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From noisy observables to accurate ground state energies: a quantum classical signal subspace approach with denoising

Hardeep Bassi, Yizhi Shen, Harish S. Bhat, Roel Van Beeumen

TL;DR

This work tackles robust ground state energy estimation from noisy quantum measurements on NISQ devices. It introduces FDODMD, a hybrid quantum-classical pipeline that applies Fourier-domain denoising to a time-series of quantum observables and then uses observable dynamic mode decomposition with signal stacking (MODMD) to identify leading eigenfrequencies associated with the GSE. The authors provide formal denoising bounds and perturbation analyses, and demonstrate that FDODMD achieves chemical-accuracy convergence with substantially fewer quantum resources than baseline ODMD, especially under high noise and low ground-state overlap. The approach shows strong potential for practical quantum chemistry on near-term hardware by delivering accurate spectral estimation with reduced quantum overhead. The method also accommodates zero-padding to boost spectral resolution and offers robust performance across a range of hyperparameters, which is crucial for real-world deployment on imperfect devices.

Abstract

We propose a hybrid quantum-classical algorithm for ground state energy (GSE) estimation that remains robust to highly noisy data and exhibits low sensitivity to hyperparameter tuning. Our approach -- Fourier Denoising Observable Dynamic Mode Decomposition (FDODMD) -- combines Fourier-based denoising thresholding to suppress spurious noise modes with observable dynamic mode decomposition (ODMD), a quantum-classical signal subspace method. By applying ODMD to an ensemble of denoised time-domain trajectories, FDODMD reliably estimates the system's eigenfrequencies. We also provide an error analysis of FDODMD. Numerical experiments on molecular systems demonstrate that FDODMD achieves convergence in high-noise regimes inaccessible to baseline methods under a limited quantum computational budget, while accelerating spectral estimation in intermediate-noise regimes. Importantly, this performance gain is entirely classical, requiring no additional quantum overhead and significantly reducing overall quantum resource demands.

From noisy observables to accurate ground state energies: a quantum classical signal subspace approach with denoising

TL;DR

This work tackles robust ground state energy estimation from noisy quantum measurements on NISQ devices. It introduces FDODMD, a hybrid quantum-classical pipeline that applies Fourier-domain denoising to a time-series of quantum observables and then uses observable dynamic mode decomposition with signal stacking (MODMD) to identify leading eigenfrequencies associated with the GSE. The authors provide formal denoising bounds and perturbation analyses, and demonstrate that FDODMD achieves chemical-accuracy convergence with substantially fewer quantum resources than baseline ODMD, especially under high noise and low ground-state overlap. The approach shows strong potential for practical quantum chemistry on near-term hardware by delivering accurate spectral estimation with reduced quantum overhead. The method also accommodates zero-padding to boost spectral resolution and offers robust performance across a range of hyperparameters, which is crucial for real-world deployment on imperfect devices.

Abstract

We propose a hybrid quantum-classical algorithm for ground state energy (GSE) estimation that remains robust to highly noisy data and exhibits low sensitivity to hyperparameter tuning. Our approach -- Fourier Denoising Observable Dynamic Mode Decomposition (FDODMD) -- combines Fourier-based denoising thresholding to suppress spurious noise modes with observable dynamic mode decomposition (ODMD), a quantum-classical signal subspace method. By applying ODMD to an ensemble of denoised time-domain trajectories, FDODMD reliably estimates the system's eigenfrequencies. We also provide an error analysis of FDODMD. Numerical experiments on molecular systems demonstrate that FDODMD achieves convergence in high-noise regimes inaccessible to baseline methods under a limited quantum computational budget, while accelerating spectral estimation in intermediate-noise regimes. Importantly, this performance gain is entirely classical, requiring no additional quantum overhead and significantly reducing overall quantum resource demands.
Paper Structure (27 sections, 1 theorem, 58 equations, 15 figures)

This paper contains 27 sections, 1 theorem, 58 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Related Work
  4. Organization
  5. Background
  6. Acquiring data from a quantum circuit
  7. Noise models
  8. Fourier data interpretation
  9. Methods
  10. Observable dynamic mode decomposition (ODMD)
  11. Frequency domain denoising
  12. Signal stacking and multi-observable ODMD (MODMD)
  13. Zero-padding in time domain
  14. Data
  15. Motivating Example
  16. ...and 12 more sections

Key Result

Theorem 7.1

The $L^2$ error between the reconstructed data $\mathbf{r}$ and noiseless data $\mathbf{s}$ is bounded above by

Figures (15)

  • Figure 1: Total workflow of FDODMD as outlined in \ref{['sec:methods']}. (Top left) The Hadamard test is used to obtain noisy measurements of the data (\ref{['eqn:doverlapnoisy']}). (Top right) The denoising procedure is then applied by taking $R$ different truncation factors in (\ref{['eqn:dift']}). The data is then stacked together according to (\ref{['eqn:signalstacked']}). (Bottom right) Here we illustrate the thresholding and denoised reconstruction procedures described in \ref{['sect:denoising']}. (Bottom left) The stacked data is used to estimate the GSE, as described in \ref{['sect:modmd']}.
  • Figure 2: In the presence of depolarizing error and shot noise, sharp peaks in the frequency domain are obfuscated. We plot the magnitude of the DFT of the noisy time series generated from (\ref{['eqn:depolarizing']}) with $(\theta, \Delta t, K_{\text{max}}, p_0) = (0.05, 1.0, 300, 0.2)$ and $\xi(t)\sim \mathcal{N}(0, \epsilon=0.01)$.
  • Figure 3: ODMD displays advantages in estimation and denoising for Cr$_2$ under depolarizing error and shot noise. See the main text in \ref{['sec:depolarizing']} for details on data generation. To compute absolute errors as a function of data length $K$, we follow the testing protocol described in \ref{['sec:data']}. (Left) Convergence behavior of basic DFT-based spectral estimation (stars) and ODMD (circles). The dashed black line indicates chemical accuracy (absolute error $< 10^{-3}$.) ODMD converges quickly to chemical accuracy, in stark contrast to spectral estimation. (Right) Convergence of the estimate $\tilde{\theta}$ of the global depolarizing error parameter $\theta$ using ODMD. Inset in the upper right is the noisy data $\{d(t_k)\}$, the underlying signal $\{s(t_k)\}$, and the ODMD-reconstructed dynamics $\{r(t_k)\}$. See \ref{['sec:depolarizing']} for details on how the reconstruction is computed. We see favorable agreement between the reconstruction and the noiseless signal, with deviations at long times due to imperfect denoising.
  • Figure 4: For a Cr$_2$ problem setting (low overlap $p_0=0.2$ and moderate noise standard deviation $\epsilon=0.1$) where both baseline ODMD (circles) and DFT-based spectral estimation (stars) fail to converge (e.g., for data length $K \leq 1000$), we see that FDODMD (squares) converges. Convergence is defined as estimating the GSE to within $10^{-3}$ (dashed line) of its true value. Data and algorithm parameters are detailed in \ref{['sec:accelbaseline']}.
  • Figure 5: Convergence comparison of FDODMD (circles) and baseline ODMD (triangles) for Cr$_2$. We choose $K_{\text{max}} = 1500$, $\Delta t = 1.0$, and $p_0 \in\{0.15, 0.2, 0.72\}$. Both methods process noisy data given by (\ref{['eqn:noisyoverlap1']}) with $\xi(t_k) \sim \mathcal{N}(0, \epsilon=0.1)$. For FDODMD, we use (\ref{['eqn:dift']}) with $\gamma \in \{1.0, 1.5, 2.0, 2.5, 3.0, 3.5\}$ to generate 6 denoised realizations. All the denoised realizations are then stacked in conjunction with the noisy data according to (\ref{['eqn:signalstacked']}). For both FDODMD and ODMD, we set $\delta = \epsilon$. Note the accelerated convergence to chemical accuracy offered by FDODMD for the lower overlap cases.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Theorem 7.1