It is essential to select efficient topology of parameterized quantum
circuits (PQCs) in variational quantum algorithms (VQAs). However, there are
problems in current circuits, i.e. optimization difficulties caused by too many
parameters or performance is hard to guarantee. How to reduce the number of
parameters (number of single-qubit rotation gates and 2-qubit gates) in PQCs
without reducing the performance has become a new challenge. To solve this
problem, we propose a novel topology, called Block-Ring (BR) topology, to
construct the PQCs. This topology allocate all qubits to several blocks,
all-to-all mode is adopt inside each block and ring mode is applied to connect
different blocks. Compared with the pure all-to-all topology circuits which own
the best power, BR topology have similar performance and the number of
parameters and 2-qubit gate reduced from 0(n^2) to 0(mn) , m is a
hyperparameter set by ourselves. Besides, we compared BR topology with other
topology circuits in terms of expressibility and entangling capability.
Considering the effects of different 2-qubit gates on circuits, we also make a
distinction between controlled X-rotation gates and controlled Z-rotation
gates. Finally, the 1- and 2-layer configurations of PQCs are taken into
consideration as well, which shows the BR's performance improvement in the
condition of multilayer circuits.

在变分量子算法中，选择高效的参数化量子电路拓扑非常重要。然而，现有电路存在的问题是参数太多导致优化困难或者性能难以保证。如何在不降低性能的情况下减少参数数量成为了新的挑战。为了解决这个问题，我们提出了一种新的拓扑结构，称为 “Block-Ring (BR) topology”，以构建参数化量子电路。这种拓扑将所有量子比特分配给若干个块，每个块内部采用全对全模式，不同块之间采用环形连接。相比于最佳性能的全对全拓扑电路，BR 拓扑具有类似的性能，并且参数数量和两比特门数量从 O (n^2) 减少到 O (mn)，其中 m 是我们自己设置的超参数。此外，我们还将 BR 拓扑与其他拓扑电路在表达能力和纠缠能力方面进行了比较。考虑到不同的两比特门对电路的影响，我们还区分了控制 X 旋转门和控制 Z 旋转门。最后，我们还考虑了 PQC 的 1 层和 2 层配置，结果表明在多层电路条件下，BR 性能有所提高。

参数化量子电路的块环连接拓扑

A Block-Ring connected Topology of Parameterized Quantum Circuits

Normalization layers (e.g., Batch Normalization, Layer Normalization) were
introduced to help with optimization difficulties in very deep nets, but they
clearly also help generalization, even in not-so-deep nets. Motivated by the
long-held belief that flatter minima lead to better generalization, this paper
gives mathematical analysis and supporting experiments suggesting that
normalization (together with accompanying weight-decay) encourages GD to reduce
the sharpness of loss surface. Here "sharpness" is carefully defined given that
the loss is scale-invariant, a known consequence of normalization.
Specifically, for a fairly broad class of neural nets with normalization, our
theory explains how GD with a finite learning rate enters the so-called Edge of
Stability (EoS) regime, and characterizes the trajectory of GD in this regime
via a continuous sharpness-reduction flow.

通过数学分析和实验证明，在深度网络中引入标准化层（例如批量标准化，层标准化）有利于优化并促进泛化，同时对于包含标准化的一类神经网络，伴随权值衰减的标准化可以鼓励梯度下降到达稳定边缘，并且对于这种情况，可以确定梯度下降的流动轨迹。

理解归一化层的泛化增益：锐化减少

Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction

Many deep neural networks trained on natural images exhibit a curious
phenomenon in common: on the first layer they learn features similar to Gabor
filters and color blobs. Such first-layer features appear not to be specific to
a particular dataset or task, but general in that they are applicable to many
datasets and tasks. Features must eventually transition from general to
specific by the last layer of the network, but this transition has not been
studied extensively. In this paper we experimentally quantify the generality
versus specificity of neurons in each layer of a deep convolutional neural
network and report a few surprising results. Transferability is negatively
affected by two distinct issues: (1) the specialization of higher layer neurons
to their original task at the expense of performance on the target task, which
was expected, and (2) optimization difficulties related to splitting networks
between co-adapted neurons, which was not expected. In an example network
trained on ImageNet, we demonstrate that either of these two issues may
dominate, depending on whether features are transferred from the bottom,
middle, or top of the network. We also document that the transferability of
features decreases as the distance between the base task and target task
increases, but that transferring features even from distant tasks can be better
than using random features. A final surprising result is that initializing a
network with transferred features from almost any number of layers can produce
a boost to generalization that lingers even after fine-tuning to the target
dataset.

本论文研究了深度神经网络中神经元的广泛性与特定性，发现转移特征会受到特化高层神经元和合作适应神经元之间优化困难性的限制，转移特征的适用性随着目标任务与基础任务的差距而降低，但即使从远任务中转移特征，也比使用随机特征好，从几乎任意数量的层中初始化网络都可以提高广义化的效果，即使经过微调后能够持续。