Rotation-Symmetric Bosonic Code Error Correction: Investigating error-correcting capabilities of the Two-Mode Binomial Code

Abstract

This project investigates the error-correcting capabilities of the Two-Mode Binomial Code, a subclass of Rotation-Symmetric Bosonic (RSB) codes designed to mitigate the trade-off between protection against photon loss and dephasing noise. The project focuses exclusively on the ability of the code to correct photon loss errors. While the Knill-Laflamme (KL) conditions for this code are known to be satisfied to the first order in the photon decay rate, its behavior at higher orders has not been previously established. We derive the two-mode photon loss error operators to the second order in κt and evaluate the satisfaction of the KL conditions for code parameters K = 2 with symmetry orders N = 2 and N = 4. By utilizing a cost function to quantify the violation of these conditions, our results demonstrate that the KL conditions are not strictly satisfied for either symmetry order. However, the N = 4 code shows a higher degree of satisfaction compared to N = 2, with optimal beam-splitter parameters found at δ = π/4 and 3π/4. This corresponds to the values required for first-order dephasing correction. These findings indicate that while exact error correction is not feasible to the second order, the N = 4 Two-Mode Binomial Code remains a promising candidate for use with approximate recovery channels such as teleportation or the Petz Recovery Map.