Spectral Filtering for Learning Quantum Dynamics

TL;DR

The problem addresses predicting quantum dynamics under linear response in high-dimensional systems, where full system identification is infeasible. The authors introduce Quantum Spectral Filtering, an improper-learning approach that uses a Slepian (DPSS) basis assembled from the Quantum Information Matrix to compress history into an effective subspace of dimension $k^* = \left\lceil \frac{\beta}{\pi} W \right\rceil$, enabling dimension-free learning. They prove an information-cutoff phenomenon and provide rigorous bounds showing the prediction error decays as $\tilde{O}(\frac{\beta W \log^2 T}{T})$ while depending only on $k^*$, not the ambient Hilbert space, with a matching lower bound in the regime of bounded bandwidth. The method is validated via Hamiltonian tomography and ablation studies that demonstrate the sharp phase transition at $k^*$ and the superiority of the Slepian basis over standard Fourier modes. This work offers a scalable avenue for forecasting quantum evolution in large systems without reconstructing the full Liouvillian, with potential impact on quantum sensing, control, and simulation.

Abstract

Learning high-dimensional quantum systems is a fundamental challenge that notoriously suffers from the curse of dimensionality. We formulate the task of predicting quantum evolution in the linear response regime as a specific instance of learning a Complex-Valued Linear Dynamical System (CLDS) with sector-bounded eigenvalues -- a setting that also encompasses modern Structured State Space Models (SSMs). While traditional system identification attempts to reconstruct full system matrices (incurring exponential cost in the Hilbert dimension), we propose Quantum Spectral Filtering, a method that shifts the goal to improper dynamic learning. Leveraging the optimal concentration properties of the Slepian basis, we prove that the learnability of such systems is governed strictly by an effective quantum dimension $k^*$, determined by the spectral bandwidth and memory horizon. This result establishes that complex-valued LDSs can be learned with sample and computational complexity independent of the ambient state dimension, provided their spectrum is bounded.

Figures (4)

• Figure 1: Quantum Information Spectrum. Singular values of $Z_W(\beta)$ for $W=100,200,400$ on a log scale. The red vertical lines indicate the theoretical Quantum Sampling limit $k^* = \lceil \frac{\beta}{\pi} W \rceil$, marking the boundary between learnable signal and noise.
• Figure 2: Robust Hamiltonian Tomography. Test MSE of the Quantum Spectral Filtering algorithm learning the dynamics of random quantum systems with window size $W=100$. We vary the filter bank size $K$ across three spectral bands: $\beta=0.2\pi$ (red), $\beta=0.5\pi$ (orange), and $\beta=0.9\pi$ (navy). Solid lines represent the mean MSE over 20 independent trials, while shaded regions indicate the 95% confidence interval. Vertical dashed lines mark the theoretical effective dimension $k^* = \lceil \frac{\beta}{\pi} W \rceil$. The sharp phase transition at $k^*$ confirms that the subspace $\mathcal{V}_{k^*}$ captures the system's dynamics with high precision, while the negligible variance post-cutoff validates the universality of this limit.
• Figure 3: Effects of the Hidden Dimension.
• Figure 4: Basis Sensitivity: Slepian vs. Fourier.

Theorems & Definitions (18)

• Definition 3.1: Effective Quantum Dimension
• Theorem 3.2: Information Cutoff
• proof : Proof Sketch
• Theorem 5.1: Generative Learning Guarantee
• Corollary 5.2: Fast Convergence Rates
• Theorem 6.1: Proof of Lemma E.4 marsden2025universal
• Lemma 6.2: Spectral Decay
• proof : Detailed Proof of Theorem \ref{['thm:regret']}
• Theorem 1.1
• proof