Certified Quantum Schrödinger Control via Hierarchical Tucker Models

1. Problem Statement

The paper addresses the challenge of feedback control for high-dimensional quantum systems governed by the Schrödinger equation.

• The Curse of Dimensionality: When a $p$-dimensional spatial domain is discretized, the state dimension $N$ grows exponentially with the number of degrees of freedom (e.g., $N = d^n$). This makes direct controller synthesis and real-time closed-loop simulation computationally intractable.
• The Limitation of Low-Rank Surrogates: While Hierarchical Tucker (HT) tensor representations offer a scalable way to approximate high-dimensional states by exploiting low-rank structures, they require fixed-rank truncation at every time step.
• The Core Gap: Existing literature focuses on HT methods for open-loop simulation. It remains unclear how the approximation errors introduced by fixed-rank truncation affect the stability and tracking performance of a closed-loop feedback system. Specifically, does a controller designed on a truncated surrogate model guarantee stability when applied to the full system?

2. Methodology

The authors develop a local robustness framework for sampled-data feedback control using fixed-rank HT projections. The methodology proceeds in four logical stages:

A. System Modeling

• Continuous to Discrete: The continuous controlled Schrödinger equation is discretized on a tensor-product grid, resulting in a high-dimensional unitary evolution.
• Tensorization: The state vector is reshaped into an order-$n$ tensor. The system is assumed to possess locality (finite-range interactions), consistent with Lieb-Robinson bounds in many-body physics.
• Control Law: A sampled-data state-feedback law $u_k = \kappa(\Psi_k)$ is applied, where the control is held constant over intervals $\Delta t$.

B. Hierarchical Tucker (HT) Truncation

• Surrogate Dynamics: The closed-loop dynamics are approximated by applying a fixed-rank HT truncation operator $\hat{\Pi}_r$ after each time-step propagation.
• Perturbation View: The truncation is modeled as a bounded additive perturbation ($e_k$) to the nominal closed-loop dynamics.
• Spectral Decay Assumption: The framework assumes that the singular values of the state tensor decay exponentially (Assumption 4) or algebraically along the trajectory. This ensures that the truncation error $\|e_k\|$ decreases exponentially with the prescribed rank $r$.

C. Stability Analysis

• Nominal Contraction: The analysis assumes the nominal (full-rank) closed-loop system admits a local contraction certificate. This means the system contracts toward a reference set $\mathcal{R}$ in a phase-invariant metric $dist_*$ (e.g., distance modulo global phase).
• Robustness to Perturbation: By treating truncation as a bounded disturbance, the authors prove that the HT-truncated system remains practically exponentially stable. The trajectories converge to a "tube" around the reference set, where the tube radius depends on the truncation error.

D. Surrogate-to-Plant Transfer

• The paper analyzes the scenario where a controller is synthesized on the surrogate model ($\Psi^r_k$) but applied to the full plant ($\Psi_k$).
• It derives bounds on the mismatch between the true plant evolution and the surrogate evolution, accounting for both truncation errors and the fact that the controller acts on an approximate state.

3. Key Contributions

1. Structured Perturbation Analysis: The paper interprets HT truncation not just as an approximation error, but as a structured, rank-dependent perturbation whose norm decays exponentially with the hierarchical rank.
2. Practical Exponential Stability: Under the assumption of nominal contraction, the authors prove that the HT-projected dynamics are practically exponentially stable. Trajectories converge to a dimension-independent tube whose radius scales as $O(e^{-c'r})$.
3. Logarithmic Rank–Accuracy Trade-off: A significant theoretical result is the derivation of a logarithmic relationship between the required rank $r$ and the desired tracking tolerance $\eta$. To achieve tolerance $\eta$, the rank need only grow as $r \sim O(\log(1/\eta))$. This implies that high accuracy can be achieved with relatively low ranks.
4. Surrogate-to-Plant Transfer Theorem: The authors establish conditions under which controllers designed on the truncated surrogate retain practical exponential tracking guarantees when deployed on the full high-dimensional system. They provide explicit bounds quantifying the "surrogate-to-plant" mismatch.
5. Algorithmic Framework: A projected closed-loop scheme is proposed using operator splitting and fixed-rank HT truncation, ensuring the control loop remains computationally tractable.

4. Results

• Theoretical Bounds:
  • Theorem 1 & 2: Prove that the asymptotic tracking error is bounded by $\frac{M_* \bar{\varepsilon}_r}{1-\rho}$, where $\bar{\varepsilon}_r \leq C_2 e^{-c'r}$. This confirms that increasing the rank exponentially reduces the steady-state error.
  • Theorem 3: Demonstrates that if the surrogate controller is contractive, the full plant trajectory remains within a bounded distance of the reference set, provided the one-step transfer error is bounded.
• Numerical Validation:
  • Setup: A $4 \times 4$ spin-1/2 lattice (Hilbert dimension $2^{16}$) with a Heisenberg drift Hamiltonian and local control.
  • Findings:
    • The full plant converges to the target state.
    • The HT-surrogate accurately tracks the full plant.
    • Rank Dependence: For ranks $r \geq 8$, the surrogate dynamics become nearly indistinguishable from the full system. The "surrogate-plant tube" (gap between full and surrogate states) drops sharply from $\approx 1.4 \times 10^{-1}$ at $r=2$ to $\lesssim 5 \times 10^{-4}$ at $r=64$.
    • Control Signals: Once the rank is sufficient ($r \geq 8$), the control signals become rank-insensitive, confirming the efficacy of the low-rank approximation.

5. Significance

• Bridging Simulation and Control: This work moves hierarchical tensor methods from purely open-loop simulation tools to certified control frameworks. It provides the mathematical guarantees necessary to trust low-rank surrogates in safety-critical or high-precision quantum control applications.
• Scalability: By proving that the required rank grows only logarithmically with accuracy, the paper demonstrates that high-dimensional quantum control is computationally feasible without sacrificing stability guarantees.
• Robustness Certification: The framework offers a rigorous way to quantify the "cost" of model reduction. Engineers can now explicitly calculate the rank needed to meet a specific tracking tolerance, rather than relying on heuristic trial-and-error.
• Applicability: The approach is relevant not only for quantum control but also for any high-dimensional PDE control problem where low-rank tensor structures (like HT or Tensor Train) are applicable and locality assumptions hold.

In summary, the paper provides a rigorous stability-certified framework that enables the use of fixed-rank Hierarchical Tucker models for the feedback control of high-dimensional quantum systems, proving that such approximations preserve stability and tracking performance with exponentially decaying errors as the rank increases.