Quantum electronic structure with early fault-tolerant quantum computers

Quantum phase estimation (QPE) run on fault-tolerant quantum devices has potential to allow the investigation of molecular electronic structure beyond what is currently achievable using classical methods. However, for chemical systems of interest the resource requirements of QPE can be very demanding. The multimodal, multi-level quantum complex exponential least squares (MM-QCELS) method [Quantum 7, 1136 (2023)] was developed to meet the likely constraints of "early fault-tolerant" quantum computers, which may be limited to few logical qubits and conservative circuit depths. Here we apply MM-QCELS to the molecular electronic structure problem and show that it can achieve levels of accuracy similar to those of QPE for the ground state using shallower circuits. We also demonstrate its ability to simultaneously estimate ground and excited states using total resources similar to those required by QPE to find only the ground state. We investigate these results across several initial state preparation methods, ranging from simple classical mean-field approaches, to more complex hybrid quantum algorithms such as the non-orthogonal quantum eigensolver [PRX Quantum 4, 030307 (2023)].