Spectral Filtering for Learning Quantum Dynamics

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.

💡 Research Summary

The paper tackles the notoriously hard problem of learning the dynamics of high‑dimensional quantum systems, whose Hilbert space grows exponentially with the number of qubits. Rather than attempting full system identification (i.e., reconstructing the Liouvillian, input, and output matrices), the authors focus on the predictive task: given a stream of past control inputs and measurement outcomes, predict the next measurement. This “improper learning” perspective aligns with recent online learning frameworks and sidesteps the exponential blow‑up inherent in quantum process tomography.

In the linear‑response regime, the quantum evolution can be vectorized into a classical linear time‑invariant (LTI) system
(y_t = \sum_{\tau=1}^{W} C A^{\tau-1} B u_{t-\tau}),
where (A) is the Liouvillian superoperator, (B) the control superoperator, and (C) the observable functional. The authors impose a spectral sector constraint on (A): all eigenvalues lie in (\mathcal{C}_\beta = {z\in\mathbb{C}\mid |z|\le 1,; |\arg(z)|\le \beta}). This guarantees stability and bounds the maximal oscillation frequency by (\beta).

The central theoretical construct is the “Quantum Information Matrix” (Z_W(\beta)), defined as the Gram matrix of all monomial trajectories (\mu_W(z) =