通过分层Tucker模型实现认证的量子薛定谔控制
Certified Quantum Schrödinger Control via Hierarchical Tucker Models
AI总结：
本文提出通过分层Tucker模型实现量子薛定谔控制的鲁棒性框架，证明了固定秩截断后系统动力学的实用指数稳定性，并建立了控制器在完整系统上保持跟踪保证的条件。
摘要：
高维薛定谔系统由于张量积离散化导致状态指数增长，使直接控制器合成和实时闭环模拟计算困难。分层Tucker（HT）张量表示提供可扩展的低秩近似，但固定秩截断对闭环稳定性的影响尚不明确。本文开发了用于采样数据反馈控制的局部鲁棒性框架，通过将每个截断视为名义闭环的有界、秩依赖扰动，并假设局部相不变收缩证书和轨迹级分层谱衰减，证明HT投影动力学在实践中是指数稳定的：轨迹收敛到一个与维度无关的管状区域，其半径随指定秩减小。进一步获得显式的对数秩-精度关系，并建立在HT截断近似模型上设计的控制器在部署到完整系统时保留实用指数跟踪保证的条件，同时建立量化由此产生的近似到植物不匹配的显式界限。一个紧凑晶格示例展示了该框架的适用性。
英文摘要：
High-dimensional Schrödinger 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.
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