Investigating Parameter Trainability in the SNAP-Displacement Protocol of a Qudit system

TL;DR

This paper investigates the trainability of SNAP-Displacement quantum gates in qudit systems, analyzing conditions that affect optimization and comparing their behavior to multi-qubit systems using advanced mathematical tools.

Contribution

It introduces new lemmas on Haar measure moments and identifies conditions where qudit systems have a trainability advantage over multi-qubit systems.

Findings

SNAP-parameter trainability shows no directional preference in the cost landscape.

New lemmas relate Haar measure moments to polynomial expectations.

Conditions are identified where qudit systems outperform multi-qubit systems in trainability.

Abstract

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.