Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

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

Oluwadara Ogunkoya, Kirsten Morris, Dog̃a Murat Kürkçüog̃lu

Published Jun 6, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ade206

Related papers

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

Oluwadara Ogunkoya, Morris, Kirsten, Dog̃a Murat Kürkçüog̃lu

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.

arXiv:2309.14942•Sep 26, 2023•0 citations

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

Oluwadara Ogunkoya, Kirsten Morris, Dog̃a Murat Kürkçüog̃lu

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.

•Dec 15, 2023•3 citations

Reducing the Cost of Qudit State Preparation by Learning a Sparse Structure for SNAP Gates

Marco Pennazio, Leonardo Ignazio Pagliochini, Riccardo Pellini +2 more

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.

quant-pharXiv:2610.21557•Oct 16, 2026•0 citations

Linking barren plateaus to effective parameters in deep rotation-gate-based parameterized quantum circuits

Yuhan Yao, Yoshihiko Hasegawa

The barren plateaus phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. Foundational analyses of barren plateaus often use methods like the Weingarten formula, which, under the global Haar random assumption, effectively links the gradient variance to the Hilbert space dimension. However, many practical circuits, such as Hardware-Efficient Ans\"atze, utilize a different, non-Haar random model based on the sampling of parameters for single-qubit rotation gates. This study introduces a direct computational method for the gradient expectation and variance, specifically for parametrized quantum circuits constructed from single-qubit rotation gates. Our analysis reveals a direct, quantitative relationship between the gradient variance, circuit depth, and number of effective parameters. Numerical simulations further confirm the validity of our theoretical results. Our findings highlight the role of effective parameters as a key indicator for gradient trainability and provide a precise framework for analyzing barren plateaus in these specific non-Haar random circuits.

arXiv:2503.05145•Physical Review A•Dec 3, 2025•0 citations

Sparse Phase Ansatzes for Resource-Efficient Qudit State Preparation via the SNAP-Displacement Protocol

Maurizio Ferrari Dacrema, Doga Murat Kurkcuoglu, Andy C. Y. Li +2 more

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 $d=64$, 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.

quant-pharXiv:2603.12203•Mar 13, 2026•0 citations

Batched Line Search Strategy for Navigating through Barren Plateaus in Quantum Circuit Training

Jakab Nádori, Gregory Morse, Barna Fülöp Villám +3 more

Variational quantum algorithms are viewed as promising candidates for demonstrating quantum advantage on near-term devices. These approaches typically involve the training of parameterized quantum circuits through a classical optimization loop. However, they often encounter challenges attributed to the exponentially diminishing gradient components, known as the barren plateau (BP) problem. This work introduces a novel optimization approach designed to alleviate the adverse effects of BPs during circuit training. In contrast to conventional gradient descent methods with a small learning parameter, our approach relies on making a finite hops along the search direction determined on a randomly chosen subsets of the free parameters. The optimization search direction, together with the range of the search, is determined by the distant features of the cost-function landscape. This enables the optimization path to navigate around barren plateaus without the need for external control mechanisms. We have successfully applied our optimization strategy to quantum circuits comprising 21 qubits and 15000 entangling gates, demonstrating robust resistance against BPs. Additionally, we have extended our optimization strategy by incorporating an evolutionary selection framework, enhancing its ability to avoid local minima in the landscape. The modified algorithm has been successfully utilized in quantum gate synthesis applications, showcasing a significantly improved efficiency in generating highly compressed quantum circuits compared to traditional gradient-based optimization approaches.

arXiv:2402.05227•Quantum•Aug 29, 2025•1 citation
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.
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.
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.
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.
The barren plateaus phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. Foundational analyses of barren plateaus often use methods like the Weingarten formula, which, under the global Haar random assumption, effectively links the gradient variance to the Hilbert space dimension. However, many practical circuits, such as Hardware-Efficient Ans\"atze, utilize a different, non-Haar random model based on the sampling of parameters for single-qubit rotation gates. This study introduces a direct computational method for the gradient expectation and variance, specifically for parametrized quantum circuits constructed from single-qubit rotation gates. Our analysis reveals a direct, quantitative relationship between the gradient variance, circuit depth, and number of effective parameters. Numerical simulations further confirm the validity of our theoretical results. Our findings highlight the role of effective parameters as a key indicator for gradient trainability and provide a precise framework for analyzing barren plateaus in these specific non-Haar random circuits.
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.
Variational quantum algorithms are viewed as promising candidates for demonstrating quantum advantage on near-term devices. These approaches typically involve the training of parameterized quantum circuits through a classical optimization loop. However, they often encounter challenges attributed to the exponentially diminishing gradient components, known as the barren plateau (BP) problem. This work introduces a novel optimization approach designed to alleviate the adverse effects of BPs during circuit training. In contrast to conventional gradient descent methods with a small learning parameter, our approach relies on making a finite hops along the search direction determined on a randomly chosen subsets of the free parameters. The optimization search direction, together with the range of the search, is determined by the distant features of the cost-function landscape. This enables the optimization path to navigate around barren plateaus without the need for external control mechanisms. We have successfully applied our optimization strategy to quantum circuits comprising 21 qubits and 15000 entangling gates, demonstrating robust resistance against BPs. Additionally, we have extended our optimization strategy by incorporating an evolutionary selection framework, enhancing its ability to avoid local minima in the landscape. The modified algorithm has been successfully utilized in quantum gate synthesis applications, showcasing a significantly improved efficiency in generating highly compressed quantum circuits compared to traditional gradient-based optimization approaches.