# Benchmarking Spiking Neural Network Learning Methods with Varying Locality

Jiaqi Lin, Sen Lu, Malyaban Bal, Abhronil Sengupta J. Lin, S. Lu, M. Bal and
A. Sengupta are with the School of Electrical Engineering and Computer
Science, The Pennsylvania State University, University Park, PA 16802, USA.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Spiking Neural Networks (SNNs), providing more realistic neuronal dynamics,
have shown to achieve performance comparable to Artificial Neural Networks
(ANNs) in several machine learning tasks. Information is processed as spikes
within SNNs in an event-based mechanism that significantly reduces energy
consumption. However, training SNNs is challenging due to the non-
differentiable nature of the spiking mechanism. Traditional approaches, such
as Backpropagation Through Time (BPTT), have shown effectiveness but comes
with additional computational and memory costs and are biologically
implausible. In contrast, recent works propose alternative learning methods
with varying degrees of locality, demonstrating success in classification
tasks. In this work, we show that these methods share similarities during the
training process, while they present a trade-off between biological
plausibility and performance. Further, this research examines the implicitly
recurrent nature of SNNs and investigates the influence of addition of
explicit recurrence to SNNs. We experimentally prove that the addition of
explicit recurrent weights enhances the robustness of SNNs. We also
investigate the performance of local learning methods under gradient and non-
gradient based adversarial attacks.

###### Index Terms:

Spiking Neural Networks, Local Learning, Training Methods, Centered Kernel
Alignment, Fisher Information.

##  I Introduction

Spiking Neural Networks (SNNs) is a type of Artificial Neural Networks (ANNs)
that is inspired by biological neurons in the brain [1, 2]. In contrast to
traditional ANNs, SNNs use a more biologically realistic model of neuron
behavior, where information is processed through the propagation of spikes
between neurons in the network [3]. The timing and rate of spikes are
influenced by the strength of synaptic connections between neurons, which can
be modified through a process called synaptic plasticity [4, 5]. In SNNs,
neuronal activities are based on discrete spikes, which occur only when the
membrane potential of a neuron exceeds a certain threshold. This event-driven
processing operates at significantly lower energy levels than traditional
computing architectures [6, 7, 8], as the neuronal and synaptic state updates
consume power only when they are actively spiking [9]. Neuromorphic hardware
implementations, like Intel’s Loihi [10] and the SpiNNaker project [11], are
specifically designed to take advantage of SNNs. The energy efficiency offered
by SNNs integrated with neuromorphic hardware provide efficient possibilities
for deploying machine learning applications in power-constrained environments,
such as mobile devices, wearable technology, and remote sensing systems, among
others.

Additionally, since membrane potentials are accumulated over time retaining
information from previous time steps, SNNs are inherently recurrent [12]. This
characteristic of SNNs allows them to naturally process time-dependent data,
rendering them suitable for tasks such as speech recognition or time-series
prediction [12, 13, 14, 15, 16]. SNNs with explicit recurrent connections have
shown to further augment their performance in speech processing tasks [13].

Although SNNs are demonstrated to provide multiple benefits from algorithms to
hardware by leveraging bio-realistic computations, Backpropagation Through
Time (BPTT), which is widely used to train SNNs, is not biologically
plausible. Firstly, BPTT assumes weight symmetry for the forward and backward
passes, which is undesirable in biological neural networks [17]. Secondly,
BPTT requires global error propagation, which is inconsistent with the local
learning mechanisms observed in biological neural networks [18, 19]. Moreover,
the explicit calculation and back-propagation of gradients and error signals
in BPTT is not directly observed in the brain [20, 21]. Lastly, BPTT maintains
a history of the membrane potential and spike history of each neuron over
time. This process can be memory-intensive and computationally expensive [22,
23, 19, 15, 24]. To solve this problem, alternative training methods, such as
Spike Timing-Dependent Plasticity (STDP) which is a specific form of Hebbian
learning [5, 25] and local learning alternatives to BPTT have been proposed
for training SNNs [13, 26, 27, 19, 28]. These methods aim to address the
challenges associated with training SNNs while still maintaining biologically
plausible learning rules and enabling more efficient and accurate learning in
spiking networks.

Figure 1: SNN training methods considered in this study from global learning
to local learning: BPTT, Feedback Alignment, e-prop, DECOLLE. ANN-SNN
conversion is also depicted on the extreme right.

Based on the the locality of training methods [27], we revisit these
alternatives to BPTT. Figure 1 distinguishes SNN training methods based on
varying levels of locality. Beginning with BPTT, which is a global learning
method, it unrolls neural dynamics over time, and it involves backpropagating
errors and learning signals through layers. The connection used for
backpropagation is the same weight used for forward propagation. An
alternative, named feedback alignment, replaces the symmetric weight
connections with random matrices during backward passes. This method removes
the need for symmetric weight during learning. E-prop introduces a type of
direct feedback alignment that provides higher biological plausibility [15].
In this method, the error calculated at the output layer is propagated
directly to each layer without propagating errors layer by layer. To obtain a
better performance, e-prop employs symmetric weights in the process of direct
propagation. We next consider a more localized method, DECOLLE, in the regime
of local error propagation [19]. In this learning rule, errors are propagated
at each layer. A cost function is assigned at each layer by a random matrix
mapping the output from each layer to the pseudo-targets. In general, all
these learning rules provide bio-plausible alternatives to global learning
rules.

##  II Related Works

###  II-A Performance Evaluation

Recently, the development of novel learning algorithms to train SNNs have
gained traction, propelled by SNNs’ unique spiking nature and energy advantage
over ANNs. Whether energy efficiency is a compromise of performance remains a
question. Ref. [29] explored such a co-design evaluation with respect to ANN
performance. Their results indicated that SNNs maintained acceptable accuracy
drops on simpler visual tasks like MNIST classification with much better
energy efficiency, while in relatively more complicated visual tasks like
CIFAR10 classification, the accuracy of SNNs degraded further. Despite this,
SNNs evaluated on neuromorphic datasets were illustrated to outperform ANNs.
Subsequent progress in SNN training have produced state-of-the-art performance
of SNNs in several image recognition benchmarks with recent works indicating
highly similar representations learned by residual SNNs and ANNs using
representational similarity measurements [30].

Moreover, the inherently recurrent nature of SNNs resembles the operation of
recurrent neural networks (RNNs) in terms of temporal and spatial neural
dynamics. Inspired by such characteristics, prior work has compared SNNs and
RNNs, highlighting the superior performance of SNNs over RNNs on neuromorphic
datasets [31]. Additionally, the work investigated the influence of temporal
resolution, leakage and reset mechanisms on the performance of SNNs,
illustrating that the exclusion of leakage and reset mechanisms negatively
impacted the performance, and the increment of temporal resolution was not
necessarily useful. In these benchmarking works, SNNs were trained with the
BPTT training method, where surrogate functions were used to overcome the non-
differentiability of spiking neurons. To quantitatively analyze the impact of
surrogate functions on the performance of SNNs, experimental results from
previous work have demonstrated that surrogate functions help recurrent SNNs
outperform conventional RNNs [14]. Additionally, this work examined the
advantage of adaptive neurons [13], where thresholds are updated based on
firing rates, and illustrated that recurrent SNNs with adaptive neurons gain
performance advantages over SNNs with fixed thresholds. Mixed usage of both
types of neurons was shown to improve the performance of SNNs over LSTM
architectures [13].

In conclusion, prior works have mainly targeted benchmarking comparable
performance of SNNs and ANNs. Some works have investigated the influence of
various SNN-specific control knobs like leakage, temporal dynamics, among
others on SNN performance. However, the majority of benchmarking efforts
optimize SNNs via the BPTT algorithm, thereby not providing insights related
to emerging SNN learning schemes offering local learning possibilities. In
contrast to prior works, we perform an extensive benchmarking analysis with a
complementary perspective where we investigate the performance of SNNs with
regards to learning methods in terms of accuracy and robustness with both
implicit and explicit recurrent architectures.

###  II-B Adversarial Attacks

The adversarial robustness of SNNs has been proven by various researchers.
Sharmin et al. [32] explored gradient-based attacks and its variants on both
the ANN-converted SNN and SNN trained from scratch and found SNN’s robustness
against adversarial attacks. Similarly, Liang et al. [33] conducted untargeted
gradient attacks, and found that SNNs require more perturbations for
successful attacks. Marchisio et al. [34] considered a deep Spiking Deep
Belief Network (SDBN) using noise attacks with the same motive of backdoor
trigger attacks [35] and reached similar conclusions. However, their studies
were either focused on a single training method or relied entirely on the
gradient. However, while at its exploration stage, many recent SNN training
methods look for alternatives to using gradient information due to high
computational costs [24, 36]. In this work, we aim to bridge this gap by
revisiting adversarial attacks on SNNs and provide a benchmarking analysis for
non-gradient poisoning attacks as well. The benchmarking effort is evaluated
corresponding to various SNN training paradigms for future researchers to make
more informed design choices.

##  III Methods

Sections A-B discusses SNN computational models and learning algorithms
considered in this work while sections C-D introduces empirical tools used to
understand the learning dynamics of various SNN topologies and learning
methods.

###  III-A Spiking Neurons

In this study, we use SNNs with Leaky-Integrate-and-Fire (LIF) neurons [14,
27]. The membrane potential $U^{l}(t)$ and synaptic currents $I^{l}(t)$ for a
layer $l$ at time step $t$ in a discrete time setting is expressed as:

$\displaystyle I^{l}(t)$ $\displaystyle=\exp(-\frac{1}{\tau_{\rm
syn}})I^{l}(t-1)+W^{l}S^{l-1}(t-1)+V^{l}S^{l}(t-1)$ $\displaystyle U^{l}(t)$
$\displaystyle=\exp(-\frac{1}{\tau_{\rm mem}})U^{l}(t-1)+I^{l}(t-1)-RV_{\rm
th}$

Here, $W^{l}$ represents the weight connection between the current layer $l$
and the previous layer $l-1$, and $V^{l}$ represents the recurrent weight
connection of current layer. $S^{l-1}(t)$ denotes the output of the previous
layer $l-1$ at time step $t$. The parameters $\exp(\frac{-1}{\tau_{\rm syn}})$
and $\exp(\frac{-1}{\tau_{\rm mem}})$ correspond to the synaptic and membrane
decay rate respectively. Each neuron maintains a membrane potential
$U^{l}(t)$, and if it surpasses the threshold $V_{\rm th}$ of the spiking
neuron, a spike will be fired. The membrane potential $U^{l}(t)$ is subtracted
by the threshold $V_{\rm th}$ after each firing event, which is captured by
the refractory period $R$.

###  III-B Training Methods

Training a neural network consists of steps to optimize the loss function $L$
with respect to the parameter set $\theta$. In this section, we will introduce
typical training algorithms to train an SNN.

####  III-B1 Backpropagation Through Time (BPTT)

Backpropagation Through Time (BPTT) is an optimization algorithm specifically
designed for training Recurrent Neural Networks (RNNs) as an extension of the
standard backpropagation algorithm to handle time-dependent sequences and
recurrent connections [22, 37]. As SNNs are inherently recurrent in time,
training SNNs through BPTT is similar to training RNNs with BPTT. The
algorithm functions by unfolding the network through time to capture the
temporal dependencies in the input data and applying the backpropagation
algorithm over time. The weight update formula for BPTT can be written as
follows:

$\Delta w^{l}_{ij}=-\eta\sum_{t=1}^{T}\frac{\partial L}{\partial
w_{ij}^{l,t}}$ (1)

where, $T$ is the total number of time steps, $\eta$ is the learning rate, and
the gradient of the loss function $L$ with respect to the weight
$w_{ij}^{l,t}$ in layer $l$ at time step $t$ can be expressed as:

$\frac{\partial L}{\partial w_{ij}^{l,t}}=\delta_{j}^{l,t}\cdot a_{i}^{l-1,t}$
(2)

where, $\delta_{j}^{l,t}$ represents the error term for neuron $j$ in layer
$l$ at time step $t$, and $a_{i}^{l-1,t}$ denotes the activation of neuron $i$
in layer $l-1$ at time step $t$. The error term $\delta_{j}^{l,t}$ can be
computed recursively as:

$\delta_{j}^{l,t}=\sum_{k=1}^{n_{l+1}}\frac{\partial L}{\partial
a_{k}^{l+1,t}}\cdot\frac{\partial a_{k}^{l+1,t}}{\partial
U_{j}^{l,t}}=\sum_{k=1}^{n_{l+1}}w_{jk}^{l+1,t}\cdot\delta_{k}^{l+1,t}\cdot\sigma^{\prime}(U_{j}^{l,t})$
(3)

where, $n_{l+1}$ is the number of neurons in layer $l+1$, $w_{jk}^{l+1,t}$ is
the weight connecting neuron $j$ in layer $l$ to neuron $k$ in layer $l+1$ at
time step $t$. $\sigma^{\prime}(U_{j}^{l,t})$ represents the derivative of the
surrogate function with respect to the pre-activation value $U_{j}^{l,t}$ for
neuron $j$ in layer $l$ at time step $t$.

####  III-B2 E-prop

Recent studies of error propagation methods in SNNs propose a new method
called e-prop [15]. Compared with BPTT, e-prop offers higher biological
plausibility, as it is inspired by the concept of synaptic eligibility traces
observed in neuroscience [38], which weighs the temporal differences of
presynaptic and postsynaptic neuronal activations. While BPTT relies on a
global learning rule that requires error information to be propagated backward
through the network between layers, e-prop aims to derive a local learning
rule, in which the error signals are propagated directly from the output layer
to hidden units. The local learning rule in e-prop makes it more suitable for
neuromorphic hardware implementations, where global communication can be a
significant challenge.

In the e-prop method, the weight updates are based on the product of
eligibility traces and learning signals. The eligibility trace $e_{ij}^{t}$
for a synapse connecting neuron $i$ to neuron $j$ at time step $t$ can be
computed by:

$e_{ij}^{t}=\frac{\partial U_{j}^{t}}{\partial w_{ij}}$ (4)

where, $U_{j}^{t}$ represents the pre-activation of neuron $j$ at time step
$t$, and $w_{ij}$ is the weight of the synapse connecting neuron $i$ to neuron
$j$. The eligibility trace captures the recent history of a synapse’s
contribution to the post-synaptic neuron’s activation. The learning signal
$z_{j}^{t}$ for neuron $j$ at time step $t$ is calculated as:

$z_{j}^{t}=\frac{\partial L}{\partial a_{j}^{t}}$ (5)

where, $L$ is the loss function and $a_{j}^{t}$ is the activation of neuron
$j$ at time step $t$. The learning signal quantifies how the activation of
neuron $j$ at time step $t$ contributes to the overall loss. The weight update
for a synapse connecting neuron $i$ to neuron $j$ can be expressed as:

$\Delta w_{ij}=\eta\sum_{t=1}^{T}z_{j}^{t}\cdot e_{ij}^{t}$ (6)

where, $\eta$ is the learning rate and $T$ is the total number of time steps
over which the network is operated.

####  III-B3 DECOLLE

Deep Continuous Local Learning (DECOLLE) is a biologically plausible online
learning algorithm for training SNNs [19]. DECOLLE combines the principles of
local learning with deep architectures, enabling the training of SNNs in a
layer-by-layer manner without the need for backpropagation or weight
transport. In DECOLLE, each layer in the network learns features using local
information, which allows the network to adapt online without relying on
global error signals. This is achieved by attaching an auxiliary cost function
to the random readouts at each layer. The random readouts $Y^{l}$ at layer $l$
is calculated as:

$Y^{l}_{i}=\sum\limits_{j}G_{ij}^{l}a^{l}_{j}$

where, $G_{ij}^{l}$ is a fixed random matrix for layer $l$ and $a^{l}_{j}$ is
the activation from neuron $j$ that connects to neuron $i$. Then, the loss
function is calculated depending on the differences between readouts
$Y^{l}_{i}$ and pseudo-target $\hat{y}_{i}$. Based on this, weight updates at
each layer are determined using a local learning rule:

$\displaystyle\Delta w_{ij}$ $\displaystyle=-\eta\frac{\partial
L^{l}}{\partial w_{ij}^{l}}=-\eta\frac{\partial L^{l}}{\partial
s_{i}^{l}}\frac{\partial s_{i}^{l}}{\partial w_{ij}^{l}}$
$\displaystyle\frac{\partial s_{i}^{l}}{\partial w_{ij}^{l}}$
$\displaystyle=\sigma^{\prime}(U^{l}_{i})(P_{j}^{l}-\rho\frac{\partial
R_{i}^{l}}{\partial w_{ij}^{l}})$

where, $\eta$ represents the learning rate, $\sigma^{\prime}(U^{l}_{i})$
denotes the surrogate gradient of the non-differentiable step function,
$P_{j}^{l}$ refers to the traces of the membrane driven solely by incoming
spikes, $R_{i}^{l}$ with a constant factor $\rho$ captures the refractory
dynamics depending on the spiking history of the neuron. DECOLLE ignores the
spiking history dependencies, considering $R_{i}^{l}$ to have a negligible
impact on the membrane potential. Consequently, the synaptic weight updates
become:

$\Delta w_{ij}=-\eta\frac{\partial L^{l}}{\partial
s_{i}^{l}}\sigma^{\prime}(U^{l}_{i})P_{j}^{l}$

Note that the locality of this algorithm is maintained by setting all non-
local gradients to zero, as expressed by the equation:

$\frac{\partial L^{l}}{\partial w^{m}_{ij}}=0,\quad\forall m\neq l.$ (7)

This constraint ensures that the learning process relies exclusively on local
information, adhering to the principles of biologically plausible learning
rules.

###  III-C Fisher Information

In statistical modeling, Fisher Information measures the quantity of
information that a given data sample holds about an unknown parameter that the
data sample depends on [39]. Previous work has applied Fisher Information to
analyze the learning dynamics of SNNs [40].

As SNNs rely on accumulating input data over multiple time steps to predict
class probabilities, the Fisher Information Matrix (FIM) is accumulated over
the time dimension to analyze the information dynamics across each time step.
The amount of Fisher Information present in SNNs at a given time step is the
sum of information across all previous time steps, from 1 to $t$.

$M_{t}=\mathbb{E}_{x\sim D}\mathbb{E}_{y\sim f}\theta(y|x_{i\leq t})\\\
\left[\nabla_{\theta}\log f_{\theta}(y|x_{i\leq t})\nabla_{\theta}\log
f_{\theta}(y|x_{i\leq t})^{T}\right]$

Here, $i\in\\{1,\ldots,T\\}$ represents the index of the time step. Direct
calculation on the Fisher Information Matrix (FIM) is computationally
intensive, due to the large number of parameters in an SNN. Consequently,
computing the trace of the FIM is a more efficient alternative [40]. Given a
dataset of $N$ training samples, we can compute the Fisher Information for
each layer $l$ at time $t$ as follows:

$F^{l}_{t}=\frac{1}{N}\sum\limits_{n=1}^{N}\lVert\nabla_{\theta}\log
f_{\theta}(y|x^{n}_{i\leq t})\rVert^{2}$

We will use this metric to quantify the relative importance of recurrent
weights in later sections.

###  III-D Centered Kernel Alignment

Centered Kernel Alignment (CKA) is used to quantify the similarity between
representations in two arbitrary layers through normalized Hilbert-Schmidt
Independence Criterion (HSIC) [41, 30]. In this work, the CKA calculation is
adopted from Li et al. [30].

Let $\textbf{R}_{\alpha}\in\mathbb{R}^{b\times\tau p_{\alpha}}$ and
$\textbf{R}_{\beta}\in\mathbb{R}^{b\times\tau p_{\beta}}$ be the
representations of layer $\alpha$ of one SNN with $p_{\alpha}$ hidden neurons
and layer $\beta$ of another SNN with $p_{\beta}$ hidden neurons over $\tau$
time steps, where $b$ is the batch size. The representations across all time
steps are concatenated in SNNs. Then CKA is defined as:

$CKA(\textbf{K},\textbf{L})=\frac{HSIC(\textbf{K},\textbf{L})}{\sqrt{HSIC(\textbf{K},\textbf{K})HSIC(\textbf{L},\textbf{L})}}.$

where, $\textbf{K}=\textbf{R}_{\alpha}\textbf{R}_{\alpha}^{T}$ and
$\textbf{L}=\textbf{R}_{\beta}\textbf{R}_{\beta}^{T}$ are the Gram matrices
with shape $b\times b$ representing the similarity between examples. The HSIC
is a statistical measure that assesses the independence of two sets of
variables [42]. HSIC is defined as following:

$HSIC(\textbf{K},\textbf{L})=\frac{1}{(m-1)^{2}}tr(\textbf{KCLC}).$

where, I is the identity matrix and O is a matrix consisting of all 1’s.
Centering matrix C can be computed by $\textbf{C}=I-\frac{1}{b}\textbf{O}$.
HSIC score of 0 means independence between two variables. CKA normalizes this
into a similarity index from 0 to 1, where higher values imply greater
similarity. We adhere to the use of an unbiased estimator for calculating HSIC
across mini-batches [30, 43, 44]. CKA metric will also be used in later
sections to empirically explain the performance differences of various SNN
learning methods in terms of accuracy and robustness.

##  IV Results

###  IV-A Experimental Setup

Experiments conducted in this study includes two datasets for investigation of
the performance of various training methods. The Neuromorphic-MNIST (N-MNIST)
dataset is a neuromorphic adaptation of the handwritten digits classification
task [45]. In contrast to the MNIST dataset [46], which consists of static
images, N-MNIST encapsulates changes in pixel values as events unfold over
time. N-MNIST is specifically designed for the evaluation of neuromorphic
algorithms and models [47, 19]. Another dataset used in this study is the
TIMIT dataset, which is a speech and audio processing sequence-to-sequence
task [48, 13, 15]. Although both N-MNIST and TIMIT datasets involve the
temporal dimension, data in TIMIT exhibits a higher degree of dependency
between time steps.

Performance and robustness measurements are conducted for two types of SNN
architectural designs. One is a feed-forward SNN (FF) and another is a
recurrent SNN (REC) architecture, which introduces additional weights to the
linear layers in FF. To classify N-MNIST dataset, both FF and REC SNNs
consisting of two hidden layers of 120 and 84 neurons are used. The TIMIT
classification task utilizes FF and REC SNNs with one hidden layer of 400
neurons. Notably, due to the configuration of e-prop, the output layer does
not contain an explicit recurrent weight connection.

###  IV-B Performance Evaluation

TABLE I: A Collection of Previous Reported Results for Each Training Method Investigated in This Study. Training | N-MNIST | TIMIT
---|---|---
Methods | REC | FF | REC | FF
BPTT | $98.7\%^{\mathrm{a}}$ | $98.2\%^{\mathrm{b}}$ | $67.1\%^{\mathrm{c}}$ |
e-prop |  |  |  $65.4\%$d |  $55.2\%$e
DECOLLE |  |  $98.9\%$f |  |
aThis architecture trained on SMNIST dataset, uses three hidden layers with
64, 256, and 256 neurons [14].
bThis architecture uses three hidden layers with 512 neurons [31].
cThis architecture uses one hidden layer with 800 neurons in an LSNN
architecture [13], which is an LSTM-equivalent architecture in the spiking
domain. In general, LSNN has better performance compared to Recurrent SNN. The
performance is obtained from [15].
dThis architecture uses one hidden layer with 800 or 400 neurons using an LSNN
architecture [15].
eWith 1 hidden layer of 1200 neurons, a feed-forward neural network trained
with e-prop achieves $49.94\%$; With 3 hidden layers, each of which consists
of 900 neurons, a feed-forward neural network trained with e-prop achieves
$55.2\%$ accuracy [15].
fThis architecture consists of 3 convolutional layers with 64, 128, and 128
channels respectively [19].

Table I presents the outcomes of previous studies that have benchmarked
performance of SNNs across two distinct types of classification tasks. These
results show promising performance of SNNs in several classification tasks in
comparison to state-of-the-art ANNs. However, due to the lack of consistency
in neural network architecture designs and variations in training sets, direct
comparison between different training methods is infeasible. To address this
problem, we have conducted a series of benchmark tests with consistent neural
network architectures and training datasets. This work aims to offer a more
comprehensive and insightful assessment of the performance of local learning
methods and the impact of explicit recurrence on SNNs.

TABLE II: Benchmark Results on N-MNIST and TIMIT Datasets. Training | N-MNIST | TIMIT
---|---|---
Methods | FF | REC | FF | REC
BPTT | 97.12% | 97.77% | 52.31% | 56.93%
e-prop | 96.80% | 96.44% | 51.12% | 54.99%
DECOLLE | 93.61% | 90.52% | 42.30% | 45.70%

In our investigations, we have observed that the introduction of recurrent
weights in the REC architecture yields significant performance improvements
when classifying the TIMIT dataset, in contrast to the FF architecture. Table
II shows SNN performance across various training methods for TIMIT and N-MNIST
classification. When trained with BPTT, the REC architecture exhibits a
notable enhancement in accuracy when compared to the FF architecture. Similar
accuracy improvements are also observed in the case of e-prop and DECOLLE.
Benefits of recurrent weights are not evident in the N-MNIST classification
task. Here, the performance of SNNs trained with BPTT and e-prop displays only
slight differences between the REC module and the FF module. Interestingly,
the explicit inclusion of recurrent weights leads to a performance degradation
in SNNs trained with DECOLLE for the N-MNIST dataset. Further investigation is
required to understand the underlying reason for this degradation.

Additionally, a more local learning method slightly degrades the performance
of SNNs, which indicates a trade-off between biological plausibility and
performance. In this study, the locality of learning method is considered to
increase from BPTT (global learning method) to e-prop and DECOLLE
successively. In N-MNIST classification, a slight accuracy difference is
observed between BPTT and e-prop in both REC and FF modules. DECOLLE exhibits
an accuracy reduction in both architectures compared with e-prop. A similar
trend is observed for the TIMIT classification task.

###  IV-C Representational Differences

In the previous section, we observed a trade-off between biological
plausibility and performance among various learning methods. Here, we
investigate whether this performance discrepancy can be attributed to
variations in the learned representations at corresponding layers across
different training methods.

CKA is a technique used to measure the similarity between two sets of data,
and it has been increasingly applied to understand the internal representation
in neural networks. Pioneering work by Cortes et al. introduced CKA as an
alignment measure for kernels in the context of support vector machines [49].
Later, Kornblith et al. adapted CKA for deep learning, using it to compare
layers across different neural network models, which provided insights into
which layers of networks learn similar features [41]. Williams et al. expanded
this analysis to study the similarity between representations learned by
convolutional neural networks [50]. Li et al. investigated how SNNs differ
from traditional ANNs in learning representations, using CKA to analyze both
spatial and temporal dimensions [30]. In this study, CKA is employed as a
metric to assess the similarity of representations across layers that are
learned using various methods with differing degrees of local learning.
Consistent with prior research, the CKA is averaged over 10 experiments with
different initializations, each of which is conducted by passing 4096
instances from the testing set of N-MNIST dataset through the corresponding
pre-trained model [30, 43]. We also compare the similarity of learnt
representations of the various training methods with ANN-SNN conversion where
the SNN is trained using conventional ANN training methods [51, 52].

Figure 2: (Left) Layer-wise similarity among training methods; (Right)
Similarity between ANN-SNN conversion method and direct SNN training methods.

Firstly, significant difference is observed between the representations
learned from ANN-SNN conversion and those obtained through other training
methods. Figure 2 plots the normalized HSIC between representations learned at
each layer of SNN converted from ANN and that of SNNs trained directly. All
the layer-wise similarities between the ANN-SNN conversion method and other
direct SNN training methods are low, underscoring the fact that SNNs and ANNs
learn very different representations.

Secondly, aside from ANN-SNN conversion, we observe that methods with a closer
level of locality exhibit a substantial degree of representational similarity
between corresponding layers. Figure 2 plots the similarity between the
representations of a layer trained by pairs of training methods. In
particular, BPTT and e-prop, which have a closer level of locality, show a
strong correlation in the representation of their first two layers based on
CKA scores. E-prop and DECOLLE pair also exhibit a similar trend. However, the
small representational similarity between corresponding layers trained by BPTT
and DECOLLE is substantiated by a low CKA score, which addresses the
performance gap between the two methods.

###  IV-D Robustness under Attack

In the previous section, we observed accuracy improvements after the addition
of recurrence in spiking architectures, especially for sequential data
processing. However, their impact on other aspects of SNN performance, like
robustness to adversarial attacks, remains unclear. In this section, we
explore this aspect along with empirical tools to explain our observations.

TABLE III: Accuracy of SNNs Trained by Methods with Varying Localities Under FGSM Attack Using N-MNIST Dataset. Epsilon | 0 | 0.001 | 0.005 | 0.01 | 0.02 | 0.05
---|---|---|---|---|---|---
Training Methods | FF
BPTT | 97.12% | 83.82% | 16.78% | 4.34% | 1.22% | 0.73%
e-prop | 96.80% | 89.90% | 41.27% | 12.33% | 3.27% | 2.87%
DECOLLE | 93.61% | 83.48% | 45.11% | 28.57% | 21.29% | 17.79%
Training Methods | REC
BPTT | 97.77% | 94.66% | 80.76% | 58.73% | 27.69% | 18.55%
e-prop | 96.44% | 93.04% | 78.28% | 60.57% | 35.07% | 18.25%
DECOLLE | 90.52% | 86.76% | 84.14% | 72.81% | 47.56% | 23.75%
TABLE IV: Accuracy and False Positive (FP) Rate of SNNs Under Targeted Backdoor Attack for Various Training Methods Using N-MNIST Dataset with 5 Different Sets of Targets and Sources. Poison Rate | 0.1 | 0.3 | 0.5 | 0.7 | 0.9
---|---|---|---|---|---
FF
Training Methods | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR
BPTT | 96.23% | 4.80% | 93.34% | 33.54% | 91.62% | 43.76% | 89.64% | 66.06% | 87.47% | 78.19%
e-prop | 96.85% | 1.93% | 93.46% | 33.35% | 92.30% | 46.94% | 89.38% | 75.09% | 87.32% | 96.95%
DECOLLE | 92.74% | 4.66% | 89.88% | 30.04% | 88.14% | 48.31% | 85.12% | 76.77% | 83.62% | 90.80%
REC
Training Methods | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR | Accuracy | ASR
BPTT | 96.43% | 2.64% | 94.19% | 22.03% | 92.24% | 43.78% | 91.22% | 62.24% | 87.38% | 90.73%
e-prop | 96.69% | 0.45% | 94.81% | 17.65% | 91.92% | 44.56% | 88.51% | 74.15% | 87.23% | 93.15%
DECOLLE | 90.21% | 0.86% | 87.41% | 27.75% | 81.55% | 88.49% | 81.29% | 88.96% | 80.81% | 91.02 %

Specifically, the Fast Gradient Sign Method (FGSM) attack leverages the sign
of gradients to create adversarial instances that maximize the loss of a
neural network, misleading it to make incorrect classification [53]. However,
due to the non-differentiable nature of SNNs, the direct calculation of
gradients is not applicable. To solve this problem, previous work has proposed
an effective framework which involves crafting an adversarial example for an
SNN by creating its ANN counterpart and then calculating gradients for the
corresponding ANN [54]. These gradients are subsequently utilized to construct
an adversarial example for the SNN. We adopt this variation of FGSM attack to
compare the robustness of various training methods and to investigate the
impact of recurrent weights on robustness.

Then, we employ a targeted backdoor attack, which is a special type of
poisoning attack. Such an attack involves the process of injecting a malicious
imperceptible trigger during the training phase. Following the trigger
mechanism established in previous work [35], we initiate the poisoning of
images by modifying 4 pixel locations at all time steps in the input data. For
each poison ratio, five random sources and targets are selected. To provide a
more robust and unbiased assessment, we calculate the average results from 5
runs with randomly assigned sources and targets, thus mitigating the potential
bias introduced by specific poisoning choices. To assess the impact of these
backdoor attacks, we examine changes in accuracy and attack success rate
(ASR), which is the proportion of labels for the source class that are
predicted as the target class. A lower ASR signifies greater robustness.

Table III reports the performance of different training methods when subjected
to FGSM attacks, where the perturbation level, denoted as “Epsilon”, varies
across a range from 0.001 to 0.05. The results of our benchmarking analysis
for backdoor attacks are illustrated in Table IV. We mainly focus on poison
rates in realistic settings below 50% as shown in prior works [55, 56, 57,
35]. Both attacks are evaluated on the N-MNIST dataset.

Enhancement of the model’s robustness is observed with the inclusion of
explicit recurrent weights. Under FGSM attacks, compared to FF models trained
by the same methods and subjected to the same “Epsilon” values, the REC model
retains more accuracy than the FF model. Besides, the robustness advantage of
explicit recurrence is also observed under backdoor attack. Specifically,
under more realistic settings with poison rates at 10% and 30%, we observe
that the REC model consistently exhibits a lower ASR compared to its
corresponding FF model, in all three training methods.

Local learning methods showed improved robustness under FGSM attacks compared
to global learning methods. Specifically, the local learning method DECOLLE
demonstrates enhanced robustness against attacks with high perturbation
values. In contrast, the global training methods experience a significant drop
in accuracy under high “Epsilon” values, which is consistent in both FF and
REC architectures. This advantage can be attributed to the local training
methods’ use of individual loss functions for each layer, which enhances
robustness in comparison to global loss functions that are vulnerable to
globally calculated gradients. Under backdoor attacks, where global gradients
are not used, such robustness gains of local learning methods are not
observed. In the previous section, we show that there is a trade-off between
biological plausibility and performance. Such a trade-off is not demonstrated
for adversarial attacks. Local learning methods retain robustness at par with
global learning methods with respect to non-gradient-based attacks and exhibit
enhanced robustness for gradient-based attacks.

###  IV-E CKA Measurements to Explain Robustness

Figure 3: Layerwise representational similarity between clean and adversarial
input data of FF architecture subtracted from REC architecture.

In this section, we further assess the robustness of SNNs from the perspective
of CKA. Prior research has quantified the robustness of SNNs directly trained
under adversarial attacks [30] and established a correlation between
robustness against adversarial attacks and the degree of representational
similarity of a particular network between clean and perturbed input images
[58]. Here, we employ the CKA metric to evaluate the robustness of SNNs
trained using each training method under FGSM attacks with the same set of
“Epsilon” values used in the previous section. Figure 3 measures the layerwise
representational similarity between clean and adversarial inputs of REC
architecture subtracted by the similarity of FF architecture under FGSM
attacks. Each similarity is computed by contrasting representations of a layer
by passing clean data and a layer at the same architectural level by passing
adversarial data. Each color corresponds to one specific layer.

It is observed that the addition of recurrent weights contributes to better
preservation of the representations learned at each layer across these
training methods. In these experiments, positive similarity differences are
obtained at all perturbation levels, revealing that REC SNNs maintain larger
representational similarity (implying more robust representations learned in
the presence of attacks) compared to FF SNNs between clean and adversarial
data. The results also reveal that global learning methods exhibit a greater
enhancement in robustness with recurrent connections, as the similarity
differences are larger in BPTT than in e-prop and DECOLLE. The trend matches
the data provided in Table III. Through this analysis, we provide evidence
that the inclusion of recurrent weights enhances the robustness of
neuromorphic architectures.

###  IV-F Relative Importance of Recurrent Weights

In the previous section, we demonstrated that there is a robustness advantage
for explicit recurrent SNNs. Here, we examine the importance of additional
recurrence in SNNs from the perspective of Fisher Information. A high Fisher
Information exhibited in weight connections is shown to play a substantial
role in making predictions for the given data [59, 60]. Specifically, the
Fisher Information in SNNs evaluates the accumulation of temporal information
during the training process. In previous work [40], normalized layer-wise
Fisher Information was calculated across time steps, providing an assessment
of the relative importance of each layer. We follow the same procedure to
calculate Fisher Information during the training process by capturing the
gradient of weights.

In Figure 4, we present the Fisher Information of linear weights in the FF
architecture, denoted as LW-FF in the figure. We then compare this Fisher
Information with both linear and recurrent weights in the REC architecture,
denoted as LW-REC and RW-REC in the graph. It is important to note that due to
the nature of e-prop, the additional recurrent weights are not added to the
last layer in the REC architecture.

Recurrent weights are demonstrated to have comparable importance with respect
to linear weights. After the addition of explicit recurrent weights, the
importance of linear weights (LW-REC) degrades in comparison to linear weights
in the FF model (LW-FF). On the other hand, the importance of recurrent
weights (RW-REC) shows comparable Fisher Information values with regards to
linear weights (LW-REC). Based on these two observations, we can conclude that
the degradation of importance in linear weights from the values in the FF
model (LW-FF) is due to reallocation of importance to linear weights (LW-REC)
and recurrent weights (RW-REC) in the REC model since Fisher Information is
normalized such that the summation of values of all layers is one.

Additionally, it is found that layerwise increments in the relative importance
of linear weights are independent of recurrent weights. Specifically, an
increase of relative importance from input layer to output layer is observed
among all training methods. Although the magnitude of this trend may vary
across different methods, this trend remains consistent independent of the
addition of extra recurrent weights (LW-REC). The magnitude variations mainly
come from the emphasis on input layers. In particular, as the locality of
learning methods shifts from global to local, the importance of the input
layer drops significantly.

Figure 4: Layerwise Fisher Information measured in SNNs trained by learning
methods with varying degrees of locality on the N-MNIST dataset.

##  V Conclusions

In this work, learning methods with varying levels of locality are compared in
terms of performance and robustness. To understand the performance disparity
of these methods, representational differences are examined through the CKA
metric. SNN learning methods with a higher degree of locality are shown to
have performance degradation but they showed better robustness under FGSM
adversarial attacks. However, the robustness advantages of local learning
methods are not observed for non-gradient-based attacks like backdoor attacks.
Additionally, we investigate the contribution of additional recurrent weights
on the performance of SNNs. Measurements on the relative importance of
recurrent weights are performed using Fisher Information. We illustrated that
recurrent weights not only have positive influences on the performance of SNNs
but also augment the robustness of SNNs. These observations are further
substantiated by the fact that recurrent weights and linear weights exhibit
comparable informative significance. In general, this work provides insights
into the selection of learning methods for SNNs and offers a benchmarking
analysis for future SNN designers. Future work can consider extension of such
benchmarking analysis to more complex architectures including convolutional
layers for image recognition tasks or attention layers for natural language
processing tasks [16].

## Acknowledgments

This material is based upon work supported in part by the U.S. Department of
Energy, Office of Science, Office of Advanced Scientific Computing Research,
under Award Number #DE-SC0021562, the National Science Foundation grants CCF
#1955815, CAREER # 2337646 and EFRI BRAID #2318101 and by Oracle Cloud credits
and related resources provided by the Oracle for Research program.

## References

  * [1] W. Maass, “Networks of spiking neurons: the third generation of neural network models,” _Neural networks_ , vol. 10, no. 9, pp. 1659–1671, 1997.
  * [2] W. Gerstner and W. M. Kistler, _Spiking neuron models: Single neurons, populations, plasticity_. Cambridge university press, 2002.
  * [3] A. Tavanaei, M. Ghodrati, S. R. Kheradpisheh, T. Masquelier, and A. Maida, “Deep learning in spiking neural networks,” _Neural networks_ , vol. 111, pp. 47–63, 2019.
  * [4] P. U. Diehl and M. Cook, “Unsupervised learning of digit recognition using spike-timing-dependent plasticity,” _Frontiers in computational neuroscience_ , vol. 9, p. 99, 2015.
  * [5] G.-q. Bi and M.-m. Poo, “Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type,” _Journal of neuroscience_ , vol. 18, no. 24, pp. 10 464–10 472, 1998.
  * [6] C. Mead, “Neuromorphic electronic systems,” _Proceedings of the IEEE_ , vol. 78, no. 10, pp. 1629–1636, 1990.
  * [7] P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura _et al._ , “A million spiking-neuron integrated circuit with a scalable communication network and interface,” _Science_ , vol. 345, no. 6197, pp. 668–673, 2014\.
  * [8] B. V. Benjamin, P. Gao, E. McQuinn, S. Choudhary, A. R. Chandrasekaran, J.-M. Bussat, R. Alvarez-Icaza, J. V. Arthur, P. A. Merolla, and K. Boahen, “Neurogrid: A mixed-analog-digital multichip system for large-scale neural simulations,” _Proceedings of the IEEE_ , vol. 102, no. 5, pp. 699–716, 2014\.
  * [9] K. Boahen, “Neuromorphic microchips,” _Scientific American_ , vol. 16, no. 3, pp. 20–27, 2006.
  * [10] M. Davies, N. Srinivasa, T.-H. Lin, G. Chinya, Y. Cao, S. H. Choday, G. Dimou, P. Joshi, N. Imam, S. Jain _et al._ , “Loihi: A neuromorphic manycore processor with on-chip learning,” _Ieee Micro_ , vol. 38, no. 1, pp. 82–99, 2018.
  * [11] S. B. Furber, F. Galluppi, S. Temple, and L. A. Plana, “The spinnaker project,” _Proceedings of the IEEE_ , vol. 102, no. 5, pp. 652–665, 2014\.
  * [12] W. Ponghiran and K. Roy, “Spiking neural networks with improved inherent recurrence dynamics for sequential learning,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 36, no. 7, 2022, pp. 8001–8008.
  * [13] G. Bellec, D. Salaj, A. Subramoney, R. Legenstein, and W. Maass, “Long short-term memory and learning-to-learn in networks of spiking neurons,” _Advances in neural information processing systems_ , vol. 31, 2018.
  * [14] B. Yin, F. Corradi, and S. M. Bohté, “Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks,” _Nature Machine Intelligence_ , vol. 3, no. 10, pp. 905–913, 2021.
  * [15] G. Bellec, F. Scherr, A. Subramoney, E. Hajek, D. Salaj, R. Legenstein, and W. Maass, “A solution to the learning dilemma for recurrent networks of spiking neurons,” _Nature communications_ , vol. 11, no. 1, p. 3625, 2020\.
  * [16] M. Bal and A. Sengupta, “Spikingbert: Distilling bert to train spiking language models using implicit differentiation,” _arXiv preprint arXiv:2308.10873_ , 2023.
  * [17] T. P. Lillicrap, D. Cownden, D. B. Tweed, and C. J. Akerman, “Random synaptic feedback weights support error backpropagation for deep learning,” _Nature communications_ , vol. 7, no. 1, p. 13276, 2016.
  * [18] P. Baldi, P. Sadowski, and Z. Lu, “Learning in the machine: the symmetries of the deep learning channel,” _Neural Networks_ , vol. 95, pp. 110–133, 2017\.
  * [19] J. Kaiser, H. Mostafa, and E. Neftci, “Synaptic plasticity dynamics for deep continuous local learning (decolle),” _Frontiers in Neuroscience_ , vol. 14, p. 424, 2020.
  * [20] F. Crick, “The recent excitement about neural networks,” _Nature_ , vol. 337, pp. 129–132, 1989.
  * [21] T. P. Lillicrap, A. Santoro, L. Marris, C. J. Akerman, and G. Hinton, “Backpropagation and the brain,” _Nature Reviews Neuroscience_ , vol. 21, no. 6, pp. 335–346, 2020.
  * [22] P. J. Werbos, “Backpropagation through time: what it does and how to do it,” _Proceedings of the IEEE_ , vol. 78, no. 10, pp. 1550–1560, 1990.
  * [23] R. J. Williams and D. Zipser, _Gradient-Based Learning Algorithms for Recurrent Networks and Their Computational Complexity_. USA: L. Erlbaum Associates Inc., 1995, p. 433–486.
  * [24] S. Lu and A. Sengupta, “Neuroevolution guided hybrid spiking neural network training,” _Frontiers in neuroscience_ , vol. 16, p. 838523, 2022.
  * [25] R. Kempter, W. Gerstner, and J. L. Van Hemmen, “Hebbian learning and spiking neurons,” _Physical Review E_ , vol. 59, no. 4, p. 4498, 1999.
  * [26] F. Zenke and S. Ganguli, “Superspike: Supervised learning in multilayer spiking neural networks,” _Neural computation_ , vol. 30, no. 6, pp. 1514–1541, 2018.
  * [27] E. O. Neftci, H. Mostafa, and F. Zenke, “Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks,” _IEEE Signal Processing Magazine_ , vol. 36, no. 6, pp. 51–63, 2019.
  * [28] M. Bal and A. Sengupta, “Sequence learning using equilibrium propagation,” _arXiv preprint arXiv:2209.09626_ , 2022.
  * [29] L. Deng, Y. Wu, X. Hu, L. Liang, Y. Ding, G. Li, G. Zhao, P. Li, and Y. Xie, “Rethinking the performance comparison between snns and anns,” _Neural networks_ , vol. 121, pp. 294–307, 2020.
  * [30] Y. Li, Y. Kim, H. Park, and P. Panda, “Uncovering the representation of spiking neural networks trained with surrogate gradient,” _arXiv preprint arXiv:2304.13098_ , 2023.
  * [31] W. He, Y. Wu, L. Deng, G. Li, H. Wang, Y. Tian, W. Ding, W. Wang, and Y. Xie, “Comparing snns and rnns on neuromorphic vision datasets: Similarities and differences,” _Neural Networks_ , vol. 132, pp. 108–120, 2020.
  * [32] S. Sharmin, P. Panda, S. S. Sarwar, C. Lee, W. Ponghiran, and K. Roy, “A comprehensive analysis on adversarial robustness of spiking neural networks,” in _2019 IJCNN_. IEEE, 2019, pp. 1–8.
  * [33] L. Liang, X. Hu, L. Deng, Y. Wu, G. Li, Y. Ding, P. Li, and Y. Xie, “Exploring adversarial attack in spiking neural networks with spike-compatible gradient,” _IEEE transactions on neural networks and learning systems_ , 2021\.
  * [34] A. Marchisio, G. Nanfa, F. Khalid, M. A. Hanif, M. Martina, and M. Shafique, “Is spiking secure? a comparative study on the security vulnerabilities of spiking and deep neural networks,” in _2020 International Joint Conference on Neural Networks (IJCNN)_. IEEE, 2020, pp. 1–8.
  * [35] T. Gu, B. Dolan-Gavitt, and S. Garg, “Badnets: Identifying vulnerabilities in the machine learning model supply chain.(2017),” _arXiv preprint arXiv:1708.06733_ , 2019.
  * [36] A. Bhargava, M. R. Rezaei, and M. Lankarany, “Gradient-free neural network training via synaptic-level reinforcement learning,” _AppliedMath_ , vol. 2, no. 2, pp. 185–195, 2022.
  * [37] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” _nature_ , vol. 323, no. 6088, pp. 533–536, 1986.
  * [38] W. Gerstner, M. Lehmann, V. Liakoni, D. Corneil, and J. Brea, “Eligibility traces and plasticity on behavioral time scales: Experimental support of neohebbian three-factor learning rules,” _Frontiers in Neural Circuits_ , vol. 12, 2018. [Online]. Available: https://www.frontiersin.org/articles/10.3389/fncir.2018.00053
  * [39] R. A. Fisher, “Theory of statistical estimation,” in _Mathematical proceedings of the Cambridge philosophical society_ , vol. 22, no. 5. Cambridge University Press, 1925, pp. 700–725.
  * [40] Y. Kim, Y. Li, H. Park, Y. Venkatesha, A. Hambitzer, and P. Panda, “Exploring temporal information dynamics in spiking neural networks,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 37, no. 7, 2023, pp. 8308–8316.
  * [41] S. Kornblith, M. Norouzi, H. Lee, and G. Hinton, “Similarity of neural network representations revisited,” in _International conference on machine learning_. PMLR, 2019, pp. 3519–3529.
  * [42] D. Greenfeld and U. Shalit, “Robust learning with the hilbert-schmidt independence criterion,” in _International Conference on Machine Learning_. PMLR, 2020, pp. 3759–3768.
  * [43] T. Nguyen, M. Raghu, and S. Kornblith, “Do wide and deep networks learn the same things? uncovering how neural network representations vary with width and depth,” _arXiv preprint arXiv:2010.15327_ , 2020.
  * [44] L. Song, A. Smola, A. Gretton, J. Bedo, and K. Borgwardt, “Feature selection via dependence maximization.” _Journal of Machine Learning Research_ , vol. 13, no. 5, 2012.
  * [45] G. Orchard, A. Jayawant, G. K. Cohen, and N. Thakor, “Converting static image datasets to spiking neuromorphic datasets using saccades,” _Frontiers in neuroscience_ , vol. 9, p. 437, 2015.
  * [46] L. Deng, “The mnist database of handwritten digit images for machine learning research [best of the web],” _IEEE signal processing magazine_ , vol. 29, no. 6, pp. 141–142, 2012.
  * [47] J. H. Lee, T. Delbruck, and M. Pfeiffer, “Training deep spiking neural networks using backpropagation,” _Frontiers in neuroscience_ , vol. 10, p. 508, 2016.
  * [48] J. S. Garofolo, L. F. Lamel, W. M. Fisher, J. G. Fiscus, and D. S. Pallett, “Darpa timit acoustic-phonetic continous speech corpus cd-rom. nist speech disc 1-1.1,” _NASA STI/Recon technical report n_ , vol. 93, p. 27403, 1993\.
  * [49] C. Cortes, M. Mohri, and A. Rostamizadeh, “Algorithms for learning kernels based on centered alignment,” _The Journal of Machine Learning Research_ , vol. 13, no. 1, pp. 795–828, 2012.
  * [50] A. H. Williams, E. Kunz, S. Kornblith, and S. Linderman, “Generalized shape metrics on neural representations,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 4738–4750, 2021.
  * [51] A. Sengupta, Y. Ye, R. Wang, C. Liu, and K. Roy, “Going deeper in spiking neural networks: Vgg and residual architectures,” _Frontiers in neuroscience_ , vol. 13, p. 95, 2019.
  * [52] S. Lu and A. Sengupta, “Exploring the connection between binary and spiking neural networks,” _Frontiers in neuroscience_ , vol. 14, 2020.
  * [53] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,” _arXiv preprint arXiv:1412.6572_ , 2014.
  * [54] S. Sharmin, P. Panda, S. S. Sarwar, C. Lee, W. Ponghiran, and K. Roy, “A comprehensive analysis on adversarial robustness of spiking neural networks,” in _2019 International Joint Conference on Neural Networks (IJCNN)_. IEEE, 2019, pp. 1–8.
  * [55] X. Chen, C. Liu, B. Li, K. Lu, and D. Song, “Targeted backdoor attacks on deep learning systems using data poisoning,” _arXiv preprint arXiv:1712.05526_ , 2017.
  * [56] Y. Adi, C. Baum, M. Cisse, B. Pinkas, and J. Keshet, “Turning your weakness into a strength: Watermarking deep neural networks by backdooring,” in _27th USENIX Security Symposium (USENIX Security 18)_ , 2018, pp. 1615–1631.
  * [57] Y. Gao, C. Xu, D. Wang, S. Chen, D. C. Ranasinghe, and S. Nepal, “Strip: A defence against trojan attacks on deep neural networks,” in _Proceedings of the 35th Annual Computer Security Applications Conference_ , 2019, pp. 113–125.
  * [58] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models resistant to adversarial attacks,” _arXiv preprint arXiv:1706.06083_ , 2017.
  * [59] A. Achille, M. Rovere, and S. Soatto, “Critical learning periods in deep networks,” in _International Conference on Learning Representations_ , 2018\.
  * [60] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska _et al._ , “Overcoming catastrophic forgetting in neural networks,” _Proceedings of the national academy of sciences_ , vol. 114, no. 13, pp. 3521–3526, 2017.


# A universal shortcut method for state transfer in quantum spin systems

Jian Xu<EMAIL_ADDRESS>College of Electronics and Information
Engineering, Guangdong Ocean University, and Guangdong Engineering Technology
Research Center of Intelligent Ocean Sensor Network and
Equipment,Zhanjiang,guangdong, 524088, China  Feng Mei  State Key Laboratory
of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy,
Shanxi University, Taiyuan, Shanxi, 030006, China  Yan-Qing Zhu
<EMAIL_ADDRESS>Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Department of Physics, and Hong Kong Institue of Quantum Science $\&$
Technology,
The University of Hong Kong, Pokfulam Road, Hong Kong, China

###### Abstract

The need for fast and robust quantum state transfer is an essential element in
scalable quantum information processing, leading to widespread interest in
shortcuts to adiabaticity for speeding up adiabatic quantum protocols.
However, shortcuts to adiabaticity for systems with more than a few levels is
occasionally challenging to compute in theory and frequently difficult to
implement in experiments. In this work, we develop a protocol for constructing
shortcuts to adiabaticity through the multi-state Landau-Zener approach and a
stricter adiabatic condition. Importantly, our protocol only requires a few
pieces of information about the energy spectrum and adjusts the evolutionary
rate of the system, making it both generic for theoretical models and friendly
for experimental implementation. As examples, we apply our protocol to state
transfer in the non-Hermitian Su-Schrieffer-Heeger (SSH) model and the
topological Thouless pump models and find that it can speed up the
manipulation speed while remaining robust to Hamiltonian errors. Furthermore,
our findings can be realized using current technology and could potentially be
extended to many-body systems, dissipation cases, or Floquet processes.
Overall, the proposed shortcut protocol offers a promising avenue for
enhancing the efficiency and reliability of quantum state transfer protocols.

_Introduction. —_ Adiabatic control over quantum states is of fundamental
importance for quantum science. Generally, adiabatic processes in quantum
systems are typically slow and steady, but this can lead to errors,
perturbations, and decoherence accumulations that can even cause the system to
escape from its confinement. To overcome this challenge, great efforts have
been made over the past two decades to obtain the optimal protocol that speeds
up the evolution process while maintaining the desired system within the
adiabatic regime. In short, most of the shortcuts rely on the addition of
auxiliary time-dependent couplings or specific time dependence of controlling
parameters of a systemBergmann ; Petr ; Guery , e.g., counterdiabatic
drivingBerry , dynamical invariantsChen , quasiadiabatic dynamicsMartinez  and
so forth.

Although shortcuts to adiabaticity is widely used in the context of few-level
systems, it has not been explored well in multi-level systems. The main
obstacle is the difficulty in acquiring and employing the counterdiabatic term
in multi-level systems, requiring extensive knowledge of the full Hamiltonian
spectral properties and the ability to control highly non-local, infinite-
range interactions. Recent progress has been made in systematically building
up the approximate counterdiabatic term using a variational methodSels . This
method has been recently applied to ground-state preparationClaeys ;
Passarelli ; Prielinger ; Barone and state transferCepaite  in specific
quantum spin systems. However, an exact and generic shortcut protocol for
precise state manipulations in these systems has yet to be developed.

Recently, quantum state transfer via topological edge states has also been
proposed and studied in the topological phases of matter since the edge modes
are quite robust against perturbation and disorder thanks to their topological
nature in the bulkFMei2018 ; Qi . Also, the quantum state transfer via the
topological edge states has been experimentally demonstrated by Rydberg
atomsS. de , semiconductor quantum dotsKiczynski , and coupled
waveguidesGabriel . Up to the present, one can only speed up this type of
state transfer through the special modulation functionBrouzos ;
Palaiodimopoulos , introducing additional coupling fieldsFelippo ; Yunlan ;
Lijun  or by topological domain wallsZurita  in Hermitian cases. So this is an
open question of whether there is a universal proposal for enhancing the
performance of state transfer via topological edge states, especially in non-
Hermitian casesCui .

Similarly, Thouless pumping via adiabatic and periodic modulation, which is
topologically protected, has been observed in different artificial systems,
e.g., metamaterialsKe ; Cerjan ; Jurgensen1 ; You ; Sun ; Jurgensen2 ,
nitrogen-vacancy centre in diamondMa ; Boyers , photonic systemsJürgensen  and
ultracold atomsLohse ; Nakajima1 ; Lu  by considering disorders Nakajima2 ,
interactionsWalter , or gains/lossesFedorova ; Dreon . Since its topological
nature is robust only under adiabatic driving paths, it is also an important
question to optimize the adiabatic process in practical implementations. Until
now, some shortcut schemes have been proposed by next to nearest neighbour
hopping and additions of a harmonic potentialSchouten , a family of finite-
frequency protocolsMalikis  and Bloch oscillationsLiu . But there is no scheme
that is friendly to experiment. Therefore, it is still a challenge to propose
a feasible and universal control protocol to optimize the speed and accuracy
of adiabatic passage.

In this work, in the same spirit, we put forward a generic strategy for
designing fast adiabatic protocols via a state transfer, which is in generic
applicable to spin systems. The theoretical derivation for the adiabatic
condition is first presented. We show that the adiabatic passage of parameter
modulation can be determined through a simple calculation once the part of the
eigenvalues of the system are obtained. More importantly, this method, which
excludes auxiliary controls, will not increase experimental complexity. To
present the advantages of our shortcut method, we apply this theory to
investigate the state transfer in the non-Hermitian SSH model and the
topological Thouless pump models. Compared to common linear parameter
modulation, our shortcut method gives optimal adiabatic passages.

_Universal shortut. —_ In general, a non-degenerate $N$-level non-Hermitian
system under a time-modulation parameter $R(t)$ can be described simply as,

$H_{N}(t)=\frac{1}{2}\left(\Omega+\delta\left[R(t)\right]\right),$ (1)

with the time-independent non-diagonal component $\Omega_{ij}$ denoting the
coupling between levels $E_{i}$ and $E_{j}$, where in generic
$\Omega_{ij}\neq\Omega_{ji}^{*}$ due to the non-Hermiticity. Meanwhile, we
take the time-modulation term as a diagonal Hermitian matrix, i.e.,
$\delta=\text{diag}(\delta_{1},\delta_{2},...,\delta_{N})$. Although many
integrable models based on this have been explored, e.g. multi-state Landau-
Zener modelsChernyak2018 ; Chernyak2021 ; Suzuki  and multi-level Landau-
Zener-Stückelberg-Majorana interference modelAshhab ; Xiao ; Ivakhnenko , a
generic method to analytically analyze the dynamics of this system has not
been known so far. The center of the difficulty is that an analytical theory
must include full information about eigenenergies and eigenstates of systems
and the calculation including the latter is usually difficult to obtain. But
in some cases, e.g. adiabatic transfer in this work, an analytical theory only
including the calculation of eigenenergies has been deduced.

Since the Landau-Zener tunneling between any two levels is independent,
without loss generality, we consider the adiabatic transfer in an effective
two-level system with the Hamiltonian is given by,

$H_{\text{eff}}(t)=\frac{1}{2}\left(\begin{array}[]{cc}\delta_{0}[R(t)]&\Omega_{0}\\\
\Omega_{0}^{\prime}&-\delta_{0}[R(t)]\\\ \end{array}\right).$ (2)

Its instantaneous spectrum reads
$E_{\pm}=\pm\sqrt{\delta_{0}^{2}+\Omega_{0}\Omega_{0}^{\prime}}/2$ with the
corresponding eigenvectors,

$\begin{split}|\phi_{+}\rangle=\sin(\theta/2)|1\rangle+\cos(\theta/2)|0\rangle,\\\
|\phi_{-}\rangle=\cos(\theta/2)|1\rangle-\sin(\theta/2)|0\rangle,\\\
\end{split}$ (3)

where the mixing angle
$\theta=\arccos\left(\delta_{0}/\sqrt{\delta_{0}^{2}+\Omega_{0}\Omega_{0}^{\prime}}\right)$,
$|1\rangle=(1,0)^{T}$, and $|0\rangle=(0,1)^{T}$. Because the system is now
non-Hermitian, we mention on its Hermitian conjugate, which satisfies
$H_{\text{eff}}^{\dagger}|\tilde{\phi}_{\pm}\rangle=\tilde{E}_{\pm}|\tilde{\phi}_{\pm}\rangle$.
The corresponding eigenvalue $\tilde{E}_{\pm}=E^{*}_{\pm}$ and
$|\tilde{\phi}_{\pm}\rangle$ takes the same form as in Eq. (3) by just
replacing $\theta$ with $\tilde{\theta}$, where
$\tilde{\theta}=\arccos\left(\delta_{0}/\sqrt{\delta_{0}^{2}+\Omega^{*}_{0}(\Omega_{0}^{\prime})^{*}}\right)$.
Note that $\theta$ and $\tilde{\theta}$ are complex and the eigenvectors
satisfy the bi-orthogonal relation, i.e.,
$\langle\tilde{\phi}_{\alpha}|\phi_{\beta}\rangle=\langle\phi_{\alpha}|\tilde{\phi}_{\beta}\rangle=\delta_{\alpha\beta}$.

For such a non-Hermitian system, the adiabatic condition is given bySM

$\left|\frac{\langle\tilde{\phi}_{\mp}|\partial
H_{\text{eff}}/\partial\delta_{0}|\phi_{\pm}\rangle}{(E_{\pm}-E_{\mp})^{2}}\frac{\partial\delta_{0}}{\partial
t}\right|e^{\int_{0}^{t}\text{Im}(\Delta E)d\tau}\ll 1,$ (4)

with the band gap $\Delta
E=E_{+}-E_{-}=\sqrt{\delta_{0}^{2}+\Omega_{0}\Omega_{0}^{\prime}}$. Since
$\partial H_{\text{eff}}/\partial\delta_{0}=\sigma_{z}/2$, we obtain a simpler
relation,

$|\langle\tilde{\phi}_{\mp}|\partial
H_{\text{eff}}/\partial\delta_{0}|\phi_{\pm}\rangle|=|\sin\theta/2|<1.$ (5)

In addition, we could vary $\Delta E$ with respect to $t$, then we have

$\partial_{t}\Delta E=\delta_{0}\partial_{t}\delta_{0}/\Delta E.$ (6)

Substituting Eqs. (5) and (6) into Eq. (4), we obtain a stricter adiabatic
condition as

$s=\left|\frac{\partial_{t}\Delta E}{\Delta E\sqrt{\Delta E^{2}-\Delta
E_{\text{min}}^{2}}}\right|e^{\int_{0}^{t}\text{Im}(\Delta E)d\tau}\ll 1,$ (7)

where $\Delta E_{\text{min}}=\sqrt{\Omega_{0}\Omega_{0}^{\prime}}$ represents
the miminal band gap when $\delta_{0}$ vanishes at critical $R(t_{c})$. Say,
$s$ is the coefficient to be determined, related to the total evolution time
of the system. Once $s\ll 1$, the system naturally satisfies the adiabatic
condition as in Eq. (4). In fact, realistic physical manipulations do more
rely on the parameter modulation, thus we prefer the following relation
deforming from $s$, i.e.,

$\partial_{t}R(t)=\left|\frac{s\Delta E\sqrt{\Delta E^{2}-\Delta
E_{\text{min}}^{2}}}{\partial_{R}\Delta
E}\right|e^{\int_{0}^{t}\text{Im}(\Delta E)d\tau}.$ (8)

In other words, we now provide a universal shortcut method (USM) that depends
only on partial information of spectrum, which is friendly for spin systems,
to speed up adiabatic passage. Therefore, we can now successfully address
previous intractable systems that conventional schemes are powerless. In the
following text, we apply the above theory and show the numerical results in
two topological models, respectively, including the 1D non-Hermitian SSH and
the well-known Thouless charge pumping models.

_Application I. —_ We consider the 1D NH SSH model in the real space whose
Hamitonian takes the form,

$\begin{split}H_{1}=\sum_{<i,j>}\left[(t_{1}+\gamma/2)c^{\dagger}_{i,A}c_{i,B}+(t_{1}-\gamma/2)c^{\dagger}_{i,B}c_{i,A}\right.\\\
\left.+t_{2}c^{\dagger}_{i,B}c_{j,A}+t_{2}c^{\dagger}_{j,A}c_{i,B}\right],\end{split}$
(9)

where a unit cell contains two sublattices labeled $A$ and $B$, $<i,j>$
denotes the nearest-neighbour unit cell and the hopping terms
$H_{\pm}=t_{1}\pm\gamma/2$, as depicted in Fig. 1(a).

This model supports the non-Hermitian skin effect and its topological zero-
modes are determined by the non-Bloch winding number as discussed in Ref.
SYao2018 . To explore the state transfer via the edge state, we consider
$H_{1}$ with $2N-1$($N\in R$) lattice sites. For this lattice chain, its
topological zero-energy edge state is well-separated from the bulk states.
Specifically, we find that there is one edge state $|L\rangle=|egg...g\rangle$
in the left end when $t_{1}\in[0,\sqrt{t_{2}^{2}+(\gamma/2)^{2}})$ and one
edge state $|R\rangle=|gg...ge\rangle$ in the right end when
$t_{1}\in(\sqrt{t_{2}^{2}+(\gamma/2)^{2}},3]$, as depicted in Fig. 1(b-c).
That is to say, as we tune $t_{1}$ from $0$ to $3$ adiabatically, there is a
zero-mode edge state transferring from the left end to the right end, i.e.
$|L\rangle\rightarrow|R\rangle$,. Note that both the left and right edge
states occupy the A-type sublattice in each unit cell, like in the Hermitian
caseFMei2018 .

Figure 1:  (a) Schematic diagram of the 1D NH SSH model where a lattice chain
consists of an odd number of lattice sites with each unit cell containing two
subalttices labelling A and B, respectively. $t_{1}$($t_{2}$) denotes the
intra(inter)-cell hopping while $+(-)\gamma/2$ is the NH intracell hopping
from $B$($A$) to $A$($B$). (b),(c) The real and imaginary parts of energy
spectrum. (d) The absolute value of the energy spectrum.

In what follows, we mainly focus on the quantum state transfer via the
topological edge states using our universal shortcut method. we mainly focus
on the quantum state transfer via the topological edge states using our
universal shortcut method. When t is varied from $0\rightarrow 3$,
$|\psi(0)\rangle=|L\rangle\rightarrow|\psi(3)\rangle$. Because the upper
branch and the lower branch of this system are symmetrical, only the absolute
value of the energy spectrum needs to be determined, as shown in Fig. 1(d).
Fig. 2(a) shows the curve of $t_{1}(T)$ and also the linear modulation of
$t_{1}$ while the optimal adiabatic modulation of $t_{1}(T)$ via our USM is
depicted in Fig.2(b). Note that the complex energy spectrum for this system
[Fig. 1(d)] can be measured in photonic/phononic crystals Bergholtz2021 ;
WZhu2023  or through the non-Hermitian absorption spectroscopy method
developed recently in cold atoms YXu2022 ; MCao2023 . Fig.2(c) shows the
fidelity $F=\langle R|\psi(3)\rangle$ as a function of evolution time $T$
under linear modulation and adiabatic passage in our shortcut method. The
adjacent adiabatic coefficient $s^{\prime}$ is shown in Fig.2(d). Moreover,
Fig.2 (e) shows that fidelity is a function of the hopping coefficient $t_{1}$
derived from the linear modulation and our shortcut method. Whether the system
is Hermitian or non-Hermitian, our method can realize faster, high fidelity
and more robust state transfer, compared with linear modulation. It is worth
mentioning that our shortcut scheme can partly suppress the dissipation in the
adiabatic process origined from the non-Hermiticity of the system since it
shortens the whole evolution time.

Figure 2: Comparison of the linear modulation and our optimized modulation,
and the lattice size $N=20$. The $H_{\pm}$($H_{+}$: blue line, $H_{-}$: red
line) vs $t_{1}/T$ for (a) the linear modulation and (b) the USE . The (c-d)
show the transfer fidelity and the adjacent adiabatic coefficient $s^{\prime}$
as a function of $T*\Omega$ for the conventional linear modulation(blue line)
and the optimal adiabatic modulation(red line) in the case of
$\gamma=\Omega/3$. (Insert: $\gamma=2\Omega/3$) (e) The transfer fidelity vs
hopping term variations for the conventional linear modulation(blue line) and
the optimal adiabatic modulation(red line) in the case of $T*\Omega=125$.

_Application II. —_ By eliminating $\gamma$ term and adding an on-site term to
$H_{1}$, we obtain the famous Rice-Mele model $H_{2}$, which is given by

$H_{2}=H_{1}(\gamma=0)+\sum_{i,m\in\\{A,B\\}}t_{m}c^{\dagger}_{i,m}c_{i,m},$
(10)

with $t_{A}=-t_{B}=\Delta(\phi)/2$. If we further set
$t_{1}=t_{0}+\delta(\phi)$ and $t_{2}=t_{0}-\delta(\phi)$ with
$(\delta,\Delta)=(\delta_{0}\cos\phi,\Delta_{0}\sin\phi)$ [$\delta_{0}$ and
$\Delta_{0}$ are constants and $\phi\in[0,2\pi)$], we obtain the Thouless
charge pump model. This topological pumping model reveals the first Chern
number $C_{1}=\frac{1}{2\pi}\int dkd\phi
F_{k\phi}=\text{sgn}(t_{0}\delta_{0}\Delta_{0})$ of the corresponding 2D
system in the $(k,\phi)$ space, which has been well studied in theories and
experiments . Consider the periodic condition in real space and after one
pumping circle, particles in the specific unit cell will be pumped to the next
unit cell with a quantified change of polarization $\Delta
P=P(\phi=2\pi)-P(\phi=0)=C_{1}$ where the lattice constant is unit. Here the
polarization $P=\langle
w(x)|\hat{x}-R|w(x)\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}A_{k}dk$ with
$|w(x)\rangle$ denotes the wannier state at unit cell $x=R$ in real space. The
Berry connection being $A_{k}=\langle
u_{k}(\phi)|i\partial_{k}|u_{k}(\phi)\rangle$ for the lower occupied Bloch
band which leads to the associate Berry curvature
$F_{k\phi}=\partial_{k}A_{\phi}-\partial_{\phi}A_{k}$.

Figure 3:  Comparison of the linear and optimal modulations, and the system
size $N=10$. The $t_{j}$($t_{1}$: blue line, $t_{2}$: red line, $t_{A}$:
yellow line, $t_{B}$: purple line) vs $t/T$ for (a) the linear modulation and
(b) the USE . (c) Energy spectrum with $\phi$ of this model. The (d-e) show
the transfer ﬁdelity and the adjacent adiabatic coefficient $s^{\prime}$ as a
function of $T*\Omega$ for the conventional linear modulation(blue line) and
the optimal adiabatic modulation(red line) in the case of
$\delta_{0}=0.6t_{0}$ and $\Delta_{0}=0.36t_{0}$. (f) The transfer fidelity vs
hopping term variations for the conventional linear modulation(blue line) and
the optimal adiabatic modulation(red line) in the case of $T*\Omega=6.8$.

In realization experiments, the adiabatic parameter is linearly modulated as
$\phi=\omega t$ and thus $\delta$ and $\Delta$ cyclically oscillate and vary
with $t$ as seen in Fig 3(a). Conventional shortcut methods, including
shortcuts to adiabaticity, quasiadiabatic dynamics and so on, can not work in
this model. We now also apply our shortcut theory to find the optimal
adiabatic path of this pumping process. Due to the half-filled Rice-Mele
model, there are only non-adiabatic transitions between the upper branch and
lower branch of energy spectra. Based on $E_{min}$ and $\Delta E$ that we
chose as plotted in Fig.3(c), the relevant parameter modulations are now
modified by our shortcut method, as shown in Fig.3(a,b). The fidelity vs
evolution time with the linear modulation and our shortcut method are shown in
Fig.3(d) and the adjacent adiabatic coefficient $s^{\prime}$ is shown in
Fig.3(e) in the same condition. Moreover, Fig.3(f) that the fidelity as a
function of $t_{j}/\Omega$ for the linear modulation and our shortcut method.
The result of this case, which is similar to the case of the 1D NH SSH model,
shows that our scheme can reduce the evolution time and enhance the robustness
without increasing the experimental complexity.

_Conclusion and outlook.—_ In summary, we propose a generic shortcut method
for exact state manipulations in quantum spin systems. Theoretically, going
beyond existing protocols for few-level systems, our shortcut only depends on
a few pieces of information about the energy spectrum. Thanks to a stricter
adiabatic condition, the universal shortcut protocol of the control parameter
has been obtained. Experimentally, our control protocol does not need
additional coupling or interaction, so this method is friendly. For example,
our protocol shows great advantages in the NH SSH and Thouless pump models
compared with the linear schemes, e.g. faster rate of evolution and more
robust against parameter error. Extending our ideas to the manipulations of
corner states in higher-order topological insulators, entangled states in
quantum many-body systems and floquet states in cycle-driven systems
constitutes another interesting line of research.

###### Acknowledgements.

This work was supported by the Natural Science Foundation of Guangdong
Province under Grant 2023A1515011212 , the Special Projects in Key Fields of
Ordinary Universities in Guangdong Province under Grant 2021ZDZX1015.

## References

  * (1) K. Bergmann, H. Theuer, and B. W. Shore, Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys. 70, 1003 (1998).
  * (2) P. Král, I. Thanopulos, and M. Shapiro, Colloquium: Coherently controlled adiabatic passage, Rev. Mod. Phys. 79, 53 (2007).
  * (3) D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martinez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  * (4) M. V. Berry, Transitionless quantum driving, J. Phys. A 42, 365303 (2009).
  * (5) X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gúery-Odelin, and J. G. Muga, Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity, Phys. Rev. Lett. 104, 063002 (2010).
  * (6) S. Martínez-Garaot, A. Ruschhaupt, J. Gillet, T. Busch, and J. G.Muga, Fast quasiadiabatic dynamics, Phys. Rev. A 92, 043406 (2015).
  * (7) D. Sels, and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, Proc. Natl. Acad. Sci. U.S.A. 114, E3909 (2017).
  * (8) P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quantum spin systems, Phys. Rev. Lett. 123, 090602 (2019).
  * (9) G. Passarelli, V. Cataudella, R. Fazio, and P. Lucignano, Counterdiabatic driving in the quantum annealing of the p-spin model: A variational approach, Phys. Rev. Research 2, 013283 (2020).
  * (10) L. Prielinger, A. Hartmann, Y. Yamashiro, K. Nishimura, W. Lechner, and H. Nishimori, Two-Parameter Counter-Diabatic Driving in Quantum Annealing, Phys. Rev. Research 3, 013227 (2021).
  * (11) F. P. Barone, O. Kiss, M. Grossi, S. Vallecorsa, and A. Mandarino, Counterdiabatic optimized driving in quantum phase sensitive models, arXiv.2311.04282 (2023).
  * (12) I. Čepaitė, A. Polkovnikov, A. J. Daley, and C. W. Duncan, Counterdiabatic Optimized Local Driving, PRX Quantum 4, 010312 (2023).
  * (13) P. Chandarana, N. N. Hegade, K. Paul, F. Albarr’́an-Arriagada, E. Solano, A. del Campo, and Xi Chen, Digitized-counterdiabatic quantum approximate optimization algorithm, Phys. Rev. Research 4, 013141 (2022).
  * (14) T. Kadowaki, H. Nishimori, Greedy parameter optimization for diabatic quantum annealing, Phil. Trans. R. Soc. A 381 20210416 (2022).
  * (15) G. B. Mbeng and W. Lechner, Rotated ansatz for approximate counterdiabatic driving, arXiv:2207.03553.
  * (16) F. Mei, G. Chen, L. Tian, S.-L. Zhu, and S. Jia, Robust quantum state transfer via topological edge states in superconducting qubit chains, Phys. Rev. A 98, 012331 (2018).
  * (17) L. Qi, G.-L. Wang, S.-T. Liu, S. Zhang, and H.-F. Wang, Engineering the topological state transfer and topological beam splitter in an even-sized Su-Schrieffer-Heeger chain, Phys. Rev. A 102, 022404 (2020).
  * (18) W.-X. Cui, L. Qi, Y. Xing, S.-T. Liu, S. Zhang, and H.-F. Wang, Localized photonic states and dynamic process in nonreciprocal coupled Su-Schrieffer-Heeger chain, Opt. Express 28, 37026 (2020).
  * (19) S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. We-ber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms, Science 365, 775 (2019).
  * (20) M. Kiczynski, S. K. Gorman, H. Geng, M. B. Donnelly, Y. Chung, Y. He, J. G. Keizer and M. Y. Simmons, Engineering topological states in atom-based semiconductor quantum dots, Nature 606, 694 (2022).
  * (21) G. Cáceres-Aravena, B. Real, D. Guzmán-Silva, P. Vildoso, I. Salinas, A. Amo, T. Ozawa,and R. A. Vicencio, Edge-to-edge topological spectral transfer in diamond photonic lattices, APL Photonics 8, 080801 (2023).
  * (22) I. Brouzos, I. Kiorpelidis, F. K. Diakonos, and G. Theocharis, Fast, robust, and amplified transfer of topological edge modes on a time-varying mechanical chain, Phys. Rev. B 102, 174312 (2020).
  * (23) N. E. Palaiodimopoulos, I. Brouzos, F. K. Diakonos, and G. Theocharis, Fast and robust quantum state transfer via a topological chain, Phys. Rev. A 103, 052409 (2021).
  * (24) F. M. D’Angelis, F. A. Pinheiro, D. Guéry-Odelin, S. Longhi, and F. Impens, Fast and robust quantum state transfer in a topological Su-Schrieffer-Heeger chain with next-to-nearest-neighbor interactions, Phys. Rev. Research 2, 033475 (2020).
  * (25) Y.-L. Ji, F.-L. Zhou, X. Chen, R. Liu, Z.-L. Li, H. Zhou, and X.-L. Peng, Counterdiabatic transfer of a quantum state in a tunable Heisenberg spin chain via the variational principle, Phys. Rev. A 105, 052422 (2022).
  * (26) L.-J. Huang, Z. Tan, H.-H. Zhong, and B. Zhu, Fast and robust quantum state transfer assisted by zero-energy interface states in a splicing Su-Schrieffer-Heeger chain, Phys. Rev. A 106, 022419 (2022).
  * (27) J. Zurita, C. E. Creffield, and G. Platero1, Fast quantum transfer mediated by topological domain walls, Quantum 7, 1043 (2023).
  * (28) Y. Ke, X. Qin, F. Mei, H. Zhong, Y. S. Kivshar, and C. Lee, Topological phase transitions and thouless pumping of light in photonic waveguide arrays, Laser and Photonics Reviews 10, 995 (2016).
  * (29) A. Cerjan, M. Wang, S. Huang, K. P. Chen, and M. C. Rechtsman, Thouless pumping in disordered photonic systems, Light: Science and Applications 9, 1 (2020).
  * (30) M. Jürgensen, S Mukherjee, and M C. Rechtsman, Quantized nonlinear Thouless pumping, Nature 596, 63 (2021).
  * (31) O. You, S. Liang, B. Xie, W. Gao, W. Ye, J. Zhu, and S. Zhang, Observation of Non-Abelian Thouless Pump, Phys. Rev. Lett. 128, 244302 (2022).
  * (32) Y.-K. Sun, X.-L. Zhang, F. Yu, Z.-N. Tian, Q.-D. Chen, and H.-B. Sun, Non-Abelian Thouless pumping in photonic waveguides, Nat. Phys. 18, 1080 (2022).
  * (33) M. Jürgensen, S. Mukherjee, C J’́crg, and M. C. Rechtsman, Quantized fractional Thouless pumping of solitons, Nat. Phy. (2023).
  * (34) W. Ma, L. Zhou, Q. Zhang, M. Li, C. Cheng, J. Geng, X. Rong, F. Shi, J. Gong, and J. Du, Experimental Observation of a Generalized Thouless Pump with a Single Spin, Phys. Rev. Lett. 120, 120501 (2018).
  * (35) E. Boyers, P.J.D. Crowley, A. Chandran, and A. O. Sushkov, Exploring 2D Synthetic Quantum Hall Physics with a Quasiperiodically Driven Qubit, Phys. Rev. Lett. 125, 160505 (2020).
  * (36) M. Jürgensen, S. Mukherjee, C. Jörg, and M. C. Rechtsman, Quantized Fractional Thouless Pumping of Solitons, Nat. Phys. 19, 420 (2023).
  * (37) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys. 12, 350 (2016).
  * (38) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological thouless pumping of ultracold fermions, Nat. Phys. 12, 296 (2016).
  * (39) H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, Geometrical pumping with a bose-einstein condensate, Phys. Rev. Lett. 116, 200402 (2016).
  * (40) S. Nakajima, N. Takei, K. Sakuma, Y. Kuno, P. Marra, and Y. Takahashi, Nat. Phy. 17, 844 (2021).
  * (41) A.S. Walter, Z. Zhu, M. Gächter, J. Min-guzzi, S. Roschinski, K. Sandholzer, K. Viebahn, and T. Esslinger, Quantization and its breakdown in a Hubbard-Thouless pump, Nat. Phys. 19, 1471 (2023).
  * (42) Z. Fedorova, H. Qiu, S. Linden, and J. Kroha, Observation of topological transport quantization by dissipation in fast Thouless pumps, Nat. Commun. 11, 1 (2020).
  * (43) D. Dreon, A. Baumg’́artner, X. Li, S. Hertlein, T. Esslinger, and T. Donner, Self-oscillating geometric pump in a dissipative atom-cavity system, Nature 608, 494 (2022).
  * (44) K. J. M. Schouten and V. Cheianov, Rapid-cycle Thouless pumping in a one-dimensional optical lattice, Phys. Rev. A 104, 063315 (2021).
  * (45) S. Malikis, and V. Cheianov, An ideal rapid-cycle Thouless pump, SciPost Phys. 12, 203 (2022).
  * (46) W. Liu, S. Hu, L. Zhang, Y. Ke, and C. Lee, Correlated topological pumping of interacting bosons assisted by Bloch oscillations, Phys. Rev. Research 5, 013020 (2023).
  * (47) V. Y Chernyak, N. A Sinitsyn, and C. Sun, A large class of solvable multistate Landau-Zener models and quantum integrability, J. Phys. A: Math. Theor. 51, 245201 (2018).
  * (48) V. Y Chernyak, and N. A Sinitsyn, Integrability in the multistate Landau-Zener model with time-quadratic commuting operators, J. Phys. A: Math. Theor. 54, 115204 (2021).
  * (49) T. Suzuki, H. Nakazato, Generalized adiabatic impulse approximation, Phys. Rev. A 105, 022211 (2022).
  * (50) S. Ashhab, Landau-Zener-Stueckelberg interferometry with driving fields in the quantum regime, J. Phys. A: Math. Theor. 50, 134002 (2017).
  * (51) L. Xiao, K.-K. Wang, X. Zhan, Z.-H. Bian, K. Kawabata, M. Ueda, W. Yi, and P. Xue, Observation of Critical Phenomena in Parity-Time-Symmetric Quantum Dynamics, Phys. Rev. Lett. 123, 240401 (2019).
  * (52) V Ivakhnenko, S. N Shevchenko, F Nori, Nonadiabatic Landau-Zener-St’́uckelberg-Majorana transitions, dynamics, and interference, Phys. Rep. 995,1 (2023).
  * (53) S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  * (54) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * (55) W. Zhu, W. Deng, Y. Liu, J. Lu, Z.-K. Lin, H.-X. Wang, X. Huang, J.-H. Jiang, Z. Liu, Topological phononic metamaterials, Rep. Prog. Phys. 86, 106501 (2023).
  * (56) S. Ibáñez, S. Martínez-Garaot, Xi Chen, E. Torrontegui, and J. G. Muga, Shortcuts to adiabaticity for non-Hermitian systems,Phys. Rev. A 84, 023415 (2011).
  * (57) K. Li and Y. Xu, Non-Hermitian Absorption Spectroscopy, Phys. Rev. Lett. 129, 093001 (2022).
  * (58) M.-M. Cao, K. Li, W.-D. Zhao, W.-X. Guo, B.-X. Qi, X.-Y. Chang, Z.-C. Zhou, Y. Xu, and L.-M. Duan, Probing Complex-Energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator, Phys. Rev. Lett. 130, 163001 (2023).