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Investigating parameter trainability in the SNAP-Displacement protocol of a qudit system.

Oluwadara Ogunkoya, Kirsten Morris, Dog̃a Murat Kürkçüog̃lu

Published Dec 15, 2023• 3 citations
Publisher

Abstract

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 SNAP -Displacement blocks. We utilize techniques similar to those in [8] and [2] 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) using Haar measure properties, 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.

DOI
10.2172/2246726

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•Aug 19, 2026•0 citations
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 SNAP -Displacement blocks. We utilize techniques similar to those in [8] and [2] along with the concept of t−t-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) using Haar measure properties, 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.
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−t-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.
The emergence of the Barren Plateau phenomenon poses a significant challenge to quantum machine learning. While most Barren Plateau analyses focus on single-qubit rotation gates, the gradient behavior of Parameterized Quantum Circuits built from multi-qubit gates remains largely unexplored. In this work, we present a general theoretical framework for analyzing the gradient properties of Parameterized Quantum Circuits with multi-qubit gates. Our method generalizes the direct computation framework, bypassing the Haar random assumption on parameters and enabling the calculation of the gradient expectation and variance. We apply this framework to single-layer and deep-layer circuits, deriving analytical results that quantify how gradient variance is co-determined by the size of the multi-qubit gate and the number of qubits, layers, and effective parameters. Numerical simulations validate our findings. Our study provides a refined framework for analyzing and optimizing Parameterized Quantum Circuits with complex multi-qubit gates.
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 [Fösel et al 2020 Efficient cavity control with snap gates arXiv:2004.14256]. Applying Variational Quantum algorithm (VQA) with observable and gate cost functions, we utilize techniques similar to those in [McClean et al 2018 Barren plateaus in quantum neural network training landscapes Nat. Commun. 9 1–6] and [Cerezo et al 2021 Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12 1–12] 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.
Preparing quantum states in bosonic qudits with SNAP--Displacement sequences requires both optimizing the gate parameters and deciding which Fock levels should be controlled by each SNAP gate. A fully parameterized ansatz assigns an independent phase to every Fock level in every gate, leading to a number of phases that grows steeply. This constitutes a problem for short to medium term systems as controlling many phases requires complex control pulses which are subject to increased control errors. Previous work reduced this cost by restricting the trainable phases to predefined sparse patterns, such as a diagonal or a grid, which however may not be optimal for a specific target state. In this work, we address this limitation by introducing a method that learns which phases are needed together with their values, using a differentiable mask. A sparsity penalty encourages the removal of unnecessary phases, while temperature annealing gradually turns the masks into binary decisions. We compare the proposed method with fully parameterized and predefined sparse ansatzes on Fourier and Haar-random target states. The evaluation considers both the minimum number of phases required to reach different infidelity thresholds and the overall trade-off between fidelity and phases required, measured through Pareto frontiers and hypervolume. The results show that learning the location of the phases together with their values provides competitive or improved fidelity--resource trade-offs, although the advantage depends on the target state and qudit dimension.
Efficient preparation of nonclassical bosonic states is a central requirement for quantum computing, simulation, and precision metrology. We study resource-efficient quantum state preparation in bosonic qudit systems using the SNAP-displacement (SD) protocol. Existing SD-based approaches typically require a large number of gates and SNAP phases, resulting in complex control pulses with longer ansatz durations and amplified impact of photon-loss and control errors. In this work, we focus on the near- to medium-term regime, in which noisy quantum devices impose trade-offs on the fidelity that can be achieved, which must be taken into account. Specifically, we propose to optimize only a subset of the SNAP phases and introduce three progressively more general sparse ansatzes. To provide fine-grained control and identify the most suitable ansatz for a given target fidelity, we further employ a scalarized multi-objective optimization that trades off fidelity against either the number of phases or the duration of the ansatz. Numerical results for several target states and qudit dimensions up to d=64d=64d=64, evaluated through the hypervolume of the Pareto frontiers, show that these sparse ansatzes achieve favorable trade-offs over the fully parameterized SD protocol in both ideal and noisy settings. The advantage is strongest and most consistent when minimizing the number of phases, while improvements in ansatz duration are smaller and more dependent on the target-state family and noise level, suggesting a practical route to more efficient near- and medium-term bosonic state preparation.
Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance Var[∂C/∂θ]\mathrm{Var}[\partial C/\partial\theta]Var[∂C/∂θ] from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts log⁡10Var\log_{10}\mathrm{Var}log10​Var with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes (n=11,12n=11,12n=11,12). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.