Stabilizer configuration interaction: Finding molecular subspaces with error detection properties

In this work, we explore a new approach to designing both algorithms and error detection codes for preparing approximate ground states of molecules. We propose a classical algorithm to find the optimal stabilizer state by using excitations of the Hartree-Fock state, followed by constructing quantum error-detection codes based on this stabilizer state using codeword-stabilized codes. Through various numerical experiments, we confirm that our method finds the best stabilizer approximations to the true ground states of molecules up to 36 qubits in size. Additionally, we construct generalized stabilizer states that offer a better approximation to the true ground states. Furthermore, for a simple noise model, we demonstrate that both the stabilizer and (some) generalized stabilizer states can be prepared with higher fidelity using the error-detection codes we construct. Our work represents a promising step toward designing algorithms for early fault-tolerant quantum computation.