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

Original: Abstract
Revised: Abstract

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

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

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

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

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

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

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

Revised: Source code is available at https://github.com/abel1231/graphQNTK .

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-parametric machine learning [52, 36, 43, 32] to classic-quantum hybrid deep leaning [7, 10, 46, 37, 14].
Revised: Quantum-based algorithms have been proposed in recent years, from naive quantum non-parametric machine learning [52, 36, 43, 32] to classic-quantum hybrid deep leaning [7, 10, 46, 37, 14].

Original: Despite that 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: Despite that 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].

Original: Inspired by the two popular classes 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 architecture that captures the structural information of graph data, such as Quantum Graph Neural Networks (QGNNs)
Revised: Inspired by the two popular classes 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 architecture that 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: [64, 7, 11, 16, 1] and Quantum Graph Kernel Methods (QGKs) [56, 3, 25, 4].

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: Some quantum subroutines for attribute encoding [5, 70] and structural encoding [64, 46] have been developed to dissolve the characteristics of the graph into the quantum model.
Revised: Some quantum subroutines for attribute encoding [5, 70] and structural encoding [64, 46] have been developed to dissolve the characteristics of the graph into the quantum model.

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

Original: It is difficult to characterize the structure information and attribute information of the graph by the quantum components without the participation of classic modules.
Revised: It is difficult to characterize the structure information and attribute information of the graph by the quantum components without the participation of classic modules.

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

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

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

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

Original: It is demonstrated that the power of quantum computing could be used to find atypical but useful patterns that classical systems are not considered to be able to generate effectively [14, 24, 28], and accelerate the training process of existing classic models [36, 43, 71].
Revised: It is demonstrated that the power of quantum computing could be used to find atypical but useful patterns that classical systems are not considered to be able to generate effectively [14, 24, 28], and accelerate the training process of existing classic models [36, 43, 71].

Original: Several quantum algorithms [52, 45, 44] based on the HHL algorithm [22] show the exponential speedup compared with their classical counterpart, with a assumption that a quantum random access memory (QRAM) [35] is accessible.
Revised: Several quantum algorithms [52, 45, 44] based on the HHL algorithm [22] show the exponential speedup compared with their classical counterpart, with a assumption that a quantum random access memory (QRAM) [35] is accessible.

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 [71], reconstruct unsupervised clustering [34] and supervised kernel classifier [43].

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

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

Original: [Figure]
Revised: [Figure]

Original: Beyond vanillaGNNs and GKs, composite graph learning studies have emerged that combine the advantages of both areas [50, 10, 18].
Revised: vanilla graph learning studies have emerged that combine the advantages of both areas [50, 10, 18].

Original: In this literature, Graph Neural Tangent Kernel (GNTK) [17] based on neural tangent kernel (NTK) [31] shines its lights on elegantly fusing GNNs and GKs, leading to new prospectives of training and analysing infinite-width
Revised: In this literature, Graph Neural Tangent Kernel (GNTK) [17] based on neural tangent kernel (NTK) [31] shines its lights on elegantly fusing GNNs and GKs, leading to new prospectives of training and analysing infinite-width

Original: GNNs.
Revised: GNNs.

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

Original: Prospectively, the barrier that conventional model is difficult to train and 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 train and 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 logic operations [8], which is profitable from the huge quantum Hilbert space to encode the normalized data.
Revised: Early research involves altering the amplitude of quantum basis states to accomplish a quantum logic operations [8], which is profitable from the huge quantum Hilbert space to encode the normalized data.

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

Original: These strategies could be helpful in the regime of training graph models with either the non-convex nature of the training procedure, or the poor scalability w.r.t.
Revised: These strategies could be helpful in the regime of training graph models with either the non-convex nature of 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 nodes and binary edges.
Revised: In this paper, we focus on quantum machine learning of graph-structured data with attributed nodes and binary edges.

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

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

Original: • Attention-enhanced GNTK.
Revised: • Attention-enhanced GNTK.

Original: Infinite-width GNN is a well established term in the GNN literature [17, 29]
Revised: Infinite-width GNN is a well established term in the GNN literature [17, 29]

Original: and GNTK [17] is a powerful tool to analyze the GNN.
Revised: and GNTK [17] is a powerful tool to analyze the GNN.

Original: However, the traditional feature aggregation of the vanilla GNTK is straightforward, limiting itself within joint neighborhoods.
Revised: However, the traditional feature aggregation of the vanilla GNTK is straightforward, limiting itself within joint neighborhoods.

Original: In this paper, a multi-head attention mechanism is introduced to properly incorporate semantic similarity information of disconnected nodes but with similar features to improve model expressiveness.
Revised: In this paper, a multi-head attention mechanism is introduced to properly incorporate semantic similarity information of disconnected nodes but with similar features to improve model expressiveness.

Original: • Kernel methods for evaluating the dynamics of wide and deep GNNs.
Revised: • Kernel methods for evaluating the dynamics of wide and deep GNNs.

Original: It is generally hard to train a deep GNN with attention, especially when the width of GNN, the width of attention layer, or the number of heads goes to large.
Revised: It is generally hard to train a deep GNN with attention, especially when the width of GNN, the width of attention layer, or the number of heads goes to large.

Original: We properly incorporate infinite-width multi-head attention into GNTK by using NTK theory.
Revised: We properly incorporate infinite-width multi-head attention into GNTK by using NTK theory.

Original: We use kernel methods to capture the dynamics of infinite-width GNN with infinite-width attention, thus avoiding the huge overhead of training a wide and deep GNN.
Revised: We use kernel methods to capture the dynamics of infinite-width GNN with infinite-width attention, thus avoiding the huge overhead of training a wide and deep GNN.

Original: • Speedup introduced by quantum computing.
Revised: • Speedup introduced by quantum computing.

Original: Although GNTK is a useful method to train an infinite-width GNN, its cost still grows quadratically with respect to the volume of data, which is intractable for large datasets.
Revised: Although GNTK is a useful method to train an infinite-width GNN, its cost still grows quadratically with respect to the volume of data, which is intractable for large datasets

Original: We re-design the Attention-enhanced GNTK by splitting it into small components and reuniting them using the quantum linear algebra subroutines.
Revised: We re-design the Attention-enhanced GNTK by splitting it into small components and reuniting them using the quantum linear algebra subroutines.

Original: The produced quantum graph kernel – GraphQNTK, theoretically reduces the computational complexity from
Revised: The produced quantum graph kernel – GraphQNTK, theoretically reduces the computational complexity from

Original: O ( N 2 ) to O ( N ) benefited from the quantum parallelism.
Revised: O ( N 2 ) to O ( N ) benefited from the quantum parallelism.

Original:  Methodology 2.1 Preliminaries
Revised:  Methodology 2.1 Preliminaries

Original: We first briefly review the most common setting for GNNs and the corresponding NTK, and by the way the notation is given.
Revised: We first briefly review the most common setting for GNNs and the corresponding NTK, and by the way the notation is given.

Original: A graph G =
Revised: A graph G =

Original: ( V, E ) is denoted by a collection of nodes V and edges
Revised: ( V, E ) is denoted by a collection of nodes V and edges

Original: E .
Revised: E .

Original: Each node has a d -dimensional feature vector h v ∈ R d , v ∈ V , and H ∈ R n × d is the feature matrix stacking all nodes features.
Revised: Each node has a d -dimensional feature vector h v ∈ R d , v ∈ V , and H ∈ R n × d is the feature matrix stacking all nodes features.

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

Original: The formulation of GNN .
Revised: The formulation of GNN .

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

Original: Here we consider a simple message passing framework [20] and the propagation of the l -th ( l ∈ [ L ] ) layer is given as:
Revised: Here we consider a simple message passing framework [20] and the propagation 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 of the l -th layer and the r -th fully-connected layers, σ is an element-wise activated function, and
Revised: where N ( u ) denotes the neighbors of u , c σ is the scaling factor, d lr is the output dimension of the l -th layer and the r -th fully-connected layers, σ is an element-wise activated function, and

Original: W lR is learnable weights performing on the input for R times of the l -th layer (equivalent to R fully-connected layers without the bias term).
Revised: W lR is learnable weights performing on the input for 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 all node features in the last layer.
Revised: For graph classification, the output is a permutation invariance function acting on the collection of all node 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 ) } Ni =1 ⊂ R d × R .

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

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 for convenience.
Revised: And we replace H ( t ) with H for convenience.

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 NTK value between x i and x ∗ .
Revised: R N is the vector whose i -th element denotes the NTK value between x i and x ∗ .

Original: It is discovered that convolutional neural networks (CNNs) with infinite-width channels and infinite number of filters also have the same behaviour [2].
Revised: It is discovered that convolutional neural networks (CNNs) with infinite-width channels and infinite number 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 designing strategy of NTK and leverages a GNN architecture to design new graph kernels, which is called GNTK.
Revised: [17] adopts the designing strategy of NTK and leverages a GNN architecture to design new graph kernels, which is called GNTK.

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

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

Original: G =
Revised: G =

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

Original: =
Revised: =

Original: ( V ′ , E ′ ) with | V | = n and | V | = n ′ , the GNTK Θ ∈ R n × n ′ and the relative covariance matrix Σ ∈ R n × n ′ in the l -th layer of the feature aggregation phase as described in Eq.after R fully-connected layers are given by
Revised: ( V ′ , E ′ ) with | V | = n and | V | = n ′ , the GNTK Θ ∈ R n × n ′ and the relative covariance matrix Σ ∈ R n × n ′ in the l -th layer of the feature aggregation phase as described in Eq.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 (cid:2)
Revised: which is an affine transformation of the input GNTK and covariance respectively where (cid:2)

Original: Θ 0 R ( G, G ′ )
Revised: Θ 0 R ( G, G ′ )

Original: (cid:3) uu ′ and (cid:2) Σ 0 R ( G, G ′ )
Revised: (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 (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 (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 feature after aggregation.
Revised: 2 are used to update the node hidden feature after aggregation.

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

Original: [Equation]
Revised: [Equation]

Original: where ˆ σ ( r ) : [ − 1 , 1] → R denotes the the conjugate activation function corresponding to the activated function σ with centered Gaussian processes of covariance at the r -th fully-connected layer, as described in [15].
Revised: where ˆ σ ( r ) : [ − 1 , 1] → R denotes the the conjugate activation function corresponding to the activated function σ with centered Gaussian processes of covariance at the r -th fully-connected layer, as described 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: (7) 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 the feature update phase denoted by Eq.
Revised: 7, the transformation of the GNTK for the feature 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 element at 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 element at 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 accelerating by a proper quantum kernel estimation algorithm.
Revised: Intuitively, calculating each element of the GNTK of fully-connected layers could be accelerating by a proper quantum kernel estimation algorithm.

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

Original: To circumvent this barrier, we derive a unitary quantum aggregation transformation to bridge the gap between quantum kernel methods and estimation of GNTK.
Revised: To circumvent this barrier, we derive a unitary quantum aggregation transformation to bridge the gap between quantum kernel methods and estimation of GNTK.

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

Original: Before giving the analytical quantum reconstruction of GNTK with multi-head attention mechanism, we first elaborate on how to integrate the transformer layer into the GNN as described in Sec. 2.1.
Revised: Before giving the analytical quantum reconstruction of GNTK with multi-head attention mechanism, we first elaborate on how to integrate the transformer layer into the GNN as described in Sec. 2.1.

Original: The resulting GraphQNTK can be efficiently reconstructed by quantum computing paradigm, which gives a quadratic speed-up over the classic estimation of GNTK.
Revised: The resulting GraphQNTK can be efficiently reconstructed by quantum computing paradigm, which gives a quadratic speed-up over the classic estimation of GNTK.

Original: The mechanism to build the GNN and estimate the GNTK is shown in Fig.
Revised: The mechanism to build the GNN and estimate the 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 each node’s neighbor to the central node 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 [39] regards the contribution of each node’s neighbor to the central node 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.

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

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

Original: To capture the global node similarity semantics of the provided graph, numerous attempts that employ transformer for graph learning have been developed [27, 51, 53, 68].
Revised: To capture the global node similarity semantics of the provided graph, numerous attempts that employ transformer for graph learning have been developed [27, 51, 53, 68].

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

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 Q l , K l , V l .
Revised: W lK ∈ R s l × s lK and W lV ∈ R s l × s lV , to the corresponding representations Q l , K l , V l .

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

Original: [Equation]
Revised: [Equation]

Revised: [Figure]

Revised: limit

Revised: [Figure]

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: multi-head attention alternative is given by

Original: [Equation]
Revised: [Equation]

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

Original: [Figure]
Revised: N × s l matrix.

Original: Let Y and Θ denote the neural network Gaussian Process Kernel (NNGP) [51] and NTK after the transformer layer, and let (cid:101) Y and (cid:101)
Revised: Let Y and Θ denote the neural network Gaussian Process Kernel (NNGP) [51] and NTK after the 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 and the 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 and the 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 dot products between Q l and K l by their dimension instead of the square root of the same quantity, i.e.,
Revised: where the under the restriction that 1) W lQ and W lK share the same weighs, and 2) scaling the dot products 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 l K .

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

Original: To efficiently estimate the element of the NTK defined by the transformer layer using quantum parallel, we slightly modify the Eq. 12 as
Revised: To efficiently estimate the element of the NTK defined by the transformer layer using quantum parallel, we slightly modify the Eq. 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 multiplication between 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 multiplication between the diagonal of those two matrix after activated operation.

Original: For simplicity of use, we consider identity function as the activated operation, i.e, ζ
Revised: For simplicity of use, we consider identity function as the activated operation, i.e, ζ

Original: = I .
Revised: = I .

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

Original: Notice that the difference between the definition of
Revised: Notice that the difference between the definition of

Original: NNGP and the covariance of NTK is that the former denotes the expectation with respect to the output before the activated operation, while the later after the denotes the expectation with respect to the output after the activated operation [31].
Revised: NNGP and the covariance of NTK is that the former denotes the expectation with respect to the output before the activated operation, while the later after the denotes the expectation with respect to the output after the activated operation [31].

Original: Consequently, we consider that Y is equal to the covariance of NTK within the transformer layer as the result of the identity activated function.
Revised: Consequently, we consider that Y is equal to the 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 tail of the each GNN layer except the first and the last layer, and the calculation of the GNTK with infinite-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 tail of the each GNN layer except the first and the last layer, and the calculation of the GNTK with infinite-width-limit attention is to insert an additional procedure after the fully connected layers.

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

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

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

Original: where T l ( G, G ′ ) is the result of matrix multiplication between the column vector of the diagonal of Σ 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 of Σ 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 ):

Revised: [Equation]

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, and generalize 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, and generalize to the multi-layer model.

Original: The following statements only consider two input graphs G =
Revised: The following statements only consider two input graphs G =

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

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

Original: The corresponding covariance is ¯ Σ ∈ R n × n ′ and ¯ Σ uu ′ .
Revised: The corresponding covariance 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: We omit the subscript R for clarity.
Revised: We omit the subscript R for clarity.

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

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

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

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

Original: [Equation]

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 to their quantum representations can be estimate efficiently by introducing an auxiliary register.
Revised: Following a similar approach to [37], the inner product between two vectors with respect to their 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 state 1 √ 2 ( | u ⟩| u ′ ⟩| 

Revised: Specifically, estimation of the inner product h ⊤ u h u ′ can be performed by constructing the state 1 √ 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 the state | 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 the state | 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 of the inner product.
Revised: 1+ h ⊤ u h u ′is the estimation of the 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 is conditioned 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 is conditioned 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 inner product 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 inner product since the transformation of each element of NTK and the covariance is not independent.

Original: To circumvent this barrier, we derive a unitary quantum aggregation transformation to approximately reconstruct the affine transformation.
Revised: To circumvent this barrier, we derive a unitary quantum aggregation transformation to approximately reconstruct 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 ⟩ → | u ⟩| u ′ ⟩ (cid:0) √ P uu ′ | 0 , g uu ′ ⟩ + √

Original: [Equation]
Revised: [Equation]

Original:  (cid:1) that is employed to estimate the inner product of two feature vectors.
Revised: − P uu ′ | 1 , g ′ uu ′ ⟩ (cid:1) that is employed to estimate the inner product of two 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

Original: [Equation]
Revised: [Equation]

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

Original: The operation ⊗ denotes the tensor product.
Revised: The operation ⊗ denotes the 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 √ ·

Original: [Equation]
Revised: [Equation]

Original:  and the detailed mathematical expression and the scalar for state normalization are omitted since the result of the affine transformation has been embedded into the amplitude of | 0 ⟩ ⊗ .
Revised: where √· | other ⟩ represents other computational basis states except for | 0 ⟩ ⊗ with amplitude √· , and the detailed mathematical expression and the scalar for state normalization are omitted since the result of the affine transformation has been embedded into the amplitude of | 0 ⟩ ⊗ .

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

Original: Similar to the inner product estimation, the quantum aggregation transformation can be performed in superposition and the resulting superposition is
Revised: Similar to the inner product estimation, the quantum aggregation 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 1 √ nn′
Revised: The obtained quantum state 1 √ nn

Original: [Equation]
Revised: [Equation]

Original: u∈ V (cid:80) u ′ ∈ V ′ 
Revised: u ′ ∈ V ′ |

Original: | u ⟩| u ′ ⟩| ¯ A uu ′ ⟩| y uu ′ ⟩ whose third register carries the approximate result after aggregation transformation as described in Eq.
Revised: u ⟩| u ′ ⟩| ¯ A uu ′ ⟩| y uu ′ ⟩ whose third register carries the approximate result after aggregation 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 a garbage state.
Revised: A uu ′ | ≤ ϵ and | y uu ′ ⟩ is a garbage 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 root of 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 root of the NTK from the previous layer, thus it is an approximate aggregation transformation.

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

Original: Quantum kernel estimation .
Revised: Quantum kernel estimation .

Original: 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 the covariance matrix in the previous fully-connected layer.
Revised: 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 the covariance 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, there exits a unitary V : (cid:80) x | x, 0 ⟩ → (cid:80) x | x, f ( x ) ⟩ for any classical function f with the same time complexity to evaluate each element of the NTK and each element of the covariance [48, 71].
Revised: In general, there exits a unitary V : (cid:80) x | x, 0 ⟩ → (cid:80) x | x, f ( x ) ⟩ for any classical function f with the same time complexity to evaluate each element of the NTK and each element of the covariance [48, 71].

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]

Revised: u

Original: [Equation]
Revised: [Equation]

Original: ⟩| y uu
Revised: u

Original: [Equation]
Revised: [Equation]

Revised: ⟩| y uu ′ ⟩ .

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

Original: [Equation]
Revised: [Equation]

Original: | u ⟩| u
Revised: | u [Equation] | u

Original: [Equation]
Revised: [Equation]

Original: ⟩| ¯ Θ uu ′ ⟩| y uu ′ ⟩ , where ¯ Θ uu ′ is the approximate estimation of its classical counterpart after R fully-connected layers.
Revised: ⟩| ¯ Θ uu ′ ⟩| y uu ′ ⟩ , where ¯ Θ uu ′ is the approximate estimation of its classical counterpart after R fully-connected layers.

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

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

Original: It means 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 ) ∈ G before input the different graphs, and we only consider the element when u = u ′ .
Revised: It means 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 ) ∈ G before input the different graphs, and we only consider the element when u = u ′ .

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

Original: ⟩| ¯ Σ uu ⟩| y uu ⟩ for graph G and 1 √ n ′ (cid:80) u ′ ∈ V ′ | u ′ ⟩| ¯ Σ u ′ u ′ ⟩| y u ′ u
Revised: ⟩| ¯ Σ uu ⟩| y uu ⟩ for graph G and [Equation] [Equation] | u ′ ⟩| ¯ Σ u ′ u ′ ⟩| y u ′ u

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

Original: Thus, estimation of the GNTK and its covariance corresponding to the multiplication part in Eq.
Revised: Thus, estimation of the GNTK and its covariance corresponding 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 the index 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 the index register, i.e. | u ⟩ and | u ′ ⟩ , which are designed by [63, 54, 41].

Original: The final GNTK after the transformer layer can be directly generated by additional quantum arithmetic operations that perform an element-wise addition between the covariance to the GNTK.
Revised: The final GNTK after the transformer layer can be directly generated by additional quantum arithmetic operations that perform an 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 the approximate NTK and its covariance are embedded into the amplitudes of a superposition.
Revised: The quantum aggregation transformation requires that the approximate 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 a superposition.
Revised: However, after the quantum kernel estimation, these matrix are embedded into the quantum basis states of a superposition.

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

Original: [Equation]
Revised: [Equation]

Original: (21) where a uu ′ =
Revised: 21) where a uu ′ =

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

Original: [Equation]
Revised: [Equation]

Original: and b 
Revised: and b uu

Original: [Equation]
Revised: [Equation]

Revised: =

Revised: [Equation]

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

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

Original: Quantum readout .
Revised: Quantum readout .

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

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

Revised: [Equation]

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 the resulting GraphQNTK between two input graphs is ¯ Θ GG 
Revised: And the resulting GraphQNTK between two input graphs is ¯ Θ GG ′ =

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

Original: [Equation]
Revised: [Equation]

Original: = (

Original: [Equation]
Revised: [Equation]

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

Original: Quantum inference to unseen data .
Revised: Quantum inference to unseen data .

Original: We assume that the test data and the label of the training set are already encoded into the QRAM such that | k ∗ ⟩ ∈ R N , the GraphQNTK between the test graph
Revised: We assume that the test data and the label of the training set are 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 × N denote the GraphQNTK.
Revised: Let ¯ Θ ∈ R N × N denote 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: 23) 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 quantum linear 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 quantum linear 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 realize the 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
Revised: To realize the 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

Original: Gershgorin circle theorem similar to [71].
Revised: Gershgorin circle theorem similar to [71].

Original: 2.4 Complexity Study
Revised: 2.4 Complexity Study

Original: In Sec. 2.3, we discuss how to approximately estimate GNTK using quantum computing paradigm between two input graphs.
Revised: In Sec. 2.3, we discuss how to approximately estimate GNTK using quantum computing paradigm between two input graphs.

Original: The time complexity is dominated by the quantum aggregation transformation procedure as it requires encoding the amplitude into an additional register, which takes
Revised: The time complexity is dominated by the quantum aggregation transformation procedure as it requires encoding the amplitude into an additional register, which takes

Original: [Table]
Revised: [Table]

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

Original: Other quantum operations including estimation of the inner product, estimation of the GNTK within the neighborhood aggregation and the fully-connected feature updating and quantum readout are totally unitary operations which can be efficiently performed under the regime of quantum computing.
Revised: Other quantum operations including estimation of the inner product, estimation of the GNTK within the neighborhood aggregation and the fully-connected feature updating and quantum readout are totally unitary operations which can be efficiently performed under the regime of quantum computing.

Original: For estimating 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: For estimating 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 O (log( Nnd )) .
Revised: The quantum runtime is O (log( Nnd )) .

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

Original: The result should be stored in QRAM in advance which will be used to update GNTK corresponding the multi-head attention as described in Eq.
Revised: The result should be stored in QRAM in advance which will be used to update GNTK corresponding the multi-head attention as described in Eq.

Original: Therefore, it takes O ( N log( Nnd )) time to train the proposed quantum graph learning model, which achieves quadratic speedup compared to the existing GKs and completed approaches with O ( N 2 ) time.
Revised: Therefore, it takes O ( N log( Nnd )) time to train the proposed quantum graph learning model, which achieves quadratic speedup compared to the existing 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 several graph classification datasets involving either discrete or continuous attributes.
Revised: We evaluate our method for both GNTK and GraphQNTK with attention mechanism on several graph classification datasets involving either discrete or continuous attributes.

Original: All the experiments are performed on a workstation with a single machine with 1TB memory, one physical CPU withcores Intel(R)
Revised: All the experiments are performed on a workstation with a single machine with 1TB memory, one physical CPU withcores Intel(R)

Original: Xeon(R) W-3175X CPU @
Revised: 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 the average 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, 67], and report the average test accuracy and its standard deviation over a 10-fold cross validation on each dataset.

Original: Source code is available at https://github.com/abel1231/graphQNTK .

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 bioinformatics datasets MUTAG, PTC, NCI1, PROTEINS and three social network datasets IMDB-BINARY, IMDBMULTI.
Revised: For graph with discrete attributes, the benchmark datasets include four bioinformatics datasets 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 to one-hot encoding representations.
Revised: For each graph, the input attributes is category of the node and they are transformed to one-hot encoding representations.

Original: For datasets where the graphs have no node features, i.e. only graph structure matters, we use degrees as input node features.
Revised: For datasets where the graphs have no node features, i.e. only graph 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 the datasets can be found in [38].
Revised: All the datasets can be found in [38].

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 [57],

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

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

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

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

Original: The data separation we use is the same as [67] for graph
Revised: The data separation we use is the same as [67] for graph

Original: [Table]
Revised: [Table]

Original: Results.
Revised: datasets with discrete attributes. For graph dataset with continuous attributes, we follow the same protocol as used in [62] to normalize the input feature vectors for a fair comparison. 3.2 Experiment Results

Original: We apply different hyper-parameter settings to L ∈ { 2 , 4 , 6 , 8 } and R ∈ { 1 , 2 , 3 } and select the model with the best averaged accuracy.
Revised: We apply different hyper-parameter settings to L ∈ { 2 , 4 , 6 , 8 } and R ∈ { 1 , 2 , 3 } and select the model with the best averaged accuracy.

Original: We test the kernel regression using SVM classifier and the regularization parameter is determined using the search protocol which is the same as the [17].
Revised: We test the kernel regression using SVM classifier and the regularization parameter is determined using the search protocol which is the same as the [17].

Original: We report the performance of the quantum approximate GNTK before and after the matrix sparsity and conditioning operations.
Revised: We report the performance of the quantum approximate GNTK before and after the matrix sparsity and conditioning operations.

Original: The numerical results are listed in Tab. 1 for datasets with discrete attributes and Tab. 2 for datasets with continuous attributes.
Revised: The numerical results are listed in Tab. 1 for datasets with discrete attributes and Tab. 2 for datasets with continuous attributes.

Original: The attention method we integrate to the infinite-width GNNs brings to an improvement in the performance of the model.
Revised: The attention method we integrate to the infinite-width GNNs brings to an improvement in the performance of the model.

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

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

Original: Moreover, our model performs better than QS-CNN on more than 60% of the datasets with discrete attributes, given the caveat that QS-CNN is a hybrid graph learning model where the contribution of the classic components (CNNs, spatial message passing) in their model cannot be ignored.
Revised: Moreover, our model performs better than QS-CNN on more than 60% of the datasets with discrete attributes, given the caveat that QS-CNN is a hybrid graph learning model where the contribution of the classic components (CNNs, spatial message passing) in their model cannot be ignored.

Original: While the matrix sparsity and conditioning operations have a great influence on the model’s performance, it can be found that the classification performance of GNTK evaluated by quantum algorithms is still comparable with that of GKs and vanilla GNNs, where a tradeoff exists between the performance of the model and the quantum computational efficiency.
Revised: While the matrix sparsity and conditioning operations have a great influence on the model’s performance, it can be found that the classification performance of GNTK evaluated by quantum algorithms is still comparable with that of GKs and vanilla GNNs, where a tradeoff exists between 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 information while using quantum parallelism to improve computing efficiency.
Revised: This paper has presented a quantum graph learning model to characterize the structural information while using quantum parallelism to improve computing efficiency.

Original: We propose quantum algorithm to approximately estimate the neural tangent kernel of the underlying graph neural network where a multi-head quantum attention mechanism is introduced to incorporate semantic similarity of nodes.
Revised: We propose quantum algorithm to approximately estimate the neural tangent kernel of the underlying graph neural network where a multi-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 of our method.
Revised: Empirical results on graph classification tasks as well as theoretical analysis show the superiority of our method.

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

Original: This work was partly supported by National Key Research and Development Program of China (2020AAA0107600), National Natural Science Foundationof China (
Revised: This work was partly supported by National Key Research and Development Program of China (2020AAA0107600), National Natural Science Foundation

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Revised: Checklist

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Original: A Related Works
Revised: A Related Works

Original: In this section, we provide background on quantum graph learning and graph neural networks that have the potential to be trained using quantum computing.
Revised: In this section, we provide background on quantum graph learning and graph neural networks that have 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 properties of both GNNs and GKs.
Revised: Graph kernel neural networks [17, 10, 18] is a class of graph learning method combining the properties of 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 [66], 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 [40].

Original: The mainstream of graph kernel neural networks
Revised: The mainstream of graph kernel neural networks

Original: [Table]
Revised: [Table]

Original: can be divided into completed and approximated approaches.
Revised: can be divided into completed and approximated approaches.

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

Original: While the later generates the approximate feature of the finite projected RKHS at the expense of information loss.
Revised: While the later generates the approximate feature of the 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 [33].

Original: In the next section, we demonstrate that the graph tangent kernel neural networks coincides with the condition of quantum parallel implementation by introducing the quantum aggregation transformation and the quantum kernel estimation techniques.
Revised: In the next section, we demonstrate that the graph tangent kernel neural networks coincides with the condition of quantum parallel implementation by introducing 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 to analyze classical data under the regime of quantum computing, since the encoding and decoding between classical vectors (or matrices) and their corresponding quantum states should be carefully designed.
Revised: It is generally nontrivial to analyze classical data under the regime of quantum computing, since the encoding and decoding between classical vectors (or matrices) and their corresponding quantum states should be carefully designed.

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

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

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

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

Original: Adiabatic evolution [64] and conditional unitary [46] are applied to evolve the quantum systems dependent on the underlying graph structure.
Revised: Adiabatic evolution [64] and conditional unitary [46] are applied to evolve the quantum systems dependent on the underlying graph structure.

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

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

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

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, the researchers exploit to encode the graph structure and node features in the quantum system through various schemes, and then process the information through quantum layers and auxiliary classical layers.
Revised: Generally speaking, the researchers exploit to encode the graph structure and node features in the quantum system through various schemes, and then process the information through quantum layers and auxiliary classical layers.

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

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

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

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

Original: We notice that there are researches which are abbreviated as QNTK [59, 42], similar to ours nominally.
Revised: 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.
Revised: But their definition is quite different from ours.

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

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

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

Original: B More Analysis B.1 Quantum Access Memory
Revised: B More Analysis B.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 as
Revised: There exists a data structure to store the entries of X into the QRAM which is stated as

Original: [Equation]
Revised: [Equation]

Original: in time T for p ∈ [ n ] .
Revised: in time T for p ∈ [ n ] .

Original: Using the binary tree QRAM architecture proposed by [35], the time T to store and readout a new element scale logarithmically with respect to both n and d .
Revised: Using the binary tree QRAM architecture proposed by [35], the time T to store and 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 quantum representations with respect to their d -dimensional classical vectors in time
Revised: A that evaluates the inner product of two quantum representations 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 map 1 √ 2 ( | p ⟩| q ⟩| 0 ⟩| 0 ⟩ + | p ⟩| q ⟩| 1 ⟩| 0 ⟩ ) → 1 √ 2 ( | p ⟩| q ⟩| 0 ⟩| | X p ⟩⟩ + | p ⟩| q ⟩| 1 ⟩| | X q ⟩⟩ ) can be performed in
Revised: By introducing an auxiliary register, with the initial state | p ⟩| q ⟩ 1 √ 2 ( | 0 ⟩ + | 1 ⟩ ) | 0 ⟩ , the map 1 √ 2 ( | p ⟩| 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 state defined by Eq.
Revised: Thus the state defined 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 proof can be found in [9].
Revised: The detailed proof can 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 in time T .
Revised: α | v, 1 ⟩ + √ 1 − α | g, 0 ⟩ for some 1 / 2 ≤ α ≤ 1 in time T .

Original: There exits a quantum algorithm that, for any ∆ > 0 and for any 1 / 2 < α 0 ≤ α , produce a state | 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 a state | 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
Revised: | x ⟩∥ ≤ √ 2∆ for some integer L in time

Original: [Equation]
Revised: [Equation]

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

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

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

Original: Here we give an intuitive illustration.
Revised: Here we give an intuitive illustration.

Original: Consider a general GNN with the neighborhood feature aggregation function
Revised: 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)
Revised: and the central node feature update function ( R fully connected layers)

Original: [Equation]
Revised: [Equation]

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

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

Original: , R ),
Revised: , R ),

Original: c σ is a scaling factor.
Revised: 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 , . . .
Revised: The infinitewidth of GNN means that the output dimension d lr of W lr goes to infinity for r = 1 , . . .

Original: , R and l = 1 , . . .
Revised: , R and l = 1 , . . .

Original: , L .
Revised: , 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].
Revised: 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.
Revised: Here we briefly explain what an infinite-width transformer looks like.

Original: Consider a single attention layer
Revised: Consider a single attention layer

Original: [Equation]
Revised: [Equation]

Original: in Eq.
Revised: in Eq.

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

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

Original: Q h , K h ,
Revised: Q h , K h ,

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

Original: s is a scaling factor.
Revised: 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.
Revised: 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
Revised: 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.
Revised: = 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
Revised: For a multi-head attention (transformer) layer, the equation is given as

Original: [Equation]
Revised: [Equation]

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

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

Original: C.2 Our contributions beyond the quantum speedup
Revised: C.2 Our contributions beyond the quantum speedup

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

Original: We hope that our model can provide technical guidance for future research, and better exploit the immense benefits of quantum computing even in a classic simulation condition.
Revised: We hope that our model can provide technical guidance for future research, and better exploit the immense benefits of quantum computing even in a classic simulation condition.

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

Original: And we hope that this novel graph learning model can bridge the gap between graph neural tangent kernel methods and attention mechanism.
Revised: And we hope that this novel graph learning model can bridge the gap between graph neural tangent kernel methods and attention mechanism.

Original: Here we briefly give an illustration.
Revised: Here we briefly give an illustration.

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

Original: The numerical results in Tab.and Tab. 2 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 capture distinct properties between different graphs.
Revised: The numerical results in Tab.and Tab. 2 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 capture distinct properties between different graphs.

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

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

Original: It shows the superiority of our model in quantum graph learning.
Revised: It shows the superiority of our model in quantum graph learning.

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

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

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

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

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: D Additional Experiments and Discussion D.1 Running time comparison
Revised: D Additional Experiments and Discussion D.1 Running time comparison

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

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

Original: All the experiments are performed on a workstation with a single machine with 1TB memory, one physical CPU withcores Intel® Xeon® W-3175X CPU @
Revised: All the experiments are performed on a workstation with a single machine with 1TB memory, one physical CPU withcores Intel® Xeon® 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: The results are shown in Tab. 4.
Revised: The results are shown in Tab. 4.

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

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

Original: It is noticed that our model.
Revised: It is noticed that our model.

Original: The runtime of our model is apparently faster than that of GIN.
Revised: The runtime of our model is apparently faster than that of GIN.

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

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

Original: We compare the graph classification accuracy bewteen GIN and our model at the same number of layers and the results are given in Tab. 5 and
Revised: We compare the graph classification accuracy bewteen GIN and our model at the same number of layers and the results are given in Tab. 5 and

Original: Tab. 6.
Revised: Tab. 6.

Original: The number in parentheses in the table indicates the number of layers.
Revised: 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 when the number of layers becomes larger.
Revised: From Tab. 5, it is shown that our model (AttentionGNTK) is more robust compared with GIN when the 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 between prediction and the ground truth.
Revised: 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 between prediction and the ground truth.

Original: While in Tab. 6, the results empirically demonstrate that our model (attentionGNTK) reaches the peak of classification accuracy when the number of layers is small, while GNTK needs more layers to reach, indicating that our model is easier to capture the global structure information of the graph.
Revised: While in Tab. 6, the results empirically demonstrate that our model (attentionGNTK) reaches the peak of classification accuracy when the number of layers is small, while GNTK needs more layers to 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.
Revised: This could be interpreted from the theoretical perspective.

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

Original: Consider G =
Revised: Consider G =

Original: QK ⊤ √ s in Eq.
Revised: QK ⊤ √ s in Eq.

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

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

Original: Then the operation
Revised: Then the operation

Original: [Equation]
Revised: [Equation]

Original: transforms the node features of the last layer to the next layer depending on the node similarity.
Revised: transforms the node features of the last layer to the next layer depending on the node similarity.

Original: This enables the model to make better use of the graph structure to transmit information and perceive topology information over long distances.
Revised: This enables the model to make better use of the graph structure to transmit information and perceive topology information over long distances.

Original: [Table]
Revised: [Table]