Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Certified Quantum Schrödinger Control via Hierarchical Tucker Models

Nahid Binandeh Dehaghani, Rafal Wisniewski, A. Pedro Aguiar

Abstract

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.

Certified Quantum Schrödinger Control via Hierarchical Tucker Models

Abstract

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.
Paper Structure (13 sections, 6 theorems, 6 equations, 1 figure)

This paper contains 13 sections, 6 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Controlled Schrödinger Model
  3. Continuous and Semi-Discrete Schrödinger Dynamics
  4. Discrete-Time Map and Nominal Assumptions
  5. Low-Rank Hierarchical Tensor Representation and Fixed-Rank Truncation
  6. Hierarchical Tucker Representation and Ranks
  7. Fixed-Rank Truncation and Spectral Decay Assumptions
  8. HT-Truncated Closed-Loop Dynamics
  9. Truncated Closed-Loop Map and Perturbation Bounds
  10. Practical Stability and Rank--Accuracy Relations
  11. Controller Design on the Surrogate HT Model and Transfer to the Full System
  12. Numerical Validation
  13. Conclusion

Key Result

Proposition 1

Under Assumption ass:HT_decay, in the uniform-rank case $r_t\equiv r$, there exist constants $C_2,c'>0$ such that $\varepsilon_{\mathbf r}(\Psi)\le C_2 e^{-c'r},$ where the constants may depend on the dimension tree $\mathcal{T}$ and the decay constants in Assumption ass:HT_decay.

Figures (1)

  • Figure 1: Full-plant convergence and HT–truncated surrogate accuracy (4$\times$4 lattice).

Theorems & Definitions (15)

  • Remark 1: Sufficient conditions
  • Remark 2
  • Proposition 1: Static HT truncation bound
  • proof
  • Proposition 2: Uniform trajectory-level truncation bound
  • Proposition 3: Uniform exponential truncation bound
  • proof
  • Theorem 1
  • proof
  • Remark 3: Robust contraction interpretation
  • ...and 5 more