The paper presents the theory and improvements of variational hybrid quantum-classical algorithms, specifically the Variational Quantum Eigensolver (VQE). The authors extend the general theory of VQE and propose algorithmic improvements for practical implementations. They develop a variational adiabatic ansatz and explore unitary coupled cluster, establishing a connection from second-order unitary coupled cluster to universal gate sets through exponential splitting relaxation. They introduce quantum variational error suppression, allowing some errors to be naturally suppressed on pre-threshold quantum devices. They also analyze truncation and correlated sampling in Hamiltonian averaging to reduce computational cost. Finally, they show how modern derivative-free optimization techniques can offer dramatic computational savings over previous methods. The paper discusses the importance of eigenvalue and optimization problems in various applications, including quantum simulation and quantum chemistry. It highlights the potential of quantum computers to solve these problems exponentially faster than classical computers. The authors review the background and notation of quantum systems and the variational principle, emphasizing the role of quantum states and Hamiltonians in solving these problems. The paper focuses on the application of VQE to quantum chemistry and fermionic Hamiltonians, discussing the mapping from fermions to qubits and the use of reference states in electronic structure calculations. It also explores the use of adiabatically parameterized states and unitary coupled cluster states in the variational approach. The authors analyze the error bounds and distributions in quantum state preparation, and discuss the use of variational principles to optimize the path of adiabatic evolution. The paper also introduces the concept of unitary coupled cluster, a method for parametrically exploring the Hilbert space of quantum states. It discusses the use of this method in quantum chemistry and its generalization to interacting two-level quantum systems. The authors highlight the importance of unitary state preparation in quantum computing and the challenges of exploring quantum state space from an arbitrary reference state. The paper concludes by emphasizing the potential of variational hybrid quantum-classical algorithms in solving complex quantum problems, and the importance of algorithmic improvements in practical implementations. The authors argue that these algorithms can be used to explore the performance of early quantum computers and offer significant computational savings through modern optimization techniques.The paper presents the theory and improvements of variational hybrid quantum-classical algorithms, specifically the Variational Quantum Eigensolver (VQE). The authors extend the general theory of VQE and propose algorithmic improvements for practical implementations. They develop a variational adiabatic ansatz and explore unitary coupled cluster, establishing a connection from second-order unitary coupled cluster to universal gate sets through exponential splitting relaxation. They introduce quantum variational error suppression, allowing some errors to be naturally suppressed on pre-threshold quantum devices. They also analyze truncation and correlated sampling in Hamiltonian averaging to reduce computational cost. Finally, they show how modern derivative-free optimization techniques can offer dramatic computational savings over previous methods. The paper discusses the importance of eigenvalue and optimization problems in various applications, including quantum simulation and quantum chemistry. It highlights the potential of quantum computers to solve these problems exponentially faster than classical computers. The authors review the background and notation of quantum systems and the variational principle, emphasizing the role of quantum states and Hamiltonians in solving these problems. The paper focuses on the application of VQE to quantum chemistry and fermionic Hamiltonians, discussing the mapping from fermions to qubits and the use of reference states in electronic structure calculations. It also explores the use of adiabatically parameterized states and unitary coupled cluster states in the variational approach. The authors analyze the error bounds and distributions in quantum state preparation, and discuss the use of variational principles to optimize the path of adiabatic evolution. The paper also introduces the concept of unitary coupled cluster, a method for parametrically exploring the Hilbert space of quantum states. It discusses the use of this method in quantum chemistry and its generalization to interacting two-level quantum systems. The authors highlight the importance of unitary state preparation in quantum computing and the challenges of exploring quantum state space from an arbitrary reference state. The paper concludes by emphasizing the potential of variational hybrid quantum-classical algorithms in solving complex quantum problems, and the importance of algorithmic improvements in practical implementations. The authors argue that these algorithms can be used to explore the performance of early quantum computers and offer significant computational savings through modern optimization techniques.