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  • Open Access

Efficient Measurement-Driven Eigenenergy Estimation with Classical Shadows

Yizhi Shen1,*,†, Alex Buzali2,†, Hong-Ye Hu2, Katherine Klymko3, Daan Camps3, Susanne F. Yelin2, and Roel Van Beeumen1

  • *Contact author: yizhis@lbl.gov
  • †These authors contributed equally to this work.
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PRX Quantum 7, 010328 – Published 10 February, 2026

DOI: https://doi.org/10.1103/74s6-3jsz

Abstract

Quantum algorithms exploiting real-time evolution under a target Hamiltonian have demonstrated remarkable efficiency in extracting key spectral information. However, the broader potential of these methods, particularly beyond ground-state calculations, is underexplored. In this work, we introduce the framework of multiobservable dynamic mode decomposition (MODMD), which combines the observable dynamic mode decomposition (DMD), a measurement-driven eigensolver tailored for near-term implementation, with classical shadow tomography. MODMD leverages random scrambling in the classical shadow technique to construct, with exponentially reduced resource requirements, a signal subspace that encodes rich spectral information. Notably, we replace typical Hadamard-test circuits with a protocol designed to predict low-rank observables, thereby broadening the use of classical shadow tomography for predicting many low-rank observables. We establish theoretical guarantees on the spectral approximation from MODMD, taking into account distinct sources of error. In the ideal case, we prove that the spectral error scales as exp(−ΔEtmax), where ΔE is the Hamiltonian spectral gap and tmax is the maximal simulation time. This analysis provides a rigorous justification of the rapid convergence observed across simulations. To demonstrate the utility of our framework, we consider its application to fundamental tasks, such as determining the low-lying, i.e., ground or excited, energies of representative many-body systems. Our work paves the path for efficient designs of measurement-driven algorithms on near-term and early fault-tolerant quantum devices.

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Popular Summary

Despite their ability to retrieve key spectral information, the potential of quantum computers beyond ground-state calculations remains underexplored. In this work, we introduce multiobservable dynamic mode decomposition (MODMD), a hybrid quantum-classical framework that employs quantum real-time evolution and randomized strategy to extract the ground- and excited-state properties of many-body systems. Our formulation unifies two complementary ingredients into a powerful and versatile approach. The first is the DMD, a data-driven method often used in fluid dynamics to characterize chaotic systems by taking snapshots at regular intervals and using them to extrapolate future behavior. The second is classical shadow tomography, a protocol that allows for the determination of many properties of a quantum system with relatively few measurements. By leveraging randomized measurements, MODMD significantly reduces the resource requirements while maintaining reliable accuracy in accessing the lowest energy levels of a quantum system.

In addition to laying the theoretical groundwork for the algorithm in the paper, we provide extensive numerical simulations to demonstrate its utility for different quantum systems and compare its performance with similar methods. These results represent an important step toward scalable eigenenergy estimation beyond the reach of classical simulation alone and pave the way for further exploration in this direction.

Article Text

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PRX Quantum 7, 010328– Published 10 February, 2026
  • Received 18 November 2024
  • Revised 25 July 2025
  • Accepted 23 December 2025
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