Tractable Infinite-Horizon Stochastic Model Predictive Control for Quantum Filtering via Eigenstate Reduction

TL;DR

The paper addresses robust control of quantum filtering dynamics under continuous measurement by formulating an infinite-horizon SMPC that becomes tractable through almost-sure eigenstate reduction. By reducing the stochastic objective to a deterministic fidelity-based cost evaluated on a one-step averaged state, it avoids Monte Carlo scenario trees and enables scalable control for multi-level quantum systems. A stability analysis and PMP-based synthesis are developed, with numerical demonstrations on angular-momentum systems and an Ising-type model showing improved convergence and scalability. The approach offers practical impact for real-time quantum feedback control under uncertainty and noise, balancing long-term objectives with computational feasibility.

Abstract

Model predictive control has shown potential to enhance the robustness of quantum control systems. In this work, we propose a tractable Stochastic Model Predictive Control (SMPC) framework for finite-dimensional quantum systems under continuous-time measurement and quantum filtering. Using the almost-sure eigenstate reduction of quantum trajectories, we prove that the infinite-horizon stochastic objective collapses to a fidelity term that is computable in closed form from the one-step averaged state. Consequently, the online SMPC step requires only deterministic propagation of the filter and a terminal fidelity evaluation. An advantage of this method is that it eliminates per-horizon Monte Carlo scenario sampling and significantly reduces computational load while retaining the essential stochastic dynamics. We establish equivalence and mean-square stability guarantees, and validate the approach on multi-level and Ising-type systems, demonstrating favorable scalability compared to sampling-based SMPC.

Figures (5)

• Figure 1: Block diagram for measurement-based feedback. The homodyne output $y(t)$ is filtered to obtain $\rho_t$, which the controller uses to compute the drive $u(t)$ applied to the plant; see \ref{['eqn:SME']}, \ref{['eqn:dy']}, \ref{['eqn:innovations']}. This corresponds to the measurement–feedback setting in sayrin2011realamini2014stabilityBouten2007An.
• Figure 2: Expected fidelity $\mathbb{E}[\Tr(\rho(t)\bar{\rho}_f)]$ over time. The SMPC strategy (blue) shows significantly improved convergence toward the target state compared to the uncontrolled case (red).
• Figure 3: Three-level system, average fidelity under Lyapunov feedback liang2019exponential, standard SMPC, and the proposed controller based on Theorem \ref{['thm:SMPC_equivalence']}. The proposed method matches the standard SMPC and avoids scenario sampling.
• Figure 4: Illustrative numerical example for $j=5$ (11 levels), corresponding to the permutation-symmetric subspace of a 10-qubit system. The process starts at $\rho_0=\mathbb{I}_d/d$; the averaged fidelity $\mathbb{E}[\Tr(\rho(t)\bar{\rho}_f)]$ over $1000$ trajectories shows sustained improvement under the proposed controller.
• Figure 5: Averaged fidelity $\mathbb{E}[\Tr(\rho_t\rho_f)]$ versus time under control with $H_u=\bigotimes_i Y_i$ and $L=\bigotimes_i Z_i$. The convergence to the target eigenstate verifies Theorem \ref{['thm:SMPC_equivalence']} under the condition of Lemma \ref{['lem:inv-subspace']}.

Theorems & Definitions (14)

• Lemma 1: Quantum State Reduction liang2019exponential
• Theorem 1: Equivalence of Infinite-Horizon SMPC
• proof
• Lemma 2
• proof
• Remark 1
• Proposition 1
• proof
• Remark 2: Control of Angular Momentum Systems
• Remark 3: Control of General Systems