Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation

Oriel Kiss; Utkarsh Azad; Borja Requena; Alessandro Roggero; David Wakeham; Juan Miguel Arrazola

doi:10.22331/q-2025-04-01-1682

Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation

Oriel Kiss^1,2,3,4, Utkarsh Azad¹, Borja Requena^1,5, Alessandro Roggero^4,6, David Wakeham¹, and Juan Miguel Arrazola¹

¹Xanadu, Toronto, ON, M5G2C8, Canada
²European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland
³Department of Nuclear and Particle Physics, University of Geneva, Geneva 1211, Switzerland
⁴Physics Department, University of Trento, Via Sommarive 14, I-38123 Trento, Italy
⁵ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
⁶INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy

Published:	2025-04-01, volume 9, page 1682
Editor:	Yu Tong
Eprint:	arXiv:2405.03754v2
Doi:	https://doi.org/10.22331/q-2025-04-01-1682
Citation:	Quantum 9, 1682 (2025).

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Abstract

We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. 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.
To 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.
Our 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.

Featured image: $\textbf{Workflow of the Algorithm.}$
The approximated cumulative distribution function (CDF) is constructed by sampling time indices and estimating the corresponding Fourier moments on a quantum computer. The ground state energy is predicted by first identifying the inflection point using the ruptures algorithm, then validating it with an ANOVA-based scheme, and finally refining the estimate by locating the point of maximum gradient within a small neighborhood.

Popular summary

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.

► BibTeX data

@article{Kiss2025earlyfaulttolerant,
  doi = {10.22331/q-2025-04-01-1682},
  url = {https://doi.org/10.22331/q-2025-04-01-1682},
  title = {Early {F}ault-{T}olerant {Q}uantum {A}lgorithms in {P}ractice: {A}pplication to {G}round-{S}tate {E}nergy {E}stimation},
  author = {Kiss, Oriel and Azad, Utkarsh and Requena, Borja and Roggero, Alessandro and Wakeham, David and Arrazola, Juan Miguel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1682},
  month = apr,
  year = {2025}
}

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