Efficient Measurement-Driven Eigenenergy Estimation with Classical Shadows

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>exp</a:mi> <a:mo></a:mo> <a:mo stretchy="false">(</a:mo> <a:mo>−</a:mo> <a:mi mathvariant="normal">Δ</a:mi> <a:mi>E</a:mi> <a:msub> <a:mi>t</a:mi> <a:mi>max</a:mi> </a:msub> <a:mo stretchy="false">)</a:mo> </a:math> , where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi mathvariant="normal">Δ</f:mi> <f:mi>E</f:mi> </f:math> is the Hamiltonian spectral gap and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>t</i:mi> <i:mi>max</i:mi> </i:msub> </i:math> 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.