Oriel Kiss, Utkarsh Azad, Alessandro Roggero, Borja Requena, David Wakeham and Juan Miguel Arazola
We explore the practicality of early fault-tolerant quantum algorithms, focusing on ground state problems. Specifically, we address the computation of the cumulative distribution function (CDF) of the spectral measure and the identification of its discontinuities. Scaling to bigger system sizes unveils three challenges: the smoothness of the CDF for large supports, the absence of tight lower bounds on the overlap with the actual ground state, and the complexity of preparing high-quality
initial states. To tackle these challenges, we introduce a signal processing technique for identifying the inflection point of the CDF. Furthermore, we offer quantitative resource estimates for the maximum time evolution needed to approximate the error function and the maximum number of samples required to identify a jump of a given size. Our claims are supported by numerical experiments conducted on a 26-qubit fully connected Heisenberg model using a truncated density-matrix renormalization group (DMRG) initial state of low bond dimension. Results show that the DMRG-converged energy aligns well with the prediction obtained with the CDF. Hence, this algorithm emerges as a viable alternative for quantum phase estimation in resource-limited scenarios.