The paper presents a hardware-efficient variational quantum eigensolver (VQE) for solving molecular structure and quantum magnetism problems on a six-qubit superconducting quantum processor. The method uses a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine to determine the ground state energy of molecules up to BeH2. The authors demonstrate the flexibility of their approach by applying it to a quantum magnetism problem, showing agreement between experiment and numerical simulations with a noisy model of the device. The results highlight the requirements for scaling the method to larger systems and aim to bridge the gap between high-performance computing and quantum hardware implementation. The VQE is based on a hardware-efficient ansatz preparation, where trial states are parameterized by quantum gates tailored to the available interactions in the quantum processor. The paper also discusses the experimental implementation details, including the device characterization, entangler characterization, and energy estimation techniques.The paper presents a hardware-efficient variational quantum eigensolver (VQE) for solving molecular structure and quantum magnetism problems on a six-qubit superconducting quantum processor. The method uses a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine to determine the ground state energy of molecules up to BeH2. The authors demonstrate the flexibility of their approach by applying it to a quantum magnetism problem, showing agreement between experiment and numerical simulations with a noisy model of the device. The results highlight the requirements for scaling the method to larger systems and aim to bridge the gap between high-performance computing and quantum hardware implementation. The VQE is based on a hardware-efficient ansatz preparation, where trial states are parameterized by quantum gates tailored to the available interactions in the quantum processor. The paper also discusses the experimental implementation details, including the device characterization, entangler characterization, and energy estimation techniques.