We investigate the feasibility of early fault-tolerant quantum algorithms for estimating ground-state energies. Specifically, we focus on computing the cumulative distribution function (CDF) of the Hamiltonian’s spectral measure and identifying its discontinuities, as illustrated in the figure. In practice, we encounter two main issues: the CDF becomes continuous for initial states with exponential support, and estimating its overlap with the true ground state is difficult.
 
 To address these issues, we propose a signal processing method that automatically extracts energy estimates without having to identify the individual discontinuities. Rather than aiming for exact ground-state energies, our approach refines classical estimates by focusing on the low-energy support of the initial state. We provide resource estimates showing a constant factor reduction in the number of samples needed to detect changes in the CDF.
 
 We test our method on a 26-qubit fully connected Heisenberg model using a low-bond-dimension DMRG initial state. Our results show that quantum predictions align well with DMRG-converged energies while requiring significantly fewer measurements than what theoretical bounds suggest. These findings indicate that CDF-based quantum algorithms can be a practical and resource-efficient alternative to quantum phase estimation, particularly in settings with limited quantum resources.