# Equation of Motion for Estimation Fidelity of Monitored Oscillating Qubits

## TL;DR

This paper develops a differential equation model to analyze how quickly the fidelity of state estimates for monitored oscillating qubits converges, showing exponential convergence with precise frequency knowledge and asymptotic behavior otherwise.

## Contribution

It introduces a novel differential equation approach to quantify the convergence of estimation fidelity in monitored oscillating qubits, including cases of frequency uncertainty.

## Key findings

- Fidelity converges exponentially fast when oscillation frequency is known.
- Derived asymptotic fidelity for imprecise frequency knowledge.
- Provides a differential equation framework for estimation analysis.

## Abstract

We study the convergence properties of state estimates of an oscillating qubit being monitored by a sequence of \textit{discrete}, unsharp measurements. Our method derives a differential equation determining the evolution of the estimation fidelity from a single incremental step. When the oscillation frequency $\Omega$ is precisely known, the estimation fidelity converges exponentially fast to unity. For imprecise knowledge of $\Omega$ we derive the asypmtotic estimation fidelity.