This paper presents a simulated quantum computation of molecular energies using a recursive phase estimation algorithm. The authors demonstrate that quantum algorithms can compute molecular ground-state energies with significantly lower computational complexity than classical methods. They show that for small molecules like H2O and LiH, quantum simulations using a modest number of qubits can achieve high precision, matching the accuracy of classical full configuration interaction (FCI) calculations. The paper discusses the challenges of simulating molecular systems on classical computers, where the computational cost grows exponentially with system size. In contrast, quantum algorithms can achieve polynomial scaling. The authors use a modified phase estimation algorithm to compute the ground-state energy of H2O and LiH, showing that the number of qubits required scales linearly with the number of basis functions, while the number of quantum gates grows polynomially with the number of qubits. The authors also address the issue of preparing a good approximate ground-state wave function using an adiabatic state preparation algorithm. This method systematically improves the overlap between the initial wave function and the exact ground state, enhancing the success probability of the phase estimation algorithm. The paper also discusses the mapping of molecular wave functions to qubit states, showing that both direct and compact mappings can be used. The compact mapping, which restricts the wave function to a subspace with fixed electron number, is more efficient and requires fewer qubits. The authors demonstrate that quantum simulations of molecular systems can be performed with a relatively small number of qubits, making them feasible for practical applications. They show that for the H2O molecule, the recursive phase estimation algorithm can achieve high precision with only four qubits in the read-out register, while the compact mapping requires eight qubits for the singlet subspace. The paper concludes that quantum simulation algorithms with 30 to 100 qubits could be among the smallest applications of quantum computing that can exceed the limitations of classical computing. The results demonstrate that quantum algorithms can provide a new way forward for exact methods in quantum chemistry.This paper presents a simulated quantum computation of molecular energies using a recursive phase estimation algorithm. The authors demonstrate that quantum algorithms can compute molecular ground-state energies with significantly lower computational complexity than classical methods. They show that for small molecules like H2O and LiH, quantum simulations using a modest number of qubits can achieve high precision, matching the accuracy of classical full configuration interaction (FCI) calculations. The paper discusses the challenges of simulating molecular systems on classical computers, where the computational cost grows exponentially with system size. In contrast, quantum algorithms can achieve polynomial scaling. The authors use a modified phase estimation algorithm to compute the ground-state energy of H2O and LiH, showing that the number of qubits required scales linearly with the number of basis functions, while the number of quantum gates grows polynomially with the number of qubits. The authors also address the issue of preparing a good approximate ground-state wave function using an adiabatic state preparation algorithm. This method systematically improves the overlap between the initial wave function and the exact ground state, enhancing the success probability of the phase estimation algorithm. The paper also discusses the mapping of molecular wave functions to qubit states, showing that both direct and compact mappings can be used. The compact mapping, which restricts the wave function to a subspace with fixed electron number, is more efficient and requires fewer qubits. The authors demonstrate that quantum simulations of molecular systems can be performed with a relatively small number of qubits, making them feasible for practical applications. They show that for the H2O molecule, the recursive phase estimation algorithm can achieve high precision with only four qubits in the read-out register, while the compact mapping requires eight qubits for the singlet subspace. The paper concludes that quantum simulation algorithms with 30 to 100 qubits could be among the smallest applications of quantum computing that can exceed the limitations of classical computing. The results demonstrate that quantum algorithms can provide a new way forward for exact methods in quantum chemistry.