  Simulating quantum systems is one of the most promising tasks where quantum
computing can potentially outperform classical computing. However, the
robustness needed for reliable simulations of medium to large systems is beyond
the reach of existing quantum devices. To address this, Quantum Krylov Subspace
(QKS) methods have been developed, enhancing the ability to perform accelerated
simulations on noisy intermediate-scale quantum computers. In this study, we
introduce and evaluate two QKS methods derived from the QDavidson algorithm, a
novel approach for determining the ground and excited states of many-body
systems. Unlike other QKS methods that pre-generate the Krylov subspace through
real- or imaginary-time evolution, QDavidson iteratively adds basis vectors
into the Krylov subspace. This iterative process enables faster convergence
with fewer iterations and necessitates shallower circuit depths, marking a
significant advancement in the field of quantum simulation.
