Efficient Measurement-Driven Eigenenergy Estimation with Classical Shadows

TL;DR

Efficiently extracting low-lying Hamiltonian eigenenergies on near-term devices is challenging due to circuit depth and noise. This work introduces MODMD, a measurement-driven framework that combines observable dynamic mode decomposition with classical shadows to build a rich, low-cost signal subspace from many observables measured with shallow quantum circuits. Theoretical guarantees based on a Koopman-operator viewpoint establish exponential convergence in favorable noise regimes, while error analyses quantify the impact of sampling, Trotterization, and depolarizing noise. Numerical demonstrations on condensed-matter and quantum-chemistry models show faster convergence for excited states and robust, scalable spectral estimates with reduced quantum resources. The approach offers a practical path toward efficient eigenenergy estimation and prospective extensions to longer-time dynamics on next-generation quantum hardware.

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 multi-observable dynamic mode decomposition (MODMD), which combines the observable dynamic mode decomposition, 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, thus marking a new application 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 $\exp(- ΔE t_{\rm max})$, where $ΔE$ is the Hamiltonian spectral gap and $t_{\rm max}$ 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.

Figures (19)

• Figure 1: MODMD for eigenenergy estimation. MODMD collects the expectations, ${\rm Tr} \left[ \rho (k\Delta t) \Gamma_{O} \right] = \Re \bra{\phi_0} O e^{-i H k\Delta t}\ket{\phi_0}$, with respect to a simple reference state $\ket{\phi_0}$. This data can be measured efficiently on a quantum processor through Hamiltonian simulations combined with shadow tomographic techniques, where the shadow circuits can be shallow with a depth logarithmic in the number of qubits. MODMD then constructs a pair of block Hankel matrices $(\mathbf{X},\mathbf{X}^{\prime})$, and computes the DMD system matrix $A$ that adopts a block companion structure. The eigenvalues $\tilde{\lambda}_n$ of $A$ converge to the true eigenphases $\lambda_n$ as the size of $\mathbf{X}$ and $\mathbf{X}^{\prime}$ increases. The low-lying energies $E_n$ are estimated as angles of $\tilde{\lambda}_{n}$, $\Tilde{E}_n = -\frac{1}{\Delta t}\text{arg}(\tilde{\lambda}_n)$.
• Figure 2: Convergence of eigenenergies for the transverse-field Ising model (TFIM). To obtain eigenenergy estimates $\tilde{E}_n$, we fix the (M)ODMD parameters $\frac{K}{d}=\frac{5}{2}$ and $\Tilde{\delta}=10^{-2}$ for constructing and thresholding the pair of data matrices $\mathbf{X},\mathbf{X}' \in \mathbb{R}^{d I \times (K+1)}$. Gaussian $\mathcal{N}(0,\varepsilon_{\rm noise}^2)$ noise with $\varepsilon_{\rm noise} = 10^{-3}$ is added independently to the real or/and imaginary parts of the matrix elements. The absolute errors, $\lvert \tilde{E}_n - E_n \rvert$, in the first four eigenenergies of the TFIM Hamiltonian are shown with respect to $K$ proportional to the non-dimensional maximal simulation time. The reference state $\ket{\phi_0}$ is an equal superposition of six computational basis states (see \ref{['app:reference state']}) and we employ a time step of $\Delta t \approx 0.08$. Shading above the solid/dashed lines shows the standard deviation across trials for each quantity. For TFIM, the model Hamiltonian is of size $32768 \times 32768$ and the largest linear system in the corresponding MODMD LS problem has size $1400 \times 501$. Left. Absolute errors from the multi-observable (MODMD) algorithm with $I = 7$ distinct observables, where the convergence results are averaged over 20 trials, each involving a Gaussian noise realization and a selection of $I-1$ random observables. Right. Absolute errors from the single-observable (ODMD) algorithm where the convergence results are averaged over 20 trials, each involving a Gaussian noise realization.
• Figure 3: Convergence of the first excited state energy of the TFIM. Absolute error in the first excited state energy is shown as a function of the spectral gap between the ground and the first excited state for fixed $K=500$. The vertical dotted line marks the noise level $\varepsilon_{\rm noise} = 10^{-3}$. Convergence results are averaged over 20 trials each involving a Gaussian noise realization and, in the MODMD case, also a selection of $I-1$ random observables. The shading above the solid/dashed lines shows the standard deviation across trials for each quantity.
• Figure 4: Convergence of eigenenergies for the lithium hydride (LiH) molecule. To obtain eigenenergy estimates $\tilde{E}_n$, we fix the (M)ODMD parameters $\frac{K}{d}=\frac{5}{2}$ and $\Tilde{\delta}=10^{-2}$ for constructing and thresholding the pair of data matrices $\mathbf{X},\mathbf{X}' \in \mathbb{R}^{d I \times (K+1)}$. Gaussian $\mathcal{N}(0,\varepsilon_{\rm noise}^2)$ noise with $\varepsilon_{\rm noise} = 10^{-3}$ is added independently to the real or/and imaginary parts of the matrix elements. The reference state $\ket{\phi_0}$ is a superposition of six Slater determinants (see \ref{['app:reference state']}) and we employ a time step of $\Delta t\approx0.39$. The absolute errors, $\lvert \tilde{E}_n - E_n \rvert$, in the first four eigenenergies of the LiH Hamiltonian are shown with respect to $K$ proportional to the non-dimensional maximal simulation time. Convergence results are averaged over 20 trials of Gaussian noise realizations with shading above the solid/dashed lines showing the standard deviation across trials for each quantity. For LiH, the model Hamiltonian is of size $1024 \times 1024$ and the largest linear system in the corresponding MODMD LS problem has size $1400 \times 501$. Left. Absolute errors from the multi-observable (MODMD) algorithm with $I = 7$ distinct observables. Right. Absolute errors from the single-observable (ODMD) algorithm.
• Figure 5: Convergence of eigenergies for LiH under varying noise level. Absolute errors in the four lowest eigenenergies of the LiH Hamiltonian are plotted against the noise level $\varepsilon_{\rm noise}$ for fixed $K=500$ and $\Tilde{\delta} = 10 \varepsilon_{\rm noise}$. All convergence results reflect an average over 20 trials of Gaussian noise realizations. The solid lines correspond to the multi-observable (MODMD) algorithm with $I = 7$ distinct observables and the dashed lines to the single-observable (ODMD) algorithm.