Investigating trainability of parameters in the Snap-Displacement protocol of a Qudit system.

SNAP and displacement protocol is universal in qudit based systems, however finding the

optimal parameters for these gates is a non-trivial problem. We analyze optimization problems

caused by the multi-modal nature of the cost function which leads to trapping in points of the local

minima and we investigate whether the SNAP and displacement protocol has the barren plateau

problem. We consider a parameterized ansatz comprising of sequence of blocks where each block

is made up of hardware operations –Snap and Displacement gates [2]. For a Variational Quantum

Algorithm (VQA) with Observable and Gate cost functions, we apply similar techniques in [1]

and [3] and t−design concept to show that (a) the trainability of a parameter does not favor any

particular direction in our cost function landscape, (b) by applying the first and second moments

properties of Haar measures we give new lemmas for the expectation of functions of particular

forms, and (c) by applying these new lemmas, we show that the chances of training a parameter

decreases polynomially in the Hilbert space dimension. Specifically, we show that we do not have

dependency on the length of the ansatz in contrast to the qubit result in [3] and also, we give a

particular case of trainability of parameters in the observable cost function.

References

[1] Jarrod R McClean et al. “Barren plateaus in quantum neural network training landscapes”.

In: Nature communications 9.1 (2018), pp. 1–6.

[2] Thomas F ¨osel et al. “Efficient cavity control with SNAP gates”. In: arXiv preprint arXiv:2004.14256

(2020).

[3] Marco Cerezo et al. “Cost function dependent barren plateaus in shallow parametrized quan-

tum circuits”. In: Nature communications 12.1 (2021), pp. 1–12.

This material is based upon work supported by the U.S. Department of Energy, Office of Science,

National Quantum Information Science Research Centers, Superconducting Quantum Materials

and Systems Center (SQMS) under contract number DE-AC02-07CH11359.”