Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations

Abstract: Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems, including $\text{H}_2$ and $\text{LiH}$ molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.