Kostant relation in filtered randomized benchmarking for passive bosonic devices
David Amaro-Alcalá
TL;DR
This work tackles the high computational and experimental cost of benchmarking passive bosonic devices, where prior methods required evaluating matrix permanents and Clebsch-Gordan coefficients and relied on photon-number-resolving detectors. By leveraging Kostant’s relation, it replaces the original filter with an immanant-based filter, enabling CG-free data analysis and expressing the relevant filter in terms of a smaller set of immanants Imm_\kappa(U). The proposed approach maintains the same data and yields a single exponential decay in a fidelity-like figure $F(\mathcal{E}) = d_\lambda^{-2}\sum_\mu d_\mu p_\mu(\mathcal{E})$, while allowing characterization with weak coherent states and intensity measurements and even tolerating gain/loss via an extended Hilbert space. Practically, this reduces both computational overhead and experimental complexity, broadening the applicability of bosonic RB to more platforms and facilitating scalable continuous-variable benchmarking.
Abstract
We reduce the cost of the current bosonic randomized benchmarking proposal. First, we introduce a filter function using immanants. With this filter, we avoid the need to compute Clebsch-Gordan coefficients. Our filter uses the same data as the original, although we propose a distinct data collection process that requires a single type of measurement. Furthermore, we argue that weak coherent states and intensity measurements are sufficient to proceed with the characterization. Our work could then allow simpler platforms to be characterized and simplify the data analysis process.