Numerical study of the performance of two-mode rotationally symmetric bosonic codes

Bosonic codes offer a promising path toward fault-tolerant quantum computing by encoding logical information in the infinite-dimensional Hilbert space of harmonic oscillators. In single-mode rotationally symmetric bosonic (RSB) codes, increasing the order of symmetry N improves protection against photon loss at the expense of dephasing resilience. Recently developed two-mode RSB codes [arXiv:2508.20647] have been proven to overcome this trade-off. However, evaluating the performance of two-mode systems presents a significant computational challenge. A crucial tool in this evaluation is the optimal fidelity, which benchmarks a code’s performance and is obtained by optimizing over all recovery operations via a Semidefinite Program (SDP). Larger N leads to states with higher photon-number support, requiring a larger Hilbert space truncation. Since the complexity of solving the SDP scales with the Hilbert space dimension, this makes the task quickly intractable. This thesis presents a computational study dedicated to pushing two-mode bosonic simulations to their numerical limits, with a particular focus on dual-rail binomial and cat encodings. We primarily utilize a computationally favorable alternative: the near-optimal fidelity metric [PRL 132, 250602], whose complexity is governed by the precision of the channel approximation rather than by the Hilbert space dimension, thereby avoiding the critical scaling associated with the SDP. The exact SDP was nonetheless solved whenever feasible, particularly for the case of pure dephasing noise by exploiting a compressed representation that rendered simulations up to N=20 tractable, significantly extending earlier studies. Our results further demonstrate how two-mode codes overcome the single-mode trade-off: for the dual-rail binomial encoding, higher N yields better performance systematically under photon loss while preserving strong dephasing resilience. Additionally, simulations of the N=K family, where K is related to the average photon number of the bosonic code, reveal unexpected scaling behavior under dephasing, opening new questions for future investigation. Simulations of the channel combining both loss and dephasing extend previous results to larger system sizes, demonstrating the capability of our framework to explore regimes previously out of reach and laying the groundwork for a deeper theoretical understanding. Finally, by simulating the dual-rail cat encoding in the large-amplitude regime, previously out of reach, we demonstrate its ability to surpass the so-called break-even point, where an error-corrected code performs as well as trivial encoding, under both pure loss and pure dephasing. This establishes a computational foundation for the systematic evaluation of two-mode cat encodings, which we leave to future work. Optimal beam-splitter angles for mixing the two modes are also identified, extending analogous results previously obtained for the dual-rail binomial encoding.