  In this study, we explore the universality of Selective Number-dependent
Arbitrary Phase (SNAP) and Displacement gates for quantum control in
qudit-based systems. However, optimizing the parameters of these gates poses a
challenging task. Our main focus is to investigate the sensitivity of training
any of the SNAP parameters in the SNAP-Displacement protocol. We analyze
conditions that could potentially lead to the Barren Plateau problem in a qudit
system and draw comparisons with multi-qubit systems. The parameterized ansatz
we consider consists of blocks, where each block is composed of hardware
operations, namely SNAP and Displacement gates \cite{fosel2020efficient}.
Applying Variational Quantum Algorithm (VQA) with observable and gate cost
functions, we utilize techniques similar to those in \cite{mcclean2018barren}
and \cite{cerezo2021cost} along with the concept of $t-$design. Through this
analysis, we make the following key observations: (a) The trainability of a
SNAP-parameter does not exhibit a preference for any particular direction
within our cost function landscape, (b) By leveraging the first and second
moments properties of Haar measures, we establish new lemmas concerning the
expectation of certain polynomial functions, and (c) utilizing these new
lemmas, we identify a general condition that indicates an expected trainability
advantage in a qudit system when compared to multi-qubit systems.
