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Barren plateaus are amplified by the dimension of qudits

Lucas Friedrich, Tiago de Souza Farias, Jonas Maziero

TL;DR

The paper addresses the challenge of barren plateaus (BP) in Variational Quantum Algorithms (VQAs) and shows that increasing qudit dimension $d'$ amplifies BP by affecting the gradient variance in relation to the Hilbert-space dimension $d = d'^n$. Using $t$-designs, the authors derive a theorem linking the gradient variance to $d'$ and provide a corollary for the common cost observable $O = |0\rangle\langle 0|$, demonstrating polynomial or exponential decay of the gradient with dimension and qudit count. They corroborate the theory with numerical experiments indicating that BP become more pronounced with higher $d'$ and larger $n$, though the observed scaling in practice may depend on the exact circuit design and the extent to which the unitary set forms a true $t$-design. The work highlights the need to tailor mitigation strategies for qudit-based VQAs and suggests that data encoding and gate decomposition choices could influence trainability in higher-dimensional quantum systems.

Abstract

Variational Quantum Algorithms (VQAs) have emerged as pivotal strategies for attaining quantum advantage in diverse scientific and technological domains, notably within Quantum Neural Networks. However, despite their potential, VQAs encounter significant obstacles, chief among them being the vanishing gradient problem, commonly referred to as barren plateaus. In this article, through meticulous analysis, we demonstrate that existing literature implicitly suggests the intrinsic influence of qudit dimensionality on barren plateaus. To instantiate these findings, we present numerical results that exemplify the impact of qudit dimensionality on barren plateaus. Therefore, despite the proposition of various error mitigation techniques, our results call for further scrutiny about their efficacy in the context of VQAs with qudits.

Barren plateaus are amplified by the dimension of qudits

TL;DR

The paper addresses the challenge of barren plateaus (BP) in Variational Quantum Algorithms (VQAs) and shows that increasing qudit dimension amplifies BP by affecting the gradient variance in relation to the Hilbert-space dimension . Using -designs, the authors derive a theorem linking the gradient variance to and provide a corollary for the common cost observable , demonstrating polynomial or exponential decay of the gradient with dimension and qudit count. They corroborate the theory with numerical experiments indicating that BP become more pronounced with higher and larger , though the observed scaling in practice may depend on the exact circuit design and the extent to which the unitary set forms a true -design. The work highlights the need to tailor mitigation strategies for qudit-based VQAs and suggests that data encoding and gate decomposition choices could influence trainability in higher-dimensional quantum systems.

Abstract

Variational Quantum Algorithms (VQAs) have emerged as pivotal strategies for attaining quantum advantage in diverse scientific and technological domains, notably within Quantum Neural Networks. However, despite their potential, VQAs encounter significant obstacles, chief among them being the vanishing gradient problem, commonly referred to as barren plateaus. In this article, through meticulous analysis, we demonstrate that existing literature implicitly suggests the intrinsic influence of qudit dimensionality on barren plateaus. To instantiate these findings, we present numerical results that exemplify the impact of qudit dimensionality on barren plateaus. Therefore, despite the proposition of various error mitigation techniques, our results call for further scrutiny about their efficacy in the context of VQAs with qudits.
Paper Structure (9 sections, 5 theorems, 65 equations, 10 figures)

This paper contains 9 sections, 5 theorems, 65 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Variational quantum algorithms
  3. Barren Plateaus
  4. Qudits
  5. Barren plateaus in qudit systems
  6. Results
  7. Conclusions
  8. Proof of Theorem 1
  9. Results in log-log scale

Key Result

Theorem 1

Let the cost function be defined in Eq. eq:cost, with $O$ being any observable, $\rho = | \psi \rangle \langle \psi |$ an initial state, and $U$ the parameterization defined in Eq. eq:parametrization, with $U_{l}$ given by Eq. eq:Ul. Then the variance of the partial derivative of the cost function i where $d=d'^{n}$ with $d'$ being the dimension of the qudits and $n$ the number of qudits used in t

Figures (10)

  • Figure 1: Illustration of the parameterizations used in this article. In this figure, each parameterization shows the form of the unitary $U_{l}$ used in Eq. \ref{['eq:parametrization']}. For parameterizations A and B, the CNOT gate is applied only between neighboring pairs of qudits, while in the parameterizations C and D it is applied between all pairs of qudits. In all these parameterizations, rotation gates $R^{\sigma}_{(j,k)}$ are applied to all qudits. The pair of variables $(j,k)$ are the indices of the Gell-Mann matrices, that we randomly choose. Moreover, $\sigma$ indicates which axis the rotation gate will be applied to, with $\sigma = X, Y, Z$. During the simulations, the $\sigma$ values are also chosen at random.
  • Figure 2: Behavior of the variance of the cost function, Eq. \ref{['eq:cost']}, with $O=|0 \rangle \langle 0|$ and parameterization $U$, Eq. \ref{['eq:parametrization']}, with $U_{l}$ given by Fig. \ref{['fig:param']}A. In this case, we can see that for sufficiently large $L$, the behavior of the variance is in accordance with Theorem \ref{['tr:1']}. However, for low values of $L$, specifically for $L=10$ and $L=15$, we can see that the behavior of the variance differs from what is expected according to Theorem \ref{['tr:1']}. As we will discuss later, this happens because the parameterization set $U$ generated does not form an exact $t$-design but rather an approximation. So, it is expected that the behavior of the variance differs from the theoretical result.
  • Figure 3: Behavior of the variance of the cost function, Eq. \ref{['eq:cost']}, with $O= |0 \rangle \langle 0|$ and parameterization $U$, Eq. \ref{['eq:parametrization']}, with $U_{l}$ shown in Fig. \ref{['fig:param']}B. In contrast to the previous case (Fig. \ref{['fig:model_1']}), here the behavior of the variance of the cost function is in accordance with Theorem \ref{['tr:1']} in all cases.
  • Figure 4: Behavior of the variance of the cost function for $U_{l}$ given by Fig. \ref{['fig:param']}C. Similar to the case seen in Fig. \ref{['fig:model_1']}, for relatively low values of $L$, the behavior of the variance differs from what is expected according to Theorem \ref{['tr:1']}.
  • Figure 5: Behavior of the variance of the cost function for the parameterization obtained using $U_{l}$ shown in Fig. \ref{['fig:param']}D. Again, we use $O= |0 \rangle \langle 0|$. As we can observe immediately, similar to the case seen in Fig. \ref{['fig:model_2']}, the behavior of $Var[\partial_{k}C]$ is in accordance with Theorem \ref{['tr:1']} in all cases analyzed.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3