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In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states.\nTo address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant factor improvement in the number of samples required to detect a specified change in CDF.\nOur numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.<\/jats:p>","DOI":"10.22331\/q-2025-04-01-1682","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T13:13:36Z","timestamp":1743513216000},"page":"1682","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7461-3342","authenticated-orcid":false,"given":"Oriel","family":"Kiss","sequence":"first","affiliation":[{"name":"Xanadu, Toronto, ON, M5G2C8, Canada"},{"name":"European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland"},{"name":"Department of Nuclear and Particle Physics, University of Geneva, Geneva 1211, Switzerland"},{"name":"Physics Department, University of Trento, Via Sommarive 14, I-38123 Trento, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7020-0305","authenticated-orcid":false,"given":"Utkarsh","family":"Azad","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G2C8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1904-4137","authenticated-orcid":false,"given":"Borja","family":"Requena","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G2C8, Canada"},{"name":"ICFO \u2013 Institut de Ci\u00e8ncies Fot\u00f2niques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8334-1120","authenticated-orcid":false,"given":"Alessandro","family":"Roggero","sequence":"additional","affiliation":[{"name":"Physics Department, University of Trento, Via Sommarive 14, I-38123 Trento, Italy"},{"name":"INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Wakeham","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G2C8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan Miguel","family":"Arrazola","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G2C8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Julia Kempe, Alexei Kitaev, and Oded Regev. 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