optimal-parametric-quantum-estimation - SKILL.md Agent Skill

name: optimal-parametric-quantum-estimation description: "Optimal control-based strategy for enhancing impulse estimation in Gaussian quantum systems via parametric modulation. Use when designing quantum sensing protocols, estimating transient disturbances in quantum systems, optimizing state preparation for impulse detection, or comparing parametric driving vs squeezing protocols. Bridges quantum control theory, optimal estimation, and non-equilibrium quantum dynamics."

Optimal Parametric Quantum Estimation

Methodology from arXiv:2605.12155 — optimal control for impulse estimation in Gaussian quantum systems.

Core Principle

Minimize impulse estimation uncertainty by casting it as a nonlinear optimal control problem over time-dependent system parameters. Dynamically shape estimation covariances through parametric modulation to maximize information gain at a known impulse time.

Key Insight

Parametric driving ≠ squeezing: Conventional squeezing protocols use periodic modulation that degrades inference of impulse-like disturbances. Optimal parametric driving reduces estimation variance by up to 2x relative to steady-state operation.

Mathematical Framework

Problem Setup

For continuously monitored linear Gaussian systems:

System dynamics: dx = A(t)x dt + B(t) dw + impulse(t) dt
Measurement: dy = C(t)x dt + D(t) dv
Kalman filter covariance evolution: dP/dt = A P + P A^T + B B^T - P C^T (D D^T)^{-1} C P

Optimal Control Formulation

minimize: Tr[P(t_impulse)]  (estimation uncertainty at impulse time)
subject to: covariance dynamics
            parameter bounds on A(t), B(t), C(t)

Implementation Steps

Model the system as linear Gaussian with time-dependent parameters
Define the impulse time t_impulse (known or estimated)
Set up the optimal control problem minimizing Tr[P(t_impulse)]
Solve using numerical optimal control (direct collocation or Pontryagin)
Apply parametric modulation to system parameters around t_impulse
Verify variance reduction against steady-state baseline

Parametric Modulation Strategy

Before impulse: Prepare non-equilibrium state that maximizes sensitivity
At impulse: System is optimally sensitive to transient disturbances
After impulse: Return to steady-state or next preparation cycle

When to Use

Quantum sensing with transient signals (force, field, displacement)
Nanomechanical resonator monitoring
Levitated nanoparticle experiments
Any Gaussian quantum system where impulse timing is known/estimable
Comparing estimation strategies: parametric driving vs continuous squeezing

When NOT to Use

Steady-state monitoring (use standard squeezing)
Unknown impulse timing (requires different approach)
Non-Gaussian systems (requires nonlinear filtering)
Periodic disturbance estimation (standard protocols suffice)

Performance Bounds

Variance reduction: up to 2x vs steady-state
Applicable systems: nanomechanical resonators, levitated nanoparticles
Requires: known impulse timing, linear Gaussian dynamics

References

See references/implementation.md for numerical optimal control setup
See references/mathematical-derivation.md for covariance dynamics derivation