Dec 05, 2023
Contents
PurposeSummaryPreliminary2 Background2.1 Classical GNN2.2 Networked Quantum Systems3 Quantum Graph Neural Network3.1 General Quantum Graph Neural Network Ansatz3.2 Quantum Graph Recurrent Neural Networks (QGRNN)3.3 Quantum Graph Convolutional Neural Networks (QGCNN)3.4 Quantum Spectral Graph Convolutional Neural Networks (QSGCNN)4 Applications & Experiments4.1 Learning Quantum Hamiltonian Dynamics with Quantum Graph Recurrent Neural Networks4.2 Quantum Graph Convolutional Neural Networks for Quantum Sensor Networks5 Conclusion & OutlookPurposeSummaryPreliminary역학수학양자컴퓨터2 Background2.1 Classical GNNLearnable parameters2.2 Networked Quantum Systems3 Quantum Graph Neural Network3.1 General Quantum Graph Neural Network Ansatztrainable variables3.2 Quantum Graph Recurrent Neural Networks (QGRNN)3.3 Quantum Graph Convolutional Neural Networks (QGCNN)3.4 Quantum Spectral Graph Convolutional Neural Networks (QSGCNN)4 Applications & Experiments4.1 Learning Quantum Hamiltonian Dynamics with Quantum Graph Recurrent Neural Networks4.2 Quantum Graph Convolutional Neural Networks for Quantum Sensor Networks5 Conclusion & Outlook
Purpose
Quantum Graph Neural Network 와 그 응용 형태의 제안과 간단한 실증
Summary
- Quantum Graph Neural Network
- Quantum Graph Recurrent Neural Network
- Quantum Graph Convolutional Neural Network
- Quantum Spectral Graph Convolutional Neural Networks (QSGCNN)
Preliminary
역학
고전역학
- 연속성
- 확실한 모델
↔ 양자역학
- 불연속적
- 이중적 & 확률적 모델
불확정성의 원리 Uncertainty Principle
Quantum State & Wavefunction
입자 파동 이중성
해밀턴 역학
자연의 움직임을 기술할 때 최소화시킬수 있는 항 "라그랑지언"을 정의하고, 라그랑지언을 통해 물리적 에너지 변화를 계산하는 방법
고전역학에서도 잘 사용되는, 하나의 방법론이다
라그랑지언을 통한 연산이 뉴턴역학적으로 물체의 운동속성을 기술하는 것 보다 양자역학적 연산을 쉽게 만들어주기 때문에 양자역학에서 쓰인다
Lagrangian L
L ≡ T - U
해밀턴 역학에서의 에너지 텀
Hamiltonian H
계의 에너지 상태를 기술하는 라그랑지언과 르장드르 변환을 통해 이어져 있는 짝 벡터이자 연산자이자 연산
수학
오일러공식
르장드르 변환
Hilbert Space
≈ linear vector space
≈ linear space ≡ euclidean space
벡터연산을 유클리드 공간의 스칼라 연산처럼 하기 위해 정의된 공간
wavefunction is in hilbert space
Spectral Clustering
데이터가 있을 때 spectrum(= eigen value) 간의 similarity matrix 를 잡아서 dimension reduction 하는 방법
라플라시안
3 차원에서의 벡터 발산과 수렴
Del operator
2 차원에서의 벡터 발산과 수렴
양자컴퓨터
Quantum Computation & Qubit Operation
2 Background
2.1 Classical GNN
- g: graph
- A: Adjacency matrix
- n: number of nodes
- X: node feature
- d: node feature dimension
- k: layer number
- H: hidden value
- : weight at layer k
- P: message propagation function depends on adjacency matrix
Learnable parameters
- : weights on layer k
- : initial embedding. (= )
2.2 Networked Quantum Systems
이 페이퍼에서는 만 사용
- g: quantum graph in hilbert space
- : vertices ≡ quantum states
- : vertice ≡ quantum state
- : edges ≡ transitions between quantum states
- : edge ≡ quantum transition
- : Hilbert space of vertices
- : Hilbert space of edges
3 Quantum Graph Neural Network
3.1 General Quantum Graph Neural Network Ansatz
- U: Quantum state
- : can be any parameterized hamiltonian whose topology of interactions is that of the problem graph
- Hamiltonian Operators
- : j state와 k state 간의 interaction
앞 term = edge, 뒷 term = vertex
- Q: Hamiltonian evolutions
- P: Repeated
trainable variables
- , : trainable variables
- , : real-valued coefficient. independent trainable parameter
3.2 Quantum Graph Recurrent Neural Networks (QGRNN)
RNN: parameter shared through the sequential inputs (over p = 1 … P)
∴ P를 무시하면 RNN과 같다
3.3 Quantum Graph Convolutional Neural Networks (QGCNN)
This is analogous to translational invariance for ordinary convolutional transformations.
In our case, permutation invariance manifests itself as a constraint on the Hamiltonian, which now should be devoid of local trainable parameters, and should only have global trainable parameters.
: tied over indices ≈ convolutional value
3.4 Quantum Spectral Graph Convolutional Neural Networks (QSGCNN)
4 Applications & Experiments
4.1 Learning Quantum Hamiltonian Dynamics with Quantum Graph Recurrent Neural Networks
4.2 Quantum Graph Convolutional Neural Networks for Quantum Sensor Networks
5 Conclusion & Outlook
여기서는 가장 간단한 예시만 보였으며, 응용가능성이 보인다.
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