Variational Quantum Algorithms (VQAs) have emerged as pivotal strategies for
attaining quantum advantages in diverse scientific and technological domains,
notably within Quantum Neural Networks. However, despite their potential, VQAs
encounter significant obstacles, chief among them being the gradient vanishing
problem, commonly referred to as barren plateaus. In this study, we unveil a
direct correlation between the dimension of qudits and the occurrence of barren
plateaus, a connection previously overlooked. 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. Additionally, despite the proposition of
various error mitigation techniques, our results call for further scrutiny
about their efficacy in the context of VQAs with qudits.

通过细致的分析，本研究揭示了多维量子比特的维度与贫瘠平台之间的直接关联，并通过数值结果展示了多维量子比特对贫瘠平台的影响，同时对基于多维量子比特的 VQA 的误差缓解技术的功效提出了进一步的审视。

基于多能级量子位的贫瘠高地

Barren plateaus induced by the dimension of qudits

Unitary $k$-designs are finite ensembles of unitary matrices that approximate
the Haar distribution over unitary matrices. Several ensembles are known to be
2-designs, including the uniform distribution over the Clifford group, but no
family of ensembles was previously known to form a 3-design. We prove that the
Clifford group is a 3-design, showing that it is a better approximation to
Haar-random unitaries than previously expected. Our proof strategy works for
any distribution of unitaries satisfying a property we call Pauli 2-mixing and
proceeds without the use of heavy mathematical machinery. We also show that the
Clifford group does not form a 4-design, thus characterizing how well random
Clifford elements approximate Haar-random unitaries. Additionally, we show that
the generalized Clifford group for qudits is not a 3-design unless the
dimension of the qudit is a power of 2.

本文证明了 Clifford 群是 3 设计，是以前所预期的逼近 Haar 随机酉矩阵更好的途径。我们的证明策略适用于任何满足我们所谓的 Pauli 2 混合属性的酉分布，且不需要使用沉重的数学工具。此外，我们还表明广义 Clifford 群不是 3 设计，除非 qudit 的维度是 2 的幂。