  Environmental noise on a controlled quantum system is generally modeled by a
dissipative Lindblad equation. This equation describes the average state of the
system via the density matrix $\rho$. One way of deriving this Lindblad
equation is by introducing a stochastic operator evolving under white noise in
the Schr\"odinger equation. However, white noise, where all noise frequencies
contribute equally in the power spectral density, is not a realistic noise
profile as lower frequencies generally dominate the spectrum. Furthermore, the
Lindblad equation does not fully describe the system as a density matrix $\rho$
does not uniquely describe a probabilistic ensemble of pure states
$\{\psi_j\}_j$. In this work, we introduce a method for solving for the full
distribution of qubit fidelity driven by important stochastic Schr\"odinger
equation cases, where qubits evolve under more realistic noise profiles, e.g.
Ornstein-Uhlenbeck noise. This allows for predictions of the mean, variance,
and higher-order moments of the fidelities of these qubits, which can be of
value when deciding on the allowed noise levels for future quantum computing
systems, e.g. deciding what quality of control systems to procure. Furthermore,
these methods will prove to be integral in the optimal control of qubit states
under (classical) control system noise.
