n=4.00\}$; (g) $\{\gamma t=0.01, \kappa t=7.2\times10^{-3}, f=0.584, s=3, r=0.3245, n=3.05\}$; (h) $\{\gamma t=0.15, \kappa t=10^{-4}, f=0.600, s=5, r=0.2309, n=4.00\}$; (i) $\{\gamma t=0.15, \kappa t=7.2\times10^{-3}, f=0.500, s=2, r=-0.0720, n=2.01\}$.
• Figure 4: (a), (b) Contour maps of near-optimal fidelity: (a) $\tilde{F}^{\mathrm{opt}}_{\mathrm{GKP}}$ and (b) $\tilde{F}^{\mathrm{opt}}_{\mathrm{NP}}$ versus loss and dephasing. (c)-(f) Corresponding bounds on the optimal fidelity, and the dashed lines are estimated by the trivial encoding using Fock states $|0\rangle$ and $|1\rangle$. (c), (e) Fidliety versus photon loss for fixed dephasing $\kappa t =0.0012$ (blue) and $0.0072$ (red). (d), (f) Fidelity versus dephasing for fixed photon loss $\gamma t=0.05$ (blue) and $0.15$ (red). Shaded regions indicate the range of optimal fidelity consistent with the two-sided bound inferred from the near-optimal fidelity.
• Figure 5: Contour map of the near-optimal fidelity difference $\Delta \tilde{F} = \tilde{F}^{\mathrm{opt}}_{\mathrm{GKP}} - \tilde{F}^{\mathrm{opt}}_{\mathrm{NP}}$ as a function of photon loss and dephasing. The green and blue curves denote the strict advantage boundaries of the GKP and NP codes, respectively. The black curve indicates the contour $\Delta\tilde{F}^{\mathrm{opt}}=0$.