The paper discusses the theory and improvements of the variational quantum eigensolver (VQE), a hybrid quantum-classical algorithm designed to solve eigenvalue and optimization problems with limited quantum resources. The authors extend the general theory of VQE, focusing on algorithmic improvements for practical implementations. Key contributions include: 1. **Variational Adiabatic Ansatz**: The authors develop a variational adiabatic ansatz, which allows for the exploration of classically intractable regions of Hilbert space through adiabatic state preparation. 2. **Unitary Coupled Cluster**: They explore unitary coupled cluster, showing a connection between second-order unitary coupled cluster and universal gate sets through the relaxation of exponential splitting. This connection opens the door to using VQE for universal quantum computation. 3. **Quantum Variational Error Suppression**: The concept of quantum variational error suppression is introduced, allowing for the natural suppression of some errors in pre-threshold quantum devices. 4. **Hamiltonian Averaging**: The paper analyzes truncation and correlated sampling in Hamiltonian averaging to reduce the cost of the procedure. 5. **Derivative-Free Optimization**: Modern derivative-free optimization techniques are shown to significantly enhance the efficacy of the VQE, offering up to three orders of magnitude in computational savings compared to previous methods. The VQE is highlighted as a robust algorithm that can adapt to various quantum hardware architectures, leveraging the strengths of specific gates or quantum operations. The ability to suppress errors without detailed knowledge of the error mechanism and the low coherence time requirements make VQE a promising candidate for early quantum computers.The paper discusses the theory and improvements of the variational quantum eigensolver (VQE), a hybrid quantum-classical algorithm designed to solve eigenvalue and optimization problems with limited quantum resources. The authors extend the general theory of VQE, focusing on algorithmic improvements for practical implementations. Key contributions include: 1. **Variational Adiabatic Ansatz**: The authors develop a variational adiabatic ansatz, which allows for the exploration of classically intractable regions of Hilbert space through adiabatic state preparation. 2. **Unitary Coupled Cluster**: They explore unitary coupled cluster, showing a connection between second-order unitary coupled cluster and universal gate sets through the relaxation of exponential splitting. This connection opens the door to using VQE for universal quantum computation. 3. **Quantum Variational Error Suppression**: The concept of quantum variational error suppression is introduced, allowing for the natural suppression of some errors in pre-threshold quantum devices. 4. **Hamiltonian Averaging**: The paper analyzes truncation and correlated sampling in Hamiltonian averaging to reduce the cost of the procedure. 5. **Derivative-Free Optimization**: Modern derivative-free optimization techniques are shown to significantly enhance the efficacy of the VQE, offering up to three orders of magnitude in computational savings compared to previous methods. The VQE is highlighted as a robust algorithm that can adapt to various quantum hardware architectures, leveraging the strengths of specific gates or quantum operations. The ability to suppress errors without detailed knowledge of the error mechanism and the low coherence time requirements make VQE a promising candidate for early quantum computers.