Quantum circuit synthesis and compilation optimization are critical for the development and implementation of quantum algorithms. This survey systematically reviews the latest research on quantum circuit synthesis, optimization, and compilation, focusing on the integration of artificial intelligence (AI) methods to enhance the efficiency and accuracy of quantum algorithm design and execution. The paper discusses the challenges and opportunities in quantum circuit design, including the synthesis of quantum circuits with specific unitary matrices, optimization of quantum circuits for different eras (NISQ and fault-tolerant), and the mapping and routing of qubits to physical qubits. It also explores various quantum circuit representations, such as the gate model, directed acyclic graphs (DAGs), phase polynomials, tensor networks, and ZX diagrams, which are essential for different synthesis and optimization methods. The paper highlights the importance of quantum architecture search (QAS) in automatically designing quantum circuits, with applications in quantum algorithms, quantum machine learning, and quantum error correction. It reviews various QAS methods, including heuristic algorithms, reinforcement learning, and sampling-based learning, and discusses their effectiveness in optimizing quantum circuits. The paper also addresses the challenges of implementing quantum circuits on current NISQ hardware, emphasizing the need for efficient and accurate quantum circuit compilation. Overall, the survey provides a comprehensive overview of the current state of quantum circuit synthesis and compilation, and highlights the potential of AI in advancing quantum computing technology.Quantum circuit synthesis and compilation optimization are critical for the development and implementation of quantum algorithms. This survey systematically reviews the latest research on quantum circuit synthesis, optimization, and compilation, focusing on the integration of artificial intelligence (AI) methods to enhance the efficiency and accuracy of quantum algorithm design and execution. The paper discusses the challenges and opportunities in quantum circuit design, including the synthesis of quantum circuits with specific unitary matrices, optimization of quantum circuits for different eras (NISQ and fault-tolerant), and the mapping and routing of qubits to physical qubits. It also explores various quantum circuit representations, such as the gate model, directed acyclic graphs (DAGs), phase polynomials, tensor networks, and ZX diagrams, which are essential for different synthesis and optimization methods. The paper highlights the importance of quantum architecture search (QAS) in automatically designing quantum circuits, with applications in quantum algorithms, quantum machine learning, and quantum error correction. It reviews various QAS methods, including heuristic algorithms, reinforcement learning, and sampling-based learning, and discusses their effectiveness in optimizing quantum circuits. The paper also addresses the challenges of implementing quantum circuits on current NISQ hardware, emphasizing the need for efficient and accurate quantum circuit compilation. Overall, the survey provides a comprehensive overview of the current state of quantum circuit synthesis and compilation, and highlights the potential of AI in advancing quantum computing technology.