Original: GraphQNTK:  Quantum Neural Tangent Kernel for Graph Data
Revised: GraphQNTK: the Quantum Neural Tangent Kernel for Graph Data

Revised: Anonymous Author(s)

Revised: AffiliationAddress

Original: Abstract
Revised: Abstract

Original: Graph Neural Networks (GNNs) and Graph Kernels (GKs) are two fundamentaltools used to analyze graph-structured data.
Revised: Graph Neural Networks (GNNs) and Graph Kernels (GKs) are two fundamentaltools used to analyze graph-structured data.

Original: Efforts have been recently made in developing a composite graph learning architecture combining the expressive powerof GNNs and the transparent trainability of GKs.
Revised: Efforts have been recently made in developing a composite graph learning architecture combining the expressive powerof GNNs and the transparent trainability of GKs.

Original: However, learning efficiency onthese models should be carefully considered as the huge computation overhead.
Revised: However, learning efficiency onthese models should be carefully considered as the huge computation overhead.

Original: Besides, their convolutional methods are often straightforward and introduce severeloss of graph structure information.
Revised: Besides, their convolutional methods are often straightforward and introduce severeloss of graph structure information.

Original: In this paper, we design a novel quantum graphlearning model to characterize the structural information while using quantumparallelism to improve computing efficiency.
Revised: In this paper, we design a novel quantum graphlearning model to characterize the structural information while using quantumparallelism to improve computing efficiency.

Original: Specifically, a quantum algorithm isproposed to approximately estimate the neural tangent kernel of the underlyinggraph neural network where a multi-head quantum attention mechanism is introduced to properly incorporate semantic similarity information of nodes into themodel.
Revised: Specifically, a quantum algorithm isproposed to approximately estimate the neural tangent kernel of the underlyinggraph neural network where a multi-head quantum attention mechanism is introduced to properly incorporate semantic similarity information of nodes into themodel.

Original: We empirically show that our method achieves competitive performanceon several graph classification benchmarks, and theoretical analysis is provided todemonstrate the superiority of our quantum algorithm.
Revised: We empirically show that our method achieves competitive performanceon several graph classification benchmarks, and theoretical analysis is provided todemonstrate the superiority of our quantum algorithm.

Original:  Introduction
Revised:  Introduction

Original: Fusing quantum computing and classic machine learning has become a promising subject of research.
Revised: Fusing quantum computing and classic machine learning has become a promising subject of research.

Original: Quantum-based algorithms have been proposed in recent years, from naive quantum non-parametricmachine learning [52, 36, 43, 32] to classic-quantum hybrid deep leaning [7, 10 ? , 37, 14].
Revised: Quantum-based algorithms have been proposed in recent years, from naive quantum non-parametricmachine learning [49, 34, 40, 30] to classic-quantum hybrid deep leaning [7, 10, 43, 35, 14].

Original: Despitethat quantum machine learning (QML) has shown its potential in many machine learning tasks,quantum computing for graph learning is still in its early stage [61].
Revised: Despitethat quantum machine learning (QML) has shown its potential in many machine learning tasks,quantum computing for graph learning is still in its early stage [57].

Original: Inspired by the two popularclasses of methods for learning on graph data, i e., Graph Neural Networks (GNNs) [20, 39, 21, 67]and Graph Kernels (GKs) [23], several works attempt to build quantum graph learning architecturethat captures the structural information of graph data, such as Quantum Graph Neural Networks(QGNNs)
Revised: Inspired by the two popularclasses of methods for learning on graph data, i e., Graph Neural Networks (GNNs) [19, 37, 20, 62]and Graph Kernels (GKs) [22], several works attempt to build quantum graph learning architecturethat captures the structural information of graph data, such as Quantum Graph Neural Networks(QGNNs)

Original: [64, 7, 11, 16, 1] and Quantum Graph Kernel Methods (QGKs) [56, 3, 25, 4].
Revised: [60, 7, 11, 16, 1] and Quantum Graph Kernel Methods (QGKs) [53, 3, 24, 4].

Original: A briefreview about quantum graph learning is illustrated in Fig.
Revised: A briefreview about quantum graph learning is illustrated in Fig.

Original: Some quantum subroutines for attribute encoding [5, 70] and structural encoding [64 ? ] have beendeveloped to dissolve the characteristics of the graph into the quantum model.
Revised: Some quantum subroutines for attribute encoding [5, 65] and structural encoding [60, 43] have beendeveloped to dissolve the characteristics of the graph into the quantum model.

Original: However, most presentquantum graph learning models are hybrid such that the expressive capability depends more on thecomplexity of the classic modules [70].
Revised: However, most presentquantum graph learning models are hybrid such that the expressive capability depends more on thecomplexity of the classic modules [65].

Original: It is difficult to characterize the structure information andattribute information of the graph by the quantum components without the participation of classicmodules.
Revised: It is difficult to characterize the structure information andattribute information of the graph by the quantum components without the participation of classicmodules.

Original: Even worse, the frequent interactions between classical systems and quantum environmentsgenerally incur additional overhead [55].
Revised: Even worse, the frequent interactions between classical systems and quantum environmentsgenerally incur additional overhead [52].

Original: It is unclear whether the introduced quantum modulecan improve the performance of the model as well as the training efficiency.
Revised: It is unclear whether the introduced quantum modulecan improve the performance of the model as well as the training efficiency.

Original: Besides, most ofexisting proposals for quantum machine learning for graphs lack a clear demonstration of a quantumsuperiority for tasks on classical datasets.
Revised: Besides, most ofexisting proposals for quantum machine learning for graphs lack a clear demonstration of a quantumsuperiority for tasks on classical datasets.

Original: Using quantum computing power to boost the trainability and expressive behaviour of classic machinelearning models provides one of the most promising direction for quantum machine learning.
Revised: Using quantum computing power to boost the trainability and expressive behaviour of classic machinelearning models provides one of the most promising direction for quantum machine learning.

Original: Itis demonstrated that the power of quantum computing could be used to find atypical but usefulpatterns that classical systems are not considered to be able to generate effectively [14, 24, 28], andaccelerate the training process of existing classic models [36, 43, 71].
Revised: Itis demonstrated that the power of quantum computing could be used to find atypical but usefulpatterns that classical systems are not considered to be able to generate effectively [14, 23, 27], andaccelerate the training process of existing classic models [34, 40, 66].

Original: Several quantum algorithms[52, 45, 44] based on the HHL algorithm [22] show the exponential speedup compared with theirclassical counterpart, with a assumption that a quantum random access memory (QRAM) [35] isaccessible.
Revised: Several quantum algorithms[49, 42, 41] based on the HHL algorithm [21] show the exponential speedup compared with theirclassical counterpart, with a assumption that a quantum random access memory (QRAM) [33] isaccessible.

Original: Recent literature employ quantum algorithms to efficiently train deep neural networks[71], reconstruct unsupervised clustering [34] and supervised kernel classifier [43].
Revised: Recent literature employ quantum algorithms to efficiently train deep neural networks[66], reconstruct unsupervised clustering [32] and supervised kernel classifier [40].

Original: It is hopeful thatquantum computing could provide a new learning paradigm.
Revised: It is hopeful thatquantum computing could provide a new learning paradigm.

Original: In addition, simulations and physicalexperiments have proved the potential of using quantum algorithms to encode and process regularclassical data such as text and image [60, 6].
Revised: In addition, simulations and physicalexperiments have proved the potential of using quantum algorithms to encode and process regularclassical data such as text and image [56, 6].

Original: Beyond vanilla GNNs and GKs, composite graph learning studies have emerged that combine theadvantages of both areas [17, 50, 10, 18].
Revised: Beyond vanilla GNNs and GKs, composite graph learning studies have emerged that combine theadvantages of both areas [17, 47, 10, 18].

Original: However, the computation overheads is extremely largedue to either the dense gram matrix [17], or the large number of substructures to be compared aftergraph decomposition [10].
Revised: However, the computation overheads is extremely largedue to either the dense gram matrix [17], or the large number of substructures to be compared aftergraph decomposition [10].

Original: Prospectively, the barrier that conventional model is difficult to trainand scale up is expected to be circumvented with the help of the uniqueness of quantum computing.
Revised: Prospectively, the barrier that conventional model is difficult to trainand scale up is expected to be circumvented with the help of the uniqueness of quantum computing.

Original: Early research involves altering the amplitude of quantum basis states to accomplish a quantum logicoperations [8], which is profitable from the huge quantum Hilbert space to encode the normalizeddata.
Revised: Early research involves altering the amplitude of quantum basis states to accomplish a quantum logicoperations [8], which is profitable from the huge quantum Hilbert space to encode the normalizeddata.

Original: Recently, simultaneous transformation of basic states in quantum superposition using quantumparallelism is regarded as a remarkable manifestation of quantum superiority, which is successfullyimplemented in classic machine learning to reduce the computational overheads [36, 37, 71].
Revised: Recently, simultaneous transformation of basic states in quantum superposition using quantumparallelism is regarded as a remarkable manifestation of quantum superiority, which is successfullyimplemented in classic machine learning to reduce the computational overheads [34, 35, 66].

Original: Thesestrategies could be helpful in the regime of training graph models with either the non-convex natureof the training procedure, or the poor scalability w.r.t.
Revised: Thesestrategies could be helpful in the regime of training graph models with either the non-convex natureof the training procedure, or the poor scalability w.r.t.

Original: training size.
Revised: training size.

Original: In this paper, we focus on quantum machine learning of graph-structured data with attributed nodesand binary edges.
Revised: In this paper, we focus on quantum machine learning of graph-structured data with attributed nodesand binary edges.

Original: Inspired by recent quantum neural network methods [37, 71] that efficientlyreconstruct the dynamics of classic neural networks using quantum computing techniques, a newquantum graph learning model is proposed which is analogue to train an infinite-width GNN withattention mechanism, where the number of heads goes to infinity.
Revised: Inspired by recent quantum neural network methods [35, 66] that efficientlyreconstruct the dynamics of classic neural networks using quantum computing techniques, a newquantum graph learning model is proposed which is analogue to train an infinite-width GNN withattention mechanism, where the number of heads goes to infinity.

Original: Our contributions are:
Revised: Our contributions are:

Original: • For the first time, we show that the infinite-width GNN, namely graph tangent neural network[17] can be trained in a quantum reconstructive paradigm, by introducing a quantum aggregationtransformation and a quantum kernel estimation adaptive to the different sizes of graph.
Revised: • For the first time, we show that the infinite-width GNN, namely graph tangent neural network[17] can be trained in a quantum reconstructive paradigm, by introducing a quantum aggregationtransformation and a quantum kernel estimation adaptive to the different sizes of graph.

Original: Thiscomposite quantum graph learning architecture preserves the expressive power of GNNs and the
Revised: Thiscomposite quantum graph learning architecture preserves the expressive power of GNNs and the

Original: transparent trainability of GKs as the composite graph learning model does.
Revised: transparent trainability of GKs as the composite graph learning model does.

Original: • A quantum-friendly self-attention mechanism(transformer) is employed to incorporate semantic 
Revised: • A quantum-friendly self-attention mechanism is employed to incorporate semantic correlation

Original: correlation information of all node pairs of the graph into the model, which can be naturally adopted by quantum algorithms with additional relaxation.
Revised: information of all node pairs of the graph into the model, which can be naturally adopted byquantum algorithms with additional relaxation.

Original: To our best acknowledge, this is the first for
Revised: To our best acknowledge, this is the first for

Original: leveraging attention mechanism in quantum graph learning.
Revised: leveraging attention mechanism in quantum graph learning.

Original: In contrast, the present quantum graphlearning algorithms can only undertake propagation between the adjoining neighbors [ ? 25, 64],and constrain themselves to specify identical priority to distinct neighbours [1, 11].•
Revised: In contrast, the present quantum graph learning algorithms can only undertake propagation between the adjoining neighbors [43, 24, 60],and constrain themselves to specify identical priority to distinct neighbours [1, 11].

Original:  We evaluate our quantum model on graph classification benchmarks.
Revised: • We evaluate our quantum model on graph classification benchmarks.

Original: The results show that ourmethod achieves competitive performance compared with the classic baselines including both
Revised: The results show that our method achieves competitive performance compared with the classic baselines including both

Original: GNNs and GKs.
Revised: GNNs and GKs.

Original: Besides, we give a theoretical analysis showing that our GraphQNTK reduces the
Revised: Besides, we give a theoretical analysis showing that our GraphQNTK reduces the

Original: running time complexity of graph kernel neural networks from O ( N 2 ) to O ( N ) .
Revised: running time complexity of graph kernel neural networks from O ( N 2 ) to O ( N ) .

Original:  Methodology
Revised:  Methodology

Revised: [Figure]

Original: 2.1 Preliminaries
Revised: 2.1 Preliminaries

Original: [Figure]
Revised: [Figure]

Original: We first briefly review the most common settingfor GNNs and the corresponding neural tangentkernel (NTK), and by the way the notation is given.
Revised: We first briefly review the most common settingfor GNNs and the corresponding neural tangentkernel (NTK), and by the way the notation is given.

Original: A graph G = ( V, E ) is denoted by a collectionof nodes V and edges E .
Revised: A graph G = ( V, E ) is denoted by a collectionof nodes V and edges E .

Original: Each node has a ddimensional feature vector h v ∈ R d , v ∈ V , and
Revised: Each node has a ddimensional feature vector h v ∈ R d , v ∈ V , and

Original: H ∈ R n × d is the feature matrix stacking all nodesfeatures.
Revised: H ∈ R n × d is the feature matrix stacking all nodesfeatures.

Original: For graph classification, we consider the dataset with a set of graphs { G 1 , . . .
Revised: For graph classification, we consider the dataset with a set of graphs { G 1 , . . .

Original: , G N } ⊆ G and their labels { y 1 , . . .
Revised: , G N } ⊆ G and their labels { y 1 , . . .

Original: , y N } ⊆ Y .
Revised: , y N } ⊆ Y .

Original: Our goal is tolearn to predict labels of unseen graphs.
Revised: Our goal is tolearn to predict labels of unseen graphs.

Original: The formulation of GNN The differences of GNNs mainly depend on the different settings ofmessage propagation process.
Revised: The formulation of GNN The differences of GNNs mainly depend on the different settings ofmessage propagation process.

Original: Here we consider a simple message passing framework [20] and thepropagation of the l -th ( l ∈ [ L ] ) layer is given as:
Revised: Here we consider a simple message passing framework [19] and thepropagation of the l -th ( l ∈ [ L ] ) layer is given as:

Original: [Equation]
Revised: [Equation]

Original: where N ( u ) denotes the neighbors of u , c σ is the scaling factor, d lr is the output dimension ofthe l -th layer and the r -th fully-connected layers, σ is an element-wise activated function, and W lR is learnable weights performing on the input for R times of the l -th layer (equivalent to Rfully-connected layers without the bias term).
Revised: where N ( u ) denotes the neighbors of u , c σ is the scaling factor, d l is the output dimension of the l -thlayer, σ is an element-wise activated function, and W lR is learnable weights performing on the inputfor R times of the l -th layer (equivalent to R fully-connected layers without the bias term).

Original: For graph classification, the output is a permutation invariance function acting on the collection of allnode features in the last layer.
Revised: For graph classification, the output is a permutation invariance function acting on the collection of allnode features in the last layer.

Original: The popular sum_pooling function is adopted: h G = (cid:80)
Revised: The popular sum_pooling function is adopted: h G = (cid:80)

Original: u ∈ V
Revised: u ∈ V

Original: h Lu . .
Revised: h Lu . .

Original: NTK of the infinite-width GNN.
Revised: NTK of the infinite-width GNN.

Original: Consider a training set { ( x i , y i ) } Ni =1 ⊂ R d × R .
Revised: Consider a training set { ( x i , y i ) } N i =1 ⊂ R d × R .

Original: When an overparameterized fully connected network f ( θ, x ) : R d → R whose width is allowed to go to infinityand parameters θ are randomly initialized and trained with gradient descent, the dynamics of thenetwork is equivalent to the kernel regression [31].
Revised: When an overparameterized fully connected network f ( θ, x ) : R d → R whose width is allowed to go to infinityand parameters θ are randomly initialized and trained with gradient descent, the dynamics of thenetwork is equivalent to the kernel regression [29].

Original: This is the so called neural tangent kernel (NTK):
Revised: This is the so called neural tangent kernel (NTK):

Original: [Equation]
Revised: [Equation]

Original: which remains constant during training, i.e., H ( t ) =
Revised: which remains constant during training, i.e., H ( t ) =

Original: H (0) .
Revised: H (0) .

Original: And we replace H ( t ) with H forconvenience.
Revised: And we replace H ( t ) with H forconvenience.

Original: The final prediction for a test datapoint x ∗ is
Revised: The final prediction for a test datapoint x ∗ is

Original: [Equation]
Revised: [Equation]

Original: where y i =
Revised: where y i =

Original: y i and k ∗ ∈
Revised: y i and k ∗ ∈

Original: R N is the vector whose i -th element denotes the NTKvalue between xi andx ∗ .
Revised: R N is the vector denoting the NTK between x ∗ and all x i in the training set.

Original: It is discovered that convolutional neural networks (CNNs) with infinite-width channels and infinitenumber of filters also have the same behaviour [2].
Revised: It is discovered that convolutional neural networks (CNNs) with infinite-width channels and infinitenumber of filters also have the same behaviour [2].

Original: Inspired by this, Du et al.
Revised: Inspired by this, Du et al.

Original: [17] adopts the designingstrategy of NTK and leverages a GNN architecture to design new graph kernels, which is calledgraph neural tangent kernel (GNTK).
Revised: [17] adopts the designingstrategy of NTK and leverages a GNN architecture to design new graph kernels, which is calledgraph neural tangent kernel (GNTK).

Original: The dynamics of training the GNTK is equivalent to train aninfinitely-wide GNN initialized with random weights trained with gradient descent.
Revised: The dynamics of training the GNTK is equivalent to train aninfinitely-wide GNN initialized with random weights trained with gradient descent.

Original: Specifically,consider two input graph G =
Revised: Specifically,consider two input graph G =

Original: ( V, E ) and G ′
Revised: ( V, E ) and G ′

Original: =
Revised: =

Original: ( V ′ , E ′ ) with | V | = n and | V | = n ′ , the GNTKΘ ∈ R n × n ′ and the relative covariance matrix Σ ∈ R n × n ′ in the l -th layer of the feature aggregationphase as described in Eq.
Revised: ( V ′ , E ′ ) with | V | = n and | V | = n ′ , the GNTKΘ ∈ R n × n ′ and the relative covariance matrix Σ ∈ R n × n ′ in the l -th layer of the feature aggregationphase as described in Eq.

Original: 1 after R fully-connected layers are given by
Revised: 1 after R fully-connected layers are given by

Original: [Equation]
Revised: [Equation]

Original: which is an affine transformation of the input GNTK and covariance respectively where 119 (cid:2) Θ 0 R ( G, G ′ ) (cid:3) uu ′ and (cid:2) Σ 0 R ( G, G ′ )
Revised: which is an affine transformation of the input GNTK and covariance respectively where 117 (cid:2) Θ 0 R ( G, G ′ ) (cid:3) uu ′ and (cid:2) Σ 0 R ( G, G ′ )

Original: (cid:3) uu ′ are both defined to be h ⊤ u h u ′ .
Revised: (cid:3) uu ′ are both defined to be h ⊤ u h u ′ .

Original: We replace them with 120 (cid:2) Θ 0 ( G, G ′ ) (cid:3) uu ′ and (cid:2) Σ 0 ( G, G ′ ) (cid:3) uu ′ respectively without ambiguity.
Revised: We replace them with 118 (cid:2) Θ 0 ( G, G ′ ) (cid:3) uu ′ and (cid:2) Σ 0 ( G, G ′ ) (cid:3) uu ′ respectively without ambiguity.

Original: The successive fully-connected layers defined in Eq.
Revised: The successive fully-connected layers defined in Eq.

Original: 2 are used to update the node hidden featureafter aggregation.
Revised: 2 are used to update the node hidden featureafter aggregation.

Original: Specifically, the GNTK of the fully-connected layer is recursively associated tothat of the previous layer, and the transformation is given by
Revised: Specifically, the GNTK of the fully-connected layer is recursively associated tothat of the previous layer, and the transformation is given by

Original: [Equation]
Revised: [Equation]

Original: [Equation] where ˆ σ ( r ) : [ − 1 , 1] → R denotes the the conjugate activation function corresponding to the activatedfunction σ with centered Gaussian processes of covariance at the r -th fully-connected layer, asdescribed in [15].
Revised: 6) where ˆ σ ( r ) : [ − 1 , 1] → R denotes the the conjugate activation function corresponding to the activatedfunction σ with centered Gaussian processes of covariance at the r -th fully-connected layer, asdescribed in [15].

Original: And the derivation of the covariance is
Revised: And the derivation of the covariance is

Original: [Equation]
Revised: [Equation]

Original: where ˆ˙ σ denotes the derivative of σ .
Revised: where ˆ˙ σ denotes the derivative of σ .

Original: Given Eq.
Revised: Given Eq.

Original: 6 and Eq.
Revised: 6 and Eq.

Original: 7, the transformation of the GNTK for thefeature update phase denoted by Eq.
Revised: 7, the transformation of the GNTK for thefeature update phase denoted by Eq.

Original: 2 is given by
Revised: 2 is given by

Original: [Equation]
Revised: [Equation]

Original: Therefore, computing each element of the GNTK (or covariance) matrix is only reliant on the elementat the same place of the GNTK (or covariance) matrix in the previous fully-connected layer.
Revised: Therefore, computing each element of the GNTK (or covariance) matrix is only reliant on the elementat the same place of the GNTK (or covariance) matrix in the previous fully-connected layer.

Original: The final
Revised: The final

Original: GNTK corresponding the two input graphs G and G ′ determined by the sum_pooling function:
Revised: GNTK corresponding the two input graphs G and G ′ determined by the sum_pooling function:

Original: [Equation]
Revised: [Equation]

Original: Intuitively, calculating each element of the GNTK of fully-connected layers could be acceleratingby a proper quantum kernel estimation algorithm.
Revised: Intuitively, calculating each element of the GNTK of fully-connected layers could be acceleratingby a proper quantum kernel estimation algorithm.

Original: However, it is indirect to realize an end-to-endspeedup for GNTK since calculating the element of GNTK requires an affine transformation.
Revised: However, it is indirect to realize an end-to-endspeedup for GNTK since calculating the element of GNTK requires an affine transformation.

Original: Tocircumvent this barrier, we derive a unitary quantum aggregation transformation to bridge the gapbetween quantum kernel methods and estimation of GNTK.
Revised: Tocircumvent this barrier, we derive a unitary quantum aggregation transformation to bridge the gapbetween quantum kernel methods and estimation of GNTK.

Original: 2.2 QNTK with Attention Mechanism
Revised: 2.2 QNTK with Attention Mechanism

Original: [Figure]
Revised: [Figure]

Original: global semantic similarity between each pairs of nodes.
Revised: global semantic similarity between each pairs of nodes.

Original: The neighbor aggregation is kept since the two nodes connected by an edge often have stronger semantic relationship.
Revised: The neighbor aggregation is kept since the two nodes connected by an edge often have stronger semantic relationship.

Original: The dynamics of the infinitewidth-limit GNN is analogous to kernel methods and we reconstruct it by quantum algorithms to estimate the kernel.
Revised: The dynamics of the infinitewidth-limit GNN is analogous to kernel methods and we reconstruct it by quantum algorithms to estimate the kernel.

Original: Before giving the analytical quantum reconstructionof GNTK with multi-head attention mechanism, we firstelaborate on how to integrate the transformer layerinto the GNN as describedin Sec. 2.1.
Revised: Before giving the analytical quantum reconstructionof GNTK with multi-head attention mechanism, we firstelaborate on how to integrate the transformer layerinto the GNN as describedin Sec. 2.1.

Original: The resulting GraphQNTK can be efficiently reconstructed by quantum computing paradigm,which gives a quadratic speedup over the classic estimationof GNTK.
Revised: The resulting GraphQNTK can be efficiently reconstructed by quantum computing paradigm,which gives a quadratic speedup over the classic estimationof GNTK.

Original: The mechanism tobuild the GNN and estimatethe GNTK is shown in Fig.
Revised: The mechanism tobuild the GNN and estimatethe GNTK is shown in Fig.

Original: GNN with multi-head attention.
Revised: GNN with multi-head attention.

Original: The aggregation process of vanilla GCN [39] regards the contribution of eachnode’s neighbor to the centralnode as equally important, which can be viewed as learning an averaged filter across the whole graph[66], leading to a great loss of structure information.
Revised: The aggregation process of vanilla GCN [37] regards the contribution of eachnode’s neighbor to the centralnode as equally important, which can be viewed as learning an averaged filter across the whole graph[61], leading to a great loss of structure information.

Original: Besides, the aggregation only is performedwithin the adjoining neighbors under the assumption that the graph is homophilous.
Revised: Besides, the aggregation only is performedwithin the adjoining neighbors under the assumption that the graph is homophilous.

Original: The method mayfail to learn effective graph structures for message passing [12].
Revised: The method mayfail to learn effective graph structures for message passing [12].

Original: To capture the global node similaritysemantics of the provided graph, numerous attempts that employ transformer for graph learning havebeen developed [27, 51, 53, 68].
Revised: To capture the global node similaritysemantics of the provided graph, numerous attempts that employ transformer for graph learning havebeen developed [26, 48, 50, 63].

Original: Consider the input feature matrix H l in ∈ R N × s l where N denotesthe number of samples and s l is the dimension of feature at layer l before implementation of thetransformer.
Revised: Consider the input feature matrix H l in ∈ R N × s l where N denotesthe number of samples and s l is the dimension of feature at layer l before implementation of thetransformer.

Original: The single transformer layer is to project the input H l in ∈ R N × s l by three matrices,i.e., W lQ ∈ R s l × s lK ,
Revised: The single transformer layer is to project the input H l in ∈ R N × s l by three matrices,i.e., W lQ ∈ R s l × s lK ,

Original: W lK ∈ R s l × s lK and W lV ∈ R s l × s lV , to the corresponding representations
Revised: W lK ∈ R s l × s lK and W lV ∈ R s l × s lV , to the corresponding representations

Original: Q l , K l , V l .
Revised: Q l , K l , V l .

Original: The formulation is given as
Revised: The formulation is given as

Original: [Equation]
Revised: [Equation]

Original: where ζ denotes an element-wise activated function.
Revised: where ζ denotes an element-wise activated function.

Original: The multi-head attention alternative is given by
Revised: The multi-head attention alternative is given by

Original: [Equation]
Revised: [Equation]

Original: where W lO ∈ R ( Ms lV ) × s l projects the N × Ms lV concatenated multi-head feature matrix back to
Revised: where W lO ∈ R ( Ms lV ) × s l projects the N × Ms lV concatenated multi-head feature matrix back to

Original: N × s l matrix.
Revised: N × s l matrix.

Original: Let Y and Θ denote the neural network Gaussian Process Kernel (NNGP) [51] and NTK afterthe transformer layer, and let (cid:101) Y and (cid:101)
Revised: Let Y and Θ denote the neural network Gaussian Process Kernel (NNGP) [48] and NTK afterthe transformer layer, and let (cid:101) Y and (cid:101)

Original: Θ be the input NNGP and NTK before the transformer layer.
Revised: Θ be the input NNGP and NTK before the transformer layer.

Original: Consider two input feature vector x and x ′ .
Revised: Consider two input feature vector x and x ′ .

Original: When the output dimension of the transformer layer andthe number of heads go to infinity, i.e., s l → ∞ , s lK → ∞ , s lV → ∞ , M → ∞ , the output NTK is:
Revised: When the output dimension of the transformer layer andthe number of heads go to infinity, i.e., s l → ∞ , s lK → ∞ , s lV → ∞ , M → ∞ , the output NTK is:

Original: [Equation]
Revised: [Equation]

Original: (12) where the under the restriction that 1) W lQ and W lK share the same weighs, and 2) scaling the dotproducts between Q l and K l by their dimension instead of the square root of the same quantity, i.e.,
Revised: (12) where the under the restriction that 1) W lQ and W lK share the same weighs, and 2) scaling the dotproducts between Q l and K l by their dimension instead of the square root of the same quantity, i.e.,

Original: G l = Q l K l ⊤ s lK .
Revised: G l = Q l K l ⊤ s lK .

Original: The detailed proof can be found in [26].
Revised: The detailed proof can be found in [25].

Original: To efficiently estimate the element of the NTK defined by the transformer layer using quantumparallel, we consider the identity activated function, i.e., ζ = I , and slightly modify the Eq.
Revised: To efficiently estimate the element of the NTK defined by the transformer layer using quantumparallel, we consider the identity activated function, i.e., ζ = I , and slightly modify the Eq.

Original: 12 as
Revised: 12 as

Original: [Equation]
Revised: [Equation]

Original: where (cid:101) T ( x , x ′ ) is the result of matrix multiplication between the column vector of the diagonal of(cid:101) Y ( x , x ) and row vector of the diagonal of (cid:101) Y ( x ′ , x ′ ) , and (cid:101) T ζ is the result of matrix multiplicationbetween the diagonal of those two matrix after activated operation.
Revised: where (cid:101) T ( x , x ′ ) is the result of matrix multiplication between the column vector of the diagonal of(cid:101) Y ( x , x ) and row vector of the diagonal of (cid:101) Y ( x ′ , x ′ ) , and (cid:101) T ζ is the result of matrix multiplicationbetween the diagonal of those two matrix after activated operation.

Original: It is reasonable to accept thismodification since in the limit of infinite width neural network the output converges in distribution toa multivariate normal with a block diagonal covariance [51].
Revised: It is reasonable to accept thismodification since in the limit of infinite width neural network the output converges in distribution toa multivariate normal with a block diagonal covariance [48].

Original: Notice that the difference between thedefinition of NNGP and the covariance of NTK is that the former denotes the expectation with respectto the output before the activated operation, while the later after the denotes the expectation withrespect to the output after the activated operation [31].
Revised: Notice that the difference between thedefinition of NNGP and the covariance of NTK is that the former denotes the expectation with respectto the output before the activated operation, while the later after the denotes the expectation withrespect to the output after the activated operation [29].

Original: Consequently, we consider that Y is equal tothe covariance of NTK within the transformer layer as the result of the identity activated function.
Revised: Consequently, we consider that Y is equal tothe covariance of NTK within the transformer layer as the result of the identity activated function.

Original: GNTK with infinite-width-limit attention.
Revised: GNTK with infinite-width-limit attention.

Original: To appropriately incorporate semantic similarity information of nodes into the model, a multi-head attention mechanism is implemented at the tailof the each GNN layer except the first and the last layer, and the calculation of the GNTK withinfinite-width-limit attention is to insert an additional procedure after the fully connected layers.
Revised: To appropriately incorporate semantic similarity information of nodes into the model, a multi-head attention mechanism is implemented at the tailof the each GNN layer except the first and the last layer, and the calculation of the GNTK withinfinite-width-limit attention is to insert an additional procedure after the fully connected layers.

Original: Forthe two input graphs G and G ′ , the formulation derived by Eq.
Revised: Forthe two input graphs G and G ′ , the formulation derived by Eq.

Original: 13 is given as
Revised: 13 is given as

Original: [Equation]
Revised: [Equation]

Original: where T l ( G, G ′ ) is the result of matrix multiplication between the column vector of the diagonal ofΣ lR ( G, G ) and row vector of the diagonal of Σ lR ( G ′ , G ′ ) .
Revised: where T l ( G, G ′ ) is the result of matrix multiplication between the column vector of the diagonal ofΣ lR ( G, G ) and row vector of the diagonal of Σ lR ( G ′ , G ′ ) .

Original: The affine transformation of the input
Revised: The affine transformation of the input

Original: GNTK corresponding to the aggregation phase as described in Eq.
Revised: GNTK corresponding to the aggregation phase as described in Eq.

Original: 5 is changed to (similar to Σ l 0 ):
Revised: 5 is changed to (similar to Σ l 0 ):

Original: [Equation]
Revised: [Equation]

Original: 2.3 The Proposed GraphQNTK
Revised: 2.3 The Proposed GraphQNTK

Original: We first show that estimating the single-layer GraphQNTK and its covariance with infinite-widthlimit attention mechanism can be efficiently reconstructed in the regime of quantum computing, andgeneralize to the multi-layer model.
Revised: We first show that estimating the single-layer GraphQNTK and its covariance with infinite-widthlimit attention mechanism can be efficiently reconstructed in the regime of quantum computing, andgeneralize to the multi-layer model.

Original: The following statements only consider two input graphs G =( V, E ) and G ′
Revised: The following statements only consider two input graphs G =( V, E ) and G ′

Original: =
Revised: =

Original: ( V ′ , E ′ ) with | V | = n and | V ′
Revised: ( V ′ , E ′ ) with | V | = n and | V ′

Original: | = n ′ , and the corresponding feature matrix H =[
Revised: | = n ′ , and the corresponding feature matrix H =[

Original: h ⊤ 1 , · · · , h ⊤ u , · · · , h ⊤ n ] ∈ R n × d and H ′
Revised: h ⊤ 1 , · · · , h ⊤ u , · · · , h ⊤ n ] ∈ R n × d and H ′

Original: =
Revised: =

Original: [ h ⊤ 1 , · · · , h ⊤ u ′ , · · · , h
Revised: [ h ⊤ 1 , · · · , h ⊤ u ′ , · · · , h

Original: ⊤
Revised: ⊤

Original: n
Revised: n

Original: ′ ] ∈ R n ′ × d .
Revised: ′ ] ∈ R n ′ × d .

Original: The approximateestimation of GNTK is denoted as ¯ Θ ∈ R n × n ′ and its element is ¯ Θ uu ′ .
Revised: The approximateestimation of GNTK is denoted as ¯ Θ ∈ R n × n ′ and its element is ¯ Θ uu ′ .

Original: The correspondingcovariance is ¯ Σ ∈ R n × n ′ and ¯ Σ uu ′ .
Revised: The correspondingcovariance is ¯ Σ ∈ R n × n ′ and ¯ Σ uu ′ .

Original: We use ¯ Θ GG ′ ∈ R to represent GraphQNTK after readout.
Revised: We use ¯ Θ GG ′ ∈ R to represent GraphQNTK after readout.

Original: Weomit the subscript R for clarity.
Revised: Weomit the subscript R for clarity.

Original: The same setting can be easily generalized to the arbitrary pair ofgraphs G, G ′ ∈ G by introducing auxiliary index registers.
Revised: The same setting can be easily generalized to the arbitrary pair ofgraphs G, G ′ ∈ G by introducing auxiliary index registers.

Original: First, we introduce the quantum datastructure accessible to the classical data, as commonly used by QML algorithms [52, 36, 34, 71].
Revised: First, we introduce the quantum datastructure accessible to the classical data, as commonly used by QML algorithms [49, 34, 32, 66].

Original: Feature encoding.
Revised: Feature encoding.

Original: Using the storage structure as stated in our proposed Theorem 1 in Appendix,the feature matrix can be prepared into the QRAM at the initialization of the algorithm.
Revised: Using the storage structure as stated in our proposed Theorem 1 in Appendix,the feature matrix can be prepared into the QRAM at the initialization of the algorithm.

Original: The dataencoding only occurs a single time and readout operation only takes logarithmic complexity timewith respect to the number of samples n and dimension of feature d .
Revised: The dataencoding only occurs a single time and readout operation only takes logarithmic complexity timewith respect to the number of samples n and dimension of feature d .

Original: The quantum representationscorresponding to the encoded feature vector and feature matrix are as follows
Revised: The quantum representationscorresponding to the encoded feature vector and feature matrix are as follows

Original: [Equation]
Revised: [Equation]

Original: Estimation of the initialized NTK.
Revised: Estimation of the initialized NTK.

Original: The empirical uncentered covariance of inputs (cid:2) Σ 0 ( G, G ′ ) (cid:3) uu ′and the initialized GNTK (cid:2) Θ 0 ( G, G ′ ) (cid:3) uu ′ is the inner product between h u and h u ′ .
Revised: The empirical uncentered covariance of inputs (cid:2) Σ 0 ( G, G ′ ) (cid:3) uu ′and the initialized GNTK (cid:2) Θ 0 ( G, G ′ ) (cid:3) uu ′ is the inner product between h u and h u ′ .

Original: Following a similar approach to [37], the inner product between two vectors with respect totheir quantum representations can be estimate efficiently by introducing an auxiliary register.
Revised: Following a similar approach to [35], the inner product between two vectors with respect totheir quantum representations can be estimate efficiently by introducing an auxiliary register.

Original: Specifically, estimation of the inner product h ⊤ u h u ′ can be performed by constructing the state1 √ 2 ( | u ⟩| u ′ ⟩| 

Revised: Specifically, estimation of the inner product h ⊤ u h u ′ can be performed by constructing the state1 √ 2 ( | u ⟩| u ′ ⟩|

Original: ⟩| | h u ⟩⟩ + | u ⟩| u ′ ⟩| 1 ⟩| |
Revised: ⟩| | h u ⟩⟩ + | u ⟩| u ′ ⟩| 1 ⟩| |

Original: h u ′ ⟩⟩ ) .
Revised: h u ′ ⟩⟩ ) .

Original: Applying a Hadamard gate on the third register gives thestate | u ⟩| u ′ ⟩ (cid:0) √ P uu ′ | 0 , g uu ′ ⟩ + √ 1 − P uu ′ | 1 , g ′ uu ′ ⟩ (cid:1) , where P uu ′ =
Revised: Applying a Hadamard gate on the third register gives thestate | u ⟩| u ′ ⟩ (cid:0) √ P uu ′ | 0 , g uu ′ ⟩ + √ 1 − P uu ′ | 1 , g ′ uu ′ ⟩ (cid:1) , where P uu ′ =

Original: 1+ h ⊤ u h u ′is the estimation ofthe inner product.
Revised: 1+ h ⊤ u h u ′is the estimation ofthe inner product.

Original: This procedure takes O (log d ) time
Revised: This procedure takes O (log d ) time

Original: and we denote this quantum operation by D 0 ,and we add a subscript to denote the corresponding conditioned operator, i.e, D 0 uu ′ represents D 0 isconditioned acting on the basis state coupled with state | 0 ⟩ → | u ⟩| u ′ ⟩ .
Revised: and we denote this quantum operation by D 0 ,and we add a subscript to denote the corresponding conditioned operator, i.e, D 0 uu ′ represents D 0 isconditioned acting on the basis state coupled with state | 0 ⟩ → | u ⟩| u ′ ⟩ .

Original: We can perform the D 0 uu ′ in superposition such that the state
Revised: We can perform the D 0 uu ′ in superposition such that the state

Original: [Equation]
Revised: [Equation]

Original: can be generated in time O (log( nd )) .
Revised: can be generated in time O (log( nd )) .

Original: Quantum aggregation transformation.
Revised: Quantum aggregation transformation.

Original: Recall that an affine transformation (refer to Eq.
Revised: Recall that an affine transformation (refer to Eq.

Original: 5 and
Revised: 5 and

Original: Eq. 15) acting on the GNTK and its covariance is relative to the neighborhood aggregation defined by
Revised: Eq. 15) acting on the GNTK and its covariance is relative to the neighborhood aggregation defined by

Original: Eq. 1.
Revised: Eq. 1.

Original: Therefore, it is indirect to realize an end-to-end speedup similar to the estimation of the innerproduct since the transformation of each element of NTK and the covariance is not independent.
Revised: Therefore, it is indirect to realize an end-to-end speedup similar to the estimation of the innerproduct since the transformation of each element of NTK and the covariance is not independent.

Original: Tocircumvent this barrier, we derive a unitary quantum aggregation transformation to approximatelyreconstruct the affine transformation.
Revised: Tocircumvent this barrier, we derive a unitary quantum aggregation transformation to approximatelyreconstruct the affine transformation.

Original: Consider the quantum operation D 0 uu ′ : | u ⟩| u ′ ⟩| 

Revised: Consider the quantum operation D 0 uu ′ : | u ⟩| u ′ ⟩|

Original: ⟩| 0 ⟩ →
Revised: ⟩| 0 ⟩ →

Original: [Equation]
Revised: [Equation]

Revised: u

Revised: [Equation]

Original: uu
Revised: uu

Original: [Equation]
Revised: [Equation]

Original: g uu
Revised: g uu

Original: [Equation]
Revised: [Equation]

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: P uu
Revised: P uu

Original: [Equation]
Revised: [Equation]

Original: | 1 , g [Equation] uu
Revised: | 1 , g ′ uu

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: that is employed to estimate the inner product oftwo feature vectors.
Revised: that is employed to estimate the inner product oftwo feature vectors.

Original: Define a unitary operator which is used to perform aggregation transformation
Revised: Define a unitary operator which is used to perform aggregation transformation

Revised: U

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: which can be generated by introducing conditional quantum evolution [22].
Revised: which can be generated by introducing conditional quantum evolution [21].

Original: The operation ⊗ denotesthe tensor product.
Revised: The operation ⊗ denotesthe tensor product.

Original: We apply the U with Hadamard gates to the given initial state, which is given as
Revised: We apply the U with Hadamard gates to the given initial state, which is given as

Original: [Equation]
Revised: [Equation]

Original: where √· | other ⟩ represents other computational basis states except for | 0 ⟩ ⊗ with amplitude √·
Revised: where √· | other ⟩ represents other computational basis states except for | 0 ⟩ ⊗ with amplitude √·

Original: [Equation]
Revised: [Equation]

Original: andthe detailed mathematical expression and the scalar for state normalization are omitted since theresult of the affine transformation has been embedded into the amplitude of | 0 ⟩ ⊗ .
Revised: andthe detailed mathematical expression and the scalar for state normalization are omitted since theresult of the affine transformation has been embedded into the amplitude of | 0 ⟩ ⊗ .

Original: The ( · ) ⊗ denotesthat there could be multiple unitary operations acting on multiple registers, depending on the numberof qubits required to encode the classic data.
Revised: The ( · ) ⊗ denotesthat there could be multiple unitary operations acting on multiple registers, depending on the numberof qubits required to encode the classic data.

Original: Similar to the inner product estimation, the quantumaggregation transformation can be performed in superposition and the resulting superposition is
Revised: Similar to the inner product estimation, the quantumaggregation transformation can be performed in superposition and the resulting superposition is

Original: [Equation]
Revised: [Equation]

Original: The amplitude √ A uu ′ can be encoded into an ancillary register by using Amplitude Estimation (Theorem 3) and Median Evaluation (Theorem 4).
Revised: The amplitude √ A uu ′ can be encoded into an ancillary register by using Amplitude Estimation (Theorem 3) and Median Evaluation (Theorem 4).

Original: The obtained quantum state
Revised: The obtained quantum state

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Table]
Revised: [Table]

Original: A uu ′ ⟩| y uu ′ ⟩ whose third register carries the approximate result afteraggregation transformation as described in Eq.
Revised: A uu ′ ⟩| y uu ′ ⟩ whose third register carries the approximate result afteraggregation transformation as described in Eq.

Original: 5 and Eq. 15, where | A uu ′ − ¯
Revised: 5 and Eq. 15, where | A uu ′ − ¯

Original: A uu ′ | ≤ ϵ and | y uu ′ ⟩ is agarbage state.
Revised: A uu ′ | ≤ ϵ and | y uu ′ ⟩ is agarbage state.

Original: The runtime is O (log( nd ) log (1 / ∆) /ϵ ) and ∆ is the proximity defined by the Median
Revised: The runtime is O (log( nd ) log (1 / ∆) /ϵ ) and ∆ is the proximity defined by the Median

Original: Evaluation.
Revised: Evaluation.

Original: Note that ¯ A uu ′ is actually the polynomial combination of the element-wise square rootof the NTK from the previous layer, thus it is an approximate aggregation transformation.
Revised: Note that ¯ A uu ′ is actually the polynomial combination of the element-wise square rootof the NTK from the previous layer, thus it is an approximate aggregation transformation.

Original: In theexperiment, we empirically show that this approximation has a restrictive effect on the performance.
Revised: In theexperiment, we empirically show that this approximation has a restrictive effect on the performance.

Original: Quantum kernel estimation For fully-connected neural network, the calculation of each element of the NTK and its covariance is only reliant on the element at the same position of thecovariance matrix in the previous fully-connected layer.
Revised: Quantum kernel estimation For fully-connected neural network, the calculation of each element of the NTK and its covariance is only reliant on the element at the same position of thecovariance matrix in the previous fully-connected layer.

Original: Besides, the affine transformation of the
Revised: Besides, the affine transformation of the

Original: GNTK and its covariance can be efficiently approximated by quantum aggregation transformation and the result has been embedded into the basis states of a superposition.
Revised: GNTK and its covariance can be efficiently approximated by quantum aggregation transformation and the result has been embedded into the basis states of a superposition.

Original: In general, thereexits a unitary V : (cid:80) x | x, 0 ⟩ → (cid:80) x | x, f ( x ) ⟩ for any classical function f with the same timecomplexity to evaluate each element of the NTK and each element of the covariance [48, 71].
Revised: In general, thereexits a unitary V : (cid:80) x | x, 0 ⟩ → (cid:80) x | x, f ( x ) ⟩ for any classical function f with the same timecomplexity to evaluate each element of the NTK and each element of the covariance [45, 66].

Original: Specifically, an oracle which operates as the same as classical function defined by Eq.
Revised: Specifically, an oracle which operates as the same as classical function defined by Eq.

Original: 8 is implemented on the third register of
Revised: 8 is implemented on the third register of

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

Original: [Equation]
Revised: [Equation]

Original: | ¯
Revised: ¯

Original: [Equation]
Revised: [Equation]

Original: uu
Revised: uu

Original: [Equation]
Revised: [Equation]

Original: ⟩| y uu
Revised: ⟩| y uu

Original: [Equation]
Revised: [Equation]

Original: The resulting NTK is
Revised: The resulting NTK is

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

Original: [Equation]
Revised: [Equation]

Revised: u

Revised: [Equation]

Original: | ¯ Θ
Revised: | ¯ Θ

Original: [Equation]
Revised: [Equation]

Original: ⟩| y 
Revised: ⟩| y uu

Original: [Equation]
Revised: [Equation]

Original: ⟩ , where ¯ Θ uu
Revised: ⟩ , where ¯ Θ uu

Original: [Equation]
Revised: [Equation]

Original: is the approximate estimation of its classicalcounterpart after R fully-connected layers.
Revised: is the approximate estimation of its classicalcounterpart after R fully-connected layers.

Original: The oracle is expected to be with the same complexityof its classical counterpart, which is associative to the number of fully-connected layers and isindependent on the number of training samples n .
Revised: The oracle is expected to be with the same complexityof its classical counterpart, which is associative to the number of fully-connected layers and isindependent on the number of training samples n .

Original: For estimation of the GNTK after a transformerlayer (Eq. 14), the covariance Σ lR ( G, G ) for any G ∈ G requires to be estimated in advance.
Revised: For estimation of the GNTK after a transformerlayer (Eq. 14), the covariance Σ lR ( G, G ) for any G ∈ G requires to be estimated in advance.

Original: Itmeans that the state 1 n (cid:80) u ∈ V (cid:80) u ′ ∈ V | u ⟩| u ′ ⟩| ¯ Σ uu ′ ⟩| y uu ′ ⟩ must be estimated for any G ( V, E ) ∈
Revised: Itmeans that the state 1 n (cid:80) u ∈ V (cid:80) u ′ ∈ V | u ⟩| u ′ ⟩| ¯ Σ uu ′ ⟩| y uu ′ ⟩ must be estimated for any G ( V, E ) ∈

Original: Gbefore input the different graphs, and we only consider the element when u = u ′ .
Revised: Gbefore input the different graphs, and we only consider the element when u = u ′ .

Original: By taking thepartial trace on the second register, we obtain the state 1 √ n (cid:80) u ∈ V | u
Revised: By taking thepartial trace on the second register, we obtain the state 1 √ n (cid:80) u ∈ V | u

Original: ⟩| ¯ Σ uu ⟩| y uu ⟩ for graph G and1 √ n ′ (cid:80) u ′ ∈ V ′ | u ′ ⟩| ¯ Σ u ′ u ′ ⟩| y u ′ u
Revised: ⟩| ¯ Σ uu ⟩| y uu ⟩ for graph G and1 √ n ′ (cid:80) u ′ ∈ V ′ | u ′ ⟩| ¯ Σ u ′ u ′ ⟩| y u ′ u

Original: ′ ⟩ for graph G ′ .
Revised: ′ ⟩ for graph G ′ .

Original: Thus, estimation of the GNTK and its covariancecorresponding to the multiplication part in Eq.
Revised: Thus, estimation of the GNTK and its covariancecorresponding to the multiplication part in Eq.

Original: 14 is given as
Revised: 14 is given as

Original: [Equation]
Revised: [Equation]

Original: This is performed by using the conditional quantum adder and the multiplier conditioned on theindex register, i.e. | u ⟩ and | u ′ ⟩ , which are designed by [63, 54, 41].
Revised: This is performed by using the conditional quantum adder and the multiplier conditioned on theindex register, i.e. | u ⟩ and | u ′ ⟩ , which are designed by [59, 51, 39].

Original: The final GNTK after thetransformer layer can be directly generated by additional quantum arithmetic operations that performan element-wise addition between the covariance to the GNTK.
Revised: The final GNTK after thetransformer layer can be directly generated by additional quantum arithmetic operations that performan element-wise addition between the covariance to the GNTK.

Original: Estimation the GNTK for multiple layers
Revised: Estimation the GNTK for multiple layers

Original: The quantum aggregation transformation requires that theapproximate NTK and its covariance are embedded into the amplitudes of a superposition.
Revised: The quantum aggregation transformation requires that theapproximate NTK and its covariance are embedded into the amplitudes of a superposition.

Original: However,after the quantum kernel estimation, these matrix are embedded into the quantum basis states of asuperposition.
Revised: However,after the quantum kernel estimation, these matrix are embedded into the quantum basis states of asuperposition.

Original: To extract them back to the amplitudes, we apply Conditional Rotation [37] on theregister containing the approximate GNTK (and the covariance), which is given by
Revised: To extract them back to the amplitudes, we apply Conditional Rotation [35] on theregister containing the approximate GNTK (and the covariance), which is given by

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where a uu ′ =
Revised: where a uu ′ =

Original: (cid:114) ¯ Θ uu ′ max uu ′ ( ¯ Θ uu ′ ) and b uu
Revised: (cid:114) ¯ Θ uu ′ max uu ′ ( ¯ Θ uu ′ ) and b uu

Original: ′ =
Revised: ′ =

Original: (cid:114)
Revised: (cid:114)

Original: [Equation]
Revised: [Equation]

Original: ′ ( ¯ Σ uu ′ ) .
Revised: ′ ( ¯ Σ uu ′ ) .

Original: We denote this quantum operation as
Revised: We denote this quantum operation as

Original: D l , l ∈ { 1 , . . .
Revised: D l , l ∈ { 1 , . . .

Original: , L } , where D L is used for the quantum readout operation.
Revised: , L } , where D L is used for the quantum readout operation.

Original: Similar to the operation
Revised: Similar to the operation

Original: D 0 , the quantum aggregation transformation can be performed by generating a unitary operator byintroducing conditional quantum evolution.
Revised: D 0 , the quantum aggregation transformation can be performed by generating a unitary operator byintroducing conditional quantum evolution.

Original: Notice that a uu ′ and b uu ′ can be viewed as √ P uu ′ in thesetting of the single-layer GraphQNTK.
Revised: Notice that a uu ′ and b uu ′ can be viewed as √ P uu ′ in thesetting of the single-layer GraphQNTK.

Original: Quantum readout The resulting NTK is embedded into the basis states of a superposition sincethe algorithm ends up in the fully-connected layers.
Revised: Quantum readout The resulting NTK is embedded into the basis states of a superposition sincethe algorithm ends up in the fully-connected layers.

Original: Similar to the classic readout operation, thesummation of all the elements of the NTK matrix at the L -th layer is required.
Revised: Similar to the classic readout operation, thesummation of all the elements of the NTK matrix at the L -th layer is required.

Original: We use Conditional
Revised: We use Conditional

Original: Rotation to extract the NTK back to the amplitude, and define a unitary O which is a generalization of the unitary U , where
Revised: Rotation to extract the NTK back to the amplitude, and define a unitary O which is a generalization of the unitary U , where

Original: O = (cid:88) v ∈ V (cid:88) v ′ ∈ V | v ⟩ (cid:12)(cid:12) v ′ (cid:11) ⟨ v | (cid:10) v ′ (cid:12)(cid:12) ⊗ D Lvv ′ .
Revised: O = (cid:88) v ∈ V (cid:88) v ′ ∈ V | v ⟩ (cid:12)(cid:12) v ′ (cid:11) ⟨ v | (cid:10) v ′ (cid:12)(cid:12) ⊗ D Lvv ′ .

Original: The unitary O sums the square root of all the elements of the GraphQNTK matrix.
Revised: The unitary O sums the square root of all the elements of the GraphQNTK matrix.

Original: And theresulting GraphQNTK between two input graphs is ¯ Θ GG ′
Revised: And theresulting GraphQNTK between two input graphs is ¯ Θ GG ′

Original: =
Revised: =

Original: ( (cid:80) u ∈ V,u ′∈ V ′ √ ¯ Θ uu ′ )
Revised: ( (cid:80) u ∈ V,u ′∈ V ′ √ ¯ Θ uu ′ )

Original: n × n ′ , where ¯ Θ uu ′ isthe GraphQNTK of the last layer.
Revised: n × n ′ , where ¯ Θ uu ′ isthe GraphQNTK of the last layer.

Original: Quantum inference to unseen data We assume that the test data and the label of the training setare already encoded into the QRAM such that | k ∗ ⟩ ∈ R N , the GraphQNTK between the test graph
Revised: Quantum inference to unseen data We assume that the test data and the label of the training setare already encoded into the QRAM such that | k ∗ ⟩ ∈ R N , the GraphQNTK between the test graph

Original: G ∗ , can be evaluated as the same way to the evaluation between the training data.
Revised: G ∗ , can be evaluated as the same way to the evaluation between the training data.

Original: Let ¯ Θ ∈ R N ×
Revised: Let ¯ Θ ∈ R N ×

Original: Ndenote the GraphQNTK.
Revised: Ndenote the GraphQNTK.

Original: The final prediction for a test datapoint G ∗ is
Revised: The final prediction for a test datapoint G ∗ is

Original: [Equation]
Revised: [Equation]

Original: which requires solving the linear equation | E ⟩ = ¯ Θ − 1 | y ⟩ and performing inner product estimation on ⟨ k ∗ | E ⟩ .
Revised: which requires solving the linear equation | E ⟩ = ¯ Θ − 1 | y ⟩ and performing inner product estimation on ⟨ k ∗ | E ⟩ .

Original: A popular quantum algorithm which is designed to solve the quantumlinear systems problem (QLSP) is developed by [13], and its runtime is O (log( N ) κs polylog ( κs/ϵ )) where s is the sparsity of matrix ¯ Θ and κ is the condition number.
Revised: A popular quantum algorithm which is designed to solve the quantumlinear systems problem (QLSP) is developed by [13], and its runtime is O (log( N ) κs polylog ( κs/ϵ )) where s is the sparsity of matrix ¯ Θ and κ is the condition number.

Original: To realizethe quantum speedup, we assume a specific sparsity pattern is created in the quantum storage that only keeps O (log N ) number of non-zero elements of the N × N GraphQNTK matrix and the well-conditioning is achieved by using Gershgorin circle theorem similar to [71].
Revised: To realizethe quantum speedup, we assume a specific sparsity pattern is created in the quantum storage that only keeps O (log N ) number of non-zero elements of the N × N GraphQNTK matrix and the well-conditioning is achieved by using Gershgorin circle theorem similar to [66].

Original: mechanism without both sparsity and well conditioning, while the
Revised: mechanism without both sparsity and well conditioning, while the

Original: GraphQNTK is the kernel after performing these two transformations to meet the conditions for the use of quantum matrix inversion.
Revised: GraphQNTK is the kernel after performing these two transformations to meet the conditions for the use of quantum matrix inversion.

Original: The results of other models are taken from [17] except QS-CNN, which we evaluate on our dataset separation 
Revised: The results of other models are taken from [17] except QS-CNN, which we evaluate on our dataset separation [Table]

Revised: 2.4 Complexity Study

Original: [Table]
Revised: [Table]

Original: In Sec. 2.3, we discuss howto approximately estimate
Revised: In Sec. 2.3, we discuss howto approximately estimate

Original: GNTK using quantum computing paradigm betweentwo input graphs.
Revised: GNTK using quantum computing paradigm betweentwo input graphs.

Original: The timecomplexity is dominated bythe quantum aggregationtransformation procedureas
Revised: The timecomplexity is dominated bythe quantum aggregationtransformation procedureas

Original: it requires encoding theamplitude into an additional register, which takes
Revised: it requires encoding theamplitude into an additional register, which takes

Original: O (log( nd ) log (1 / ∆) /ϵ )time.
Revised: O (log( nd ) log (1 / ∆) /ϵ )time.

Original: Other quantumoperations including estimation of the inner product,estimation of the GNTKwithin the neighborhoodaggregation and the fully-connected feature updating and quantum readout are totally unitaryoperations which can be efficiently performed under the regime of quantum computing.
Revised: Other quantumoperations including estimation of the inner product,estimation of the GNTKwithin the neighborhoodaggregation and the fully-connected feature updating and quantum readout are totally unitaryoperations which can be efficiently performed under the regime of quantum computing.

Original: Forestimating GNTK of each pairs of the graphs ( G, G ′ ) where G, G ′ ∈ G , each element of GraphQNTK¯ Θ can be generated simultaneously by introducing auxiliary index registers.
Revised: Forestimating GNTK of each pairs of the graphs ( G, G ′ ) where G, G ′ ∈ G , each element of GraphQNTK¯ Θ can be generated simultaneously by introducing auxiliary index registers.

Original: The quantum runtime is
Revised: The quantum runtime is

Original: O (log( Nnd )) .
Revised: O (log( Nnd )) .

Original: However, evaluating GNTK of the infinite-width-limit attention requires computingthe kernel where the input is two same graphs, which can be implemented in time O ( N ) .
Revised: However, evaluating GNTK of the infinite-width-limit attention requires computingthe kernel where the input is two same graphs, which can be implemented in time O ( N ) .

Original: The resultshould be stored in QRAM in advance which will be used to update GNTK corresponding themulti-head attention as described in Eq.
Revised: The resultshould be stored in QRAM in advance which will be used to update GNTK corresponding themulti-head attention as described in Eq.

Original: Therefore, it takes O ( N log( Nnd )) time to train theproposed quantum graph learning model, which achieves quadratic speedup compared to the existing
Revised: Therefore, it takes O ( N log( Nnd )) time to train theproposed quantum graph learning model, which achieves quadratic speedup compared to the existing

Original: GKs and completed approaches with O ( N 2 ) time.
Revised: GKs and completed approaches with O ( N 2 ) time.

Original:  Experiments
Revised:  Experiments

Original: We evaluate our method for both GNTK and GraphQNTK with attention mechanism on severalgraph classification datasets involving either discrete or continuous attributes.
Revised: We evaluate our method for both GNTK and GraphQNTK with attention mechanism on severalgraph classification datasets involving either discrete or continuous attributes.

Original: All the experimentsare performed on a workstation with a single machine with 1TB memory, one physical CPU with 28cores Intel(R) Xeon(R) W-3175X CPU @
Revised: All the experimentsare performed on a workstation with a single machine with 1TB memory, one physical CPU with 28cores Intel(R) Xeon(R) W-3175X CPU @

Original: 3.10GHz, and a single GPU (Nvidia Quadro RTX 8000).
Revised: 3.10GHz, and a single GPU (Nvidia Quadro RTX 8000).

Original: For our method and all the compared models, We follow the same setting as [17, 67], and report theaverage test accuracy and its standard deviation over a 10-fold cross validation on each dataset.
Revised: For our method and all the compared models, We follow the same setting as [17, 62], and report theaverage test accuracy and its standard deviation over a 10-fold cross validation on each dataset.

Original: 3.1 Experiments setup
Revised: 3.1 Experiments setup

Original: Datasets.
Revised: Datasets.

Original: For graph with discrete attributes, the benchmark datasets include four bioinformaticsdatasets MUTAG, PTC, NCI1, PROTEINS and three social network datasets IMDB-BINARY, IMDBMULTI.
Revised: For graph with discrete attributes, the benchmark datasets include four bioinformaticsdatasets MUTAG, PTC, NCI1, PROTEINS and three social network datasets IMDB-BINARY, IMDBMULTI.

Original: For each graph, the input attributes is category of the node and they are transformed toone-hot encoding representations.
Revised: For each graph, the input attributes is category of the node and they are transformed toone-hot encoding representations.

Original: For datasets where the graphs have no node features, i.e. onlygraph structure matters, we use degrees as input node features.
Revised: For datasets where the graphs have no node features, i.e. onlygraph structure matters, we use degrees as input node features.

Original: For graph with continuous attributes,we selcect four benchmark datasets including ENZYMES, PROTEINS full, BZR, COX2.
Revised: For graph with continuous attributes,we selcect four benchmark datasets including ENZYMES, PROTEINS full, BZR, COX2.

Original: All thedatasets can be found in [38].
Revised: All thedatasets can be found in [36].

Original: The statistic information of the datasets are given in Tab. 3 in Appendix.
Revised: The statistic information of the datasets are given in Tab. 3 in Appendix.

Original: Compared baselines.
Revised: Compared baselines.

Original: We compare our method with state-of-the-art GKs such as WL kernel [57],
Revised: We compare our method with state-of-the-art GKs such as WL kernel [54],

Original: AWL [30], RetGK [69], GNTK [17], WWL [62], and GNNs including GCN [39], PatchySAN [49],
Revised: AWL [28], RetGK [64], GNTK [17], WWL [58], and GNNs including GCN [37], PatchySAN [46],

Original: GCKN [10], GraphSAGE [21] and GIN [67].
Revised: GCKN [10], GraphSAGE [20] and GIN [62].

Original: For quantum graph learning, there are very few baselineavailable.
Revised: For quantum graph learning, there are very few baselineavailable.

Original: We report the performance of the quantum walk based subgraph convolutional neuralnetwork (QS-CNN) developed by [70].
Revised: We report the performance of the quantum walk based subgraph convolutional neuralnetwork (QS-CNN) developed by [65].

Original: The data separation we use is the same as [67] for graphdatasets with discrete attributes.
Revised: The data separation we use is the same as [62] for graphdatasets with discrete attributes.

Original: For graph dataset with continuous attributes, we follow the sameprotocol as used in [62] to normalize the input feature vectors for a fair comparison.
Revised: For graph dataset with continuous attributes, we follow the sameprotocol as used in [58] to normalize the input feature vectors for a fair comparison.

Original: The accuracies of other models are taken from [10].
Revised: The accuracies of other models are taken from [10].

Original: We only take the results of GCKN under the supervised learning for a fail comparison.
Revised: We only take the results of GCKN under the supervised learning for a fail comparison.

Original: We utilize the similar settings that preprocess the continuous node features to a normalized feature vector as in [62] for fair comparison (Note that the data encoded into the QRAM requires normalization, thus it is reasonable to use this data-prepossessing operation).
Revised: We utilize the similar settings that preprocess the continuous node features to a normalized feature vector as in [58] for fair comparison (Note that the data encoded into the QRAM requires normalization, thus it is reasonable to use this data-prepossessing operation).

Original: [Table]
Revised: [Table]

Original: Results.
Revised: Results.

Original: We apply different hyper-parameter settings to L ∈ { 2 , 4 , 6 , 8 } and
Revised: We apply dif- [Table] ferent hyper-parameter settings to L ∈ { 2 , 4 , 6 , 8 } and

Original: R ∈ { 1 , 2 , 3 } and selectthe model with the best averaged accuracy.
Revised: R ∈ { 1 , 2 , 3 } and selectthe model with the best averaged accuracy.

Original: We testthe kernel regression using
Revised: We testthe kernel regression using

Original: SVM classifier and the regularization parameter is determined using the searchprotocol which is the sameas the [17].
Revised: SVM classifier and the regularization parameter is determined using the searchprotocol which is the sameas the [17].

Original: We report theperformance of the quantum approximate GNTK before and after the matrixsparsity and conditioningoperations.
Revised: We report theperformance of the quantum approximate GNTK before and after the matrixsparsity and conditioningoperations.

Original: The numericalresults are listed in Tab.
Revised: The numericalresults are listed in Tab.

Original: 2.3for datasets with discrete attributes and Tab. 3.1 for datasets with continuous attributes.
Revised: 2.3for datasets with discrete attributes and Tab. 3.1 for datasets with continuous attributes.

Original: The attentionmethod we integrate to the infinite-width GNNs brings to an improvement in the performance ofthe model.
Revised: The attentionmethod we integrate to the infinite-width GNNs brings to an improvement in the performance ofthe model.

Original: The results show that the GNTK with attention mechanism achieves better classificationaccuracy for graph data with medium number of nodes and edges.
Revised: The results show that the GNTK with attention mechanism achieves better classificationaccuracy for graph data with medium number of nodes and edges.

Original: It is demonstrated that the infinitewidth-limit attention captures global node similarity semantics and learns effective structure of theprovided graph, which brings an remarkable accuracy improvement of the model compared withthe vanilla GNTK [17].
Revised: It is demonstrated that the infinitewidth-limit attention captures global node similarity semantics and learns effective structure of theprovided graph, which brings an remarkable accuracy improvement of the model compared withthe vanilla GNTK [17].

Original: Moreover, our model performs better than QS-CNN on more than 60% ofthe datasets with discrete attributes, given the caveat that QS-CNN is a hybrid graph learning modelwhere the contribution of the classic components (CNNs, spatial message passing) in their modelcannot be ignored.
Revised: Moreover, our model performs better than QS-CNN on more than 60% ofthe datasets with discrete attributes, given the caveat that QS-CNN is a hybrid graph learning modelwhere the contribution of the classic components (CNNs, spatial message passing) in their modelcannot be ignored.

Original: While the matrix sparsity and conditioning operations have a great influence onthe model’s performance, it can be found that the classification performance of GNTK evaluated byquantum algorithms is still comparable with that of GKs and vanilla GNNs, where a tradeoff existsbetween the performance of the model and the quantum computational efficiency.
Revised: While the matrix sparsity and conditioning operations have a great influence onthe model’s performance, it can be found that the classification performance of GNTK evaluated byquantum algorithms is still comparable with that of GKs and vanilla GNNs, where a tradeoff existsbetween the performance of the model and the quantum computational efficiency.

Original:  Conclusion and Broader Impact
Revised:  Conclusion and Broader Impact

Original: This paper has presented a quantum graph learning model to characterize the structural informationwhile using quantum parallelism to improve computing efficiency.
Revised: This paper has presented a quantum graph learning model to characterize the structural informationwhile using quantum parallelism to improve computing efficiency.

Original: We propose quantum algorithm toapproximately estimate the neural tangent kernel of the underlying graph neural network where amulti-head quantum attention mechanism is introduced to incorporate semantic similarity of nodes.
Revised: We propose quantum algorithm toapproximately estimate the neural tangent kernel of the underlying graph neural network where amulti-head quantum attention mechanism is introduced to incorporate semantic similarity of nodes.

Original: Empirical results on graph classification tasks as well as theoretical analysis show the superiority ofour method.
Revised: Empirical results on graph classification tasks as well as theoretical analysis show the superiority ofour method.

Original: The limitation of the paper is that currently it only addresses graph-level embedding andwe leave node-level quantum learning for future work.
Revised: The limitation of the paper is that currently it only addresses graph-level embedding andwe leave node-level quantum learning for future work.

Original: Our work may raise concerns for encryption,privacy protection etc.
Revised: Our work may raise concerns for encryption,privacy protection etc.

Original: when the quantum hardware become more feasible.
Revised: when the quantum hardware become more feasible.

Original: Checklist
Revised: Checklist

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Revised: [Table]

Original: A Related Works
Revised: A Related Works

Original: In this section, we provide background on quantum graph learning and graph neural networks thathave the potential to be trained using quantum computing.
Revised: In this section, we provide background on quantum graph learning and graph neural networks thathave the potential to be trained using quantum computing.

Original: A.1 Graph Kernel Neural Network
Revised: A.1 Graph Kernel Neural Network

Original: Graph kernel neural networks [17, 10, 18] is a class of graph learning method combining the propertiesof both GNNs and GKs.
Revised: Graph kernel neural networks [17, 10, 18] is a class of graph learning method combining the propertiesof both GNNs and GKs.

Original: Forward process of the model tends to transmit node information like GNNs[66], layer by layer, whereas the node (or graph) features live in the implicit reproducing kernel
Revised: Forward process of the model tends to transmit node information like GNNs[61], layer by layer, whereas the node (or graph) features live in the implicit reproducing kernel

Original: Hilbert space (RKHS) of a specific kernel [40].
Revised: Hilbert space (RKHS) of a specific kernel [38].

Original: The mainstream of graph kernel neural networkscan be divided into completed and approximated approaches.
Revised: The mainstream of graph kernel neural networkscan be divided into completed and approximated approaches.

Original: For the former, the output is a kernelmatrix where each entry denotes the similarity of graph pairs, afterwards support vector machin isused to perform classification or regression task.
Revised: For the former, the output is a kernelmatrix where each entry denotes the similarity of graph pairs, afterwards support vector machin isused to perform classification or regression task.

Original: While the later generates the approximate feature ofthe finite projected RKHS at the expense of information loss.
Revised: While the later generates the approximate feature ofthe finite projected RKHS at the expense of information loss.

Original: We consider the completed approaches for the basis of our proposed quantum graph learning model,since the access to the explicit feature information requires measuring the relative quantum representations, which incurs quantum collapse [33].
Revised: We consider the completed approaches for the basis of our proposed quantum graph learning model,since the access to the explicit feature information requires measuring the relative quantum representations, which incurs quantum collapse [31].

Original: In the next section, we demonstrate that the graphtangent kernel neural networks coincides with the condition of quantum parallel implementation byintroducing the quantum aggregation transformation and the quantum kernel estimation techniques.
Revised: In the next section, we demonstrate that the graphtangent kernel neural networks coincides with the condition of quantum parallel implementation byintroducing the quantum aggregation transformation and the quantum kernel estimation techniques.

Original: A.2 Quantum Graph Learning
Revised: A.2 Quantum Graph Learning

Original: Quantum graph learning aims at leveraging quantum physics to extract graph structural information,bringing up new possibilities for quantum computing applications.
Revised: Quantum graph learning aims at leveraging quantum physics to extract graph structural information,bringing up new possibilities for quantum computing applications.

Original: It is generally nontrivial toanalyze classical data under the regime of quantum computing, since the encoding and decodingbetween classical vectors (or matrices) and their corresponding quantum states should be carefullydesigned.
Revised: It is generally nontrivial toanalyze classical data under the regime of quantum computing, since the encoding and decodingbetween classical vectors (or matrices) and their corresponding quantum states should be carefullydesigned.

Original: In addition, encoding the irregular graph data and diverse structure topology may incurdifferent configurations of quantum models.
Revised: In addition, encoding the irregular graph data and diverse structure topology may incurdifferent configurations of quantum models.

Original: Advanced contributions has developed some techniquesto overcome these issues.
Revised: Advanced contributions has developed some techniquesto overcome these issues.

Original: A hybrid graph learning method developed by [70, 16] encode the structureinformation and generate a new adjacent matrix evaluating by the using quantum walk.
Revised: A hybrid graph learning method developed by [65, 16] encode the structureinformation and generate a new adjacent matrix evaluating by the using quantum walk.

Original: The resultingadjacent information captures the global topological arrangement information for graph substructures.
Revised: The resultingadjacent information captures the global topological arrangement information for graph substructures.

Original: Adiabatic evolution [64] and conditional unitary [ ? ] are applied to evolve the quantum systemsdependent on the underlying graph structure.
Revised: Adiabatic evolution [60] and conditional unitary [43] are applied to evolve the quantum systemsdependent on the underlying graph structure.

Original: In addition, the node attribute is encoded usingvariational circuit [1] or a quantum random access memory [58].
Revised: In addition, the node attribute is encoded usingvariational circuit [1] or a quantum random access memory [55].

Original: Processing the encoded quantumrepresentation of the original graph can be realized via either a naive quantum algorithm [64] or ahybrid method [11].
Revised: Processing the encoded quantumrepresentation of the original graph can be realized via either a naive quantum algorithm [60] or ahybrid method [11].

Original: Then a post-processing operation is performed to further analyze the quantumoutput.
Revised: Then a post-processing operation is performed to further analyze the quantumoutput.

Original: A brief review about quantum graph learning is illustrated in Fig.
Revised: A brief review about quantum graph learning is illustrated in Fig.

Original: Generally speaking, theresearchers exploit to encode the graph structure and node features in the quantum system throughvarious schemes, and then process the information through quantum layers and auxiliary classicallayers.

Original: Finally, the (quantum) results are decoded through post-processing.

Original: However, most quantumgraph learning models requires that adjustable parameters in the quantum algorithm need to beupdated frequently, where takes great computational overheads.
Revised: However, most quantumgraph learning models requires that adjustable parameters in the quantum algorithm need to beupdated frequently, where takes great computational overheads.

Original: Moreover, the classical componentsin post-processing may dominate the performance of the model, thus weakening the role of the
Revised: Moreover, the classical componentsin post-processing may dominate the performance of the model, thus weakening the role of thequantum part.

Original: [Table] quantum part. In this paper, We seek to establish a parameter-free quantum graph learning model tomaximize the efficacy of quantum computing.
Revised: In this paper, We seek to establish a parameter-free quantum graph learning model tomaximize the efficacy of quantum computing.

Original: We notice that there are researches which are abbreviated as QNTK [59, 42], similar to ours nominally.

Original: But their definition is quite different from ours.

Original: The motivation of these two papers is to analyze thetrainability and expressive power of variational quantum circuits through NTK.

Original: In contrast, in ourwork, QNTK is a metric measuring the similarity of two input graphs.

Original: In this context, NTK is thekernel that captures the dynamics of infinite-width GNNs, as well as the multi-head attention wherethe number of heads and the dimension of output go to infinity.

Original: B More AnalysisB.1 Quantum Access Memory
Revised: B More AnalysisB.1 Quantum Access Memory

Original: Theorem 1 Let | X p ⟩ = 1 ∥
Revised: Theorem 1 Let | X p ⟩ = 1 ∥

Original: X p ∥ (cid:80) d − 1 q =

Revised: X p ∥ (cid:80) d − 1 q =

Original: X
Revised: X

Original: pq | j ⟩ denotes the amplitude encoding of the p -th row of data
Revised: pq | j ⟩ denotes the amplitude encoding of the p -th row of data

Original: X ∈ R n × d .
Revised: X ∈ R n × d .

Original: There exists a data structure to store the entries of X into the QRAM which is stated asi) | p ⟩ | 0 ⟩ → | p ⟩ | X p ⟩ii) | 0 ⟩
Revised: There exists a data structure to store the entries of X into the QRAM which is stated as

Original: → 1 ∥ X ∥ F (cid:80) p

Original: ∥ X p ∥ | p ⟩in time T for p ∈ [ n ] .
Revised: [Equation] in time T for p ∈ [ n ] .

Original: Using the binary tree QRAM architecture proposed by [35], the time T to storeand readout a new element scale logarithmically with respect to both n and d .
Revised: Using the binary tree QRAM architecture proposed by [33], the time T to storeand readout a new element scale logarithmically with respect to both n and d .

Original: B.2 Inner Product Estimation
Revised: B.2 Inner Product Estimation

Original: Theorem 2 There exists a quantum operation
Revised: Theorem 2 There exists a quantum operation

Original: A that evaluates the inner product of two quantumrepresentations with respect to their d -dimensional classical vectors in time
Revised: A that evaluates the inner product of two quantumrepresentations with respect to their d -dimensional classical vectors in time

Original: O (log d ) .
Revised: O (log d ) .

Original: Proof.
Revised: Proof.

Original: By introducing an auxiliary register, with the initial state | p ⟩| q ⟩ 1 √ 2 ( | 0 ⟩ + | 1 ⟩ ) | 0 ⟩ , the map1 √ 2 ( | p ⟩|
Revised: By introducing an auxiliary register, with the initial state | p ⟩| q ⟩ 1 √ 2 ( | 0 ⟩ + | 1 ⟩ ) | 0 ⟩ , the map1 √ 2 ( | p ⟩|

Original: q ⟩| 0 ⟩| 0 ⟩ + | p ⟩| q ⟩| 1 ⟩| 0 ⟩ ) → 1 √ 2 ( | p ⟩| q ⟩| 0 ⟩| | X p ⟩⟩ + | p ⟩| q ⟩| 1 ⟩| | X q ⟩⟩ ) can be performed in
Revised: q ⟩| 0 ⟩| 0 ⟩ + | p ⟩| q ⟩| 1 ⟩| 0 ⟩ ) → 1 √ 2 ( | p ⟩| q ⟩| 0 ⟩| | X p ⟩⟩ + | p ⟩| q ⟩| 1 ⟩| | X q ⟩⟩ ) can be performed in

Original: O (log d ) for two quantum representations | X p ⟩ and | X q ⟩ with respect to their classical vectors
Revised: O (log d ) for two quantum representations | X p ⟩ and | X q ⟩ with respect to their classical vectors

Original: X p ∈ R d and X q ∈ R d .
Revised: X p ∈ R d and X q ∈ R d .

Original: Applying a Hadamard gate on the third register, the state becomes
Revised: Applying a Hadamard gate on the third register, the state becomes

Original: [Equation]
Revised: [Equation]

Original: The probability of measuring 0 on the third register is given by P pq = 1+ ⟨ X p | X q ⟩.
Revised: The probability of measuring 0 on the third register is given by P pq = 1+ ⟨ X p | X q ⟩.

Original: Thus the statedefined by Eq.
Revised: Thus the statedefined by Eq.

Original: 24 can be reformulated as | p ⟩| q ⟩ (cid:0)(cid:112) P pq | 0 , g pq ⟩ + (cid:112) 1 − P pq (cid:12)(cid:12) 1 , g ′ pq (cid:11)(cid:1) where | g pq ⟩and (cid:12)(cid:12) g ′ pq (cid:11) are garbage states.
Revised: 24 can be reformulated as | p ⟩| q ⟩ (cid:0)(cid:112) P pq | 0 , g pq ⟩ + (cid:112) 1 − P pq (cid:12)(cid:12) 1 , g ′ pq (cid:11)(cid:1) where | g pq ⟩and (cid:12)(cid:12) g ′ pq (cid:11) are garbage states.

Original: B.3 Amplitude Estimation
Revised: B.3 Amplitude Estimation

Original: Theorem 3 Given a unitary operator U such that U : | 0 ⟩ (cid:55)→ √ p | y ⟩| 0 ⟩ + √ 1 − p | y ′ ⟩ | 1 ⟩ in time T ,where p > 0 is the probability of measuring 0, it is possible to obtain the state | y ⟩| 0 ⟩ using O ( T √ p )queries to U , or to estimate p with relative error δ using O ( Tδ √ p ) queries to U .
Revised: Theorem 3 Given a unitary operator U such that U : | 0 ⟩ (cid:55)→ √ p | y ⟩| 0 ⟩ + √ 1 − p | y ′ ⟩ | 1 ⟩ in time T ,where p > 0 is the probability of measuring 0, it is possible to obtain the state | y ⟩| 0 ⟩ using O ( T √ p )queries to U , or to estimate p with relative error δ using O ( Tδ √ p ) queries to U .

Original: The detailed proofcan be found in [9].
Revised: The detailed proofcan be found in [9].

Original: B.4 Median Evaluation
Revised: B.4 Median Evaluation

Original: Theorem 4 Consider a unitary U : | 0 ⊗ m ⟩ (cid:55)→ √
Revised: Theorem 4 Consider a unitary U : | 0 ⊗ m ⟩ (cid:55)→ √

Original: α | v, 1 ⟩ + √ 1 − α | g, 0 ⟩ for some 1 / 2 ≤ α ≤ 1 intime T .
Revised: α | v, 1 ⟩ + √ 1 − α | g, 0 ⟩ for some 1 / 2 ≤ α ≤ 1 intime T .

Original: There exits a quantum algorithm that, for any ∆ > 0 and for any 1 / 2 < α 0 ≤ α , produce astate | ψ ⟩ such that ∥| ψ ⟩ − (cid:12)(cid:12) 

Revised: There exits a quantum algorithm that, for any ∆ > 0 and for any 1 / 2 < α 0 ≤ α , produce astate | ψ ⟩ such that ∥| ψ ⟩ − (cid:12)(cid:12)

Original: ⊗ mL (cid:11)
Revised: ⊗ mL (cid:11)

Original: | x ⟩∥ ≤ √ 2∆ for some integer L in time[Equation]
Revised: | x ⟩∥ ≤ √ 2∆ for some integer L in time

Original: Refer to [65] for a detailed proof.

Original: C Additional clarification of the proposed modelC.1 Definition of the infinite-width GNN and multi-head attention

Original: Graph neural tangent kernel (GNTK) is a generalization of NTK from infinite-width fully connectedneural network to graph neural network, which is a well-established approach in recent literature andthus we miss some detailed explanation in our paper for space saving (and also due to the complexityof the details which otherwise will cost a lot of space), and the details can be found in [17].

Original: Here wegive an intuitive illustration.

Original: Consider a general GNN with the neighborhood feature aggregation function

Original: [Equation]
Revised: [Equation]

Original: and the central node feature update function ( R fully connected layers)

Original: [Equation]

Original: where h lu denotes the feature vector of node u in the l -th layer, N ( u ) denotes the neighborhood ofnode u , W lr ∈ R d lr ×

Original: d lr +1 is the weight matrix ( r = 1 , . . .

Original: , R ),

Original: c σ is a scaling factor.

Original: The infinitewidth of GNN means that the output dimension d lr of W lr goes to infinity for r = 1 , . . .

Original: , R andl = 1 , . . .

Original: , L .

Original: Apart from the infinite-width GNN, the infinite-width transformer, which is exploited by us to enhance the GNTK, has also been investigated in [26].

Original: Here we briefly explain what an infinite-width transformer looks like.

Original: Consider a single attention layer

Original: [Equation]

Original: in Eq.

Original: 10, where H ∈ R n × d is the node feature matrix

Original: and we ignore the superscript and the subscript for simplicity.

Original: Q h , K h ,

Original: V h ∈ R d × d ′ are weight matrices corresponding to Query, Key and Value, respectively.

Original: s is a scaling factor.

Original: The square matrix G h ∈ R n × n can be viewed as a matrix whose element corresponds to the similarity of each pair of nodes.

Original: Then the operation (cid:98) H h

Original: = ζ (cid:0) G h (cid:1) V h transforms the node features of the last layer to the next layer depending on the node similarity.

Original: For a multi-head attention (transformer) layer, the equation is given as

Original: [Equation]

Original: where W O ∈ R Hd ′ × d ′′ is the weight matrix.

Original: The infinite-width transformer means that the outputdimension d ′ of Q h , K h , V h , the output dimension d ′′ of W O , and the number of heads H go toinfinity.

Original: [Table]

Original: C.2 Our contributions beyond the quantum speedup

Original: Although the current scale of quantum hardware can not support the application of large-scalequantum algorithms, it still can not stop the widespread attention of academia and industry toquantum computing, especially in quantum machine learning [46, 19].

Original: We hope that our model canprovide technical guidance for future research, and better exploit the immense benefits of quantumcomputing even in a classic simulation condition.

Original: Apart from quantum speedup, there are additional contributions of our proposed model, which havebeen illustrated in Line 66 to Line 80 in the introduction section.

Original: And we hope that this novelgraph learning model can bridge the gap between graph neural tangent kernel methods and attentionmechanism.

Original: Here we briefly give an illustration.

Original: Better performance compared with state-of-art graph models.

Original: The numerical results in Tab. 2.3and Tab.

Original: 3.1 demonstrate that our model (AttentionGNTK) outperforms GNTK on 7 out of 10 datasets,reflecting that the introduced infinite-width multi-head attention is useful and can better capturedistinct properties between different graphs.

Original: Furthermore, we compare our model with anotherquantum-inspired graph learner QS-CNN.

Original: Our model surpasses QS-CNN on 4 out of 6 datasets andperforms similarly on the rest 2 datasets.

Original: It shows the superiority of our model in quantum graphlearning.

Original: More robust compared with GNNs when the number of layers becomes larger.

Original: As illustratedin Tab. 5, when the number of layers is larger than 5 (but less than 10), Our model is more resistant tothe oversmooth problem than GIN.

Original: Fewer layers for reaching the peak of classification accuracy compared with GNTK.

Original: Asillustrated in Tab. 6, our model (attentionGNTK) reaches the peak of classification accuracy whenthe number of layers is small, while GNTK needs more layers to reach, indicating that our model iseasier to capture the global structure information of the graph.

Original: D Additional Experiments and DiscussionD.1 Running time comparison

Original: We supplement the training time of our model on four selected datasets and compare it with two othermodels.

Original: To make a fair comparison, we set the layers of all the models to 2.

Original: All the experiments areperformed on a workstation with a single machine with 1TB memory, one physical CPU with 28cores

Original: Intel® Xeon® W-3175X CPU @

Original: 3.10GHz, and a single GPU (Nvidia Quadro RTX 8000).

Original: Theresults are shown in Tab. 4.

Original: Although the speedup introduced by the quantum algorithm depends onthe quantum devices, it shows that our proposed model still has a computational overhead reductionwhen training on classic computers.

Original: The running time is slightly higher than that of GNTK which isa lack of attention mechanism.

Original: It is noticed that our model.

Original: The runtime of our model is apparentlyfaster than that of GIN.

Original: It is worth mentioning that the quadratic quantum speedup will be realized when the quantumhardware becomes more feasible.

Original: D.2 Model sensitivity to the number of layers

Original: In the main body of the paper, we report the best classification accuracy of the model when thethe number of layers L is selected from { 2 , 4 , 6 , 8 } .

Original: We compare the graph classification accuracy

Original: [Table]

Original: [Table]

Original: 82.3 ± 2.

Original: bewteen GIN and our model at the same number of layers and the results are given in Tab. 5 and

Original: Tab. 6.

Original: The number in parentheses in the table indicates the number of layers.

Original: From Tab. 5, it is shown that our model (AttentionGNTK) is more robust compared with GIN whenthe number of layers becomes larger.

Original: The main reason is that an additional feature aggregation, e.g.,the transformer, can slow down the convergence rate, which is consistent with the observations in[29] that connectivity enhancement can help wide and deep GNNs to avoid a discrepancy betweenprediction and the ground truth.

Original: While in Tab. 6, the results empirically demonstrate that our model (attentionGNTK) reaches thepeak of classification accuracy when the number of layers is small, while GNTK needs more layersto reach, indicating that our model is easier to capture the global structure information of the graph.

Original: This could be interpreted from the theoretical perspective.

Original: The transformer (Eq. 12) captured thesemantic information between each pair of (connected and disconnected) nodes with similar features.

Original: Consider G =

Original: QK ⊤ √ s in Eq.

Original: 10, where Q and K are linear transformation of the node feature matrixand we ignore the superscript and the subscript for simplicity.

Original: The G can be viewed as a matrixwhose element corresponds to the similarity of each pair of nodes.

Original: Then the operation (cid:98) H = ζ ( G ) Vtransforms the node features of the last layer to the next layer depending on the node similarity.

Original: Thisenables the model to make better use of the graph structure to transmit information and perceivetopology information over long distances.