Certified Quantum Schr\"odinger Control via Hierarchical Tucker Models

TL;DR

This paper introduces a robustness framework for controlling high-dimensional Schrödinger systems using Hierarchical Tucker tensor models, ensuring stability and accuracy despite low-rank truncations.

Contribution

It develops a local robustness analysis for HT-based feedback control, linking fixed-rank truncation effects to stability and tracking guarantees in quantum systems.

Findings

HT truncation induces bounded perturbations that preserve practical exponential stability.

Explicit relation between tensor rank and control accuracy is established.

Controllers on HT surrogates maintain stability and tracking on full systems under certain conditions.

Abstract

High-dimensional Schr\"odinger systems arising from tensor-product discretizations suffer from exponential state growth, making direct controller synthesis and real-time closed-loop simulation computationally challenging. Hierarchical Tucker (HT) tensor representations offer scalable low-rank surrogates, but the impact of fixed-rank truncation on closed-loop stability is not well understood. This paper develops a local robustness framework for sampled-data feedback control implemented with fixed-rank HT projections. By viewing each truncation as a bounded, rank-dependent perturbation of the nominal closed loop, and assuming a local phase-invariant contraction certificate together with trajectory-level hierarchical spectral decay, we show that the HT-projected dynamics are practically exponentially stable: trajectories converge to a dimension-independent tube whose radius decreases with the prescribed rank. We further obtain an explicit logarithmic rank-accuracy relation and establish conditions under which controllers designed on the HT-truncated surrogate model retain practical exponential tracking guarantees when deployed on the full system, together with an explicit bound quantifying the resulting surrogate-to-plant mismatch. A compact lattice example demonstrates the applicability of the framework.