Kostant relation in filtered randomized benchmarking for passive bosonic devices

TL;DR

This paper introduces two filter functions for bosonic randomized benchmarking that reduce computational complexity and variance, enabling more efficient and accurate device characterization.

Contribution

The authors propose novel filter functions based on immanants and characters of SU(n), simplifying variance calculations and improving efficiency in bosonic benchmarking.

Findings

Character filter has low, constant variance.

Numerical results show effective benchmarking with loss and gain.

Scheme with weak coherent states yields accurate estimates.

Abstract

We aim to reduce the cost of the current bosonic randomized benchmarking proposal. To do this, we introduce two filter functions: one uses immanants, the other uses characters of the special unitary group. These filters avoid computing Clebsch-Gordan coefficients and yield simple variance expressions. The character filter is not only efficient to compute, but also has a constant, low variance. Our filters rely on the same data as the original proposal. We also discuss an example with photon loss and gain. Our numerical evidence shows that a scheme using weak coherent states and intensity measurements can yield estimates close to those obtained without loss or gain. Our work could support simpler platform characterization and streamline data analysis.