eigenstate-reduction-quantum-smcp

name	eigenstate-reduction-quantum-smcp
description	Tractable Infinite-Horizon Stochastic Model Predictive Control for Quantum Filtering via Eigenstate Reduction. Uses almost-sure eigenstate reduction of quantum trajectories to collapse infinite-horizon stochastic objective to closed-form fidelity term. Eliminates per-horizon Monte Carlo sampling while retaining stochastic dynamics. Use when designing quantum SMPC controllers, quantum filtering-based control, mean-square stability analysis, or tractable stochastic optimal control for quantum systems.
metadata	{"arxiv_id":"2511.05916","published":"2025-11-08","authors":"Yunyan Lee, Ian R. Petersen, Daoyi Dong","tags":["quantum-control","MPC","stochastic-control","quantum-filtering","eigenstate-reduction","optimal-control","systems-engineering"]}

Eigenstate-Reduction Quantum SMPC

Overview

Stochastic Model Predictive Control (SMPC) for quantum systems traditionally requires per-horizon Monte Carlo scenario sampling, which is computationally prohibitive. This methodology uses the almost-sure eigenstate reduction of quantum trajectories under continuous measurement to collapse the infinite-horizon stochastic objective to a closed-form fidelity term computable from the one-step averaged state.

Core Mathematical Insight

Eigenstate Reduction Theorem

Under continuous-time measurement, quantum trajectories almost surely converge to eigenstates of the measurement operator. This means the infinite-horizon expected cost:

J = E[∑ γ^k · cost(ρ_k, u_k)]

collapses to:

J = F(ρ̄_1)  (closed-form fidelity from one-step averaged state)

where ρ̄_1 is the one-step averaged (expected) state after applying the control.

Computational Reduction

Approach	Complexity	Scalability
Sampling-based SMPC	O(N_scenarios × N_horizon)	Poor for long horizons
Eigenstate-reduction SMPC	O(1) terminal evaluation	Linear in system dimension

Algorithm

Step 1: Quantum Filter Propagation

ρ̄_{k+1} = 𝔼[ρ_{k+1} | ρ_k, u_k]  (deterministic)

Propagate the averaged density matrix deterministically — no sampling needed.

Step 2: Terminal Fidelity Evaluation

F_term = Tr(ρ_target · ρ̄_N)

Evaluate fidelity against target state at the prediction horizon endpoint.

Step 3: Optimization

min_u  -F_term(ρ̄_N(u))
s.t.   physical constraints on u (amplitude, bandwidth, etc.)

Solve the deterministic optimization problem.

Step 4: Receding Horizon Implementation

Apply first control step, re-measure, re-optimize.

Stability Guarantees

• Equivalence: The reduced objective is equivalent to the full stochastic objective in the infinite-horizon limit
• Mean-square stability: Closed-loop system is mean-square stable under the receding horizon policy
• Validated on multi-level systems and Ising-type coupled systems

Application Workflow

1. Define system: Hamiltonian H₀, control Hamiltonians H_c, measurement operators M
2. Set target: Desired quantum state or gate operation
3. Configure filter: Quantum stochastic differential equation for averaged state propagation
4. Terminal cost: Fidelity with respect to target state
5. Solve: Standard optimization (gradient-based or derivative-free)
6. Validate: Compare with sampling-based SMPC for correctness

Key Advantages

1. No Monte Carlo sampling: Eliminates the dominant computational bottleneck
2. Closed-form terminal: Fidelity computable analytically from one-step averaged state
3. Mean-square stability: Theoretical guarantees preserved
4. Scalability: Linear scaling vs exponential for sampling-based approaches
5. Generality: Applicable to finite-dimensional quantum systems under continuous measurement

Pitfalls

• Eigenstate reduction assumption: Requires continuous measurement to drive trajectories to eigenstates. Intermittent or weak measurement may not satisfy the convergence condition.
• Finite-dimensional systems: Theory established for finite-dimensional Hilbert spaces. Extension to infinite dimensions requires additional analysis.
• Measurement backaction: The approach assumes the measurement operator structure is known and correctly modeled.
• Horizon truncation: While theoretically infinite-horizon, practical implementation uses finite N — verify convergence.

Related Skills

• vf-qctrl-llm-quantum-control — LLM-driven quantum control (alternative approach)
• quantum-control-pulse-software — Pulse-level control software framework
• model-predictive-quantum-control — General MPC for quantum systems
• discounted-mpc-control — Discounted MPC stability analysis (classical analogue)

Activation: stochastic model predictive control, quantum SMPC, quantum filtering, eigenstate reduction, quantum trajectory, continuous measurement, mean-square stability, quantum optimal control, receding horizon quantum control