Amplitude damping, T1 decay and engineered dephasing

A qubit is prepared in $\lvert1\rangle$ and then idles for $t=10\,\mu\mathrm s$. Its energy-relaxation time is $T_1=50\,\mu\mathrm s$ and there is no additional pure dephasing ($T_2=2T_1$). Over the idle period the qubit undergoes the amplitude-damping channel

$\rho\mapsto K_0\rho K_0^\dagger+K_1\rho K_1^\dagger,\qquad K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix},\quad K_1=\begin{pmatrix}0&\sqrt\gamma\\0&0\end{pmatrix}.$

Enter the damping parameter $\gamma$ of this channel and the probability $P(1)$ of measuring $1$ after the idle period. Each value is graded with absolute tolerance $10^{-3}$; exact expressions such as 1-exp(-0.2) are accepted (and then compared to $10^{-9}$).

Starting from $\lvert0\rangle$, build a one-qubit circuit whose output is the mixed state

$\sigma=\begin{pmatrix}0.5&0.3\\0.3&0.5\end{pmatrix}=\tfrac12\big(I+0.6\,X\big).$

You may use one gate, either $H$ or $R_Y$, together with any number of noise channels from amplitude damping, phase damping and depolarizing (the gate limit counts gates only, not channels). The circuit is simulated exactly on the density-matrix backend, with the engine conventions of the lesson: phase damping $\rho_{01}\mapsto\sqrt{1-\lambda}\,\rho_{01}$, amplitude damping $K_0=\mathrm{diag}(1,\sqrt{1-\gamma})$, $K_1=\sqrt\gamma\lvert0\rangle\langle1\rvert$, and depolarizing $\rho\mapsto(1-\tfrac{4p}3)\rho+\tfrac{4p}3\tfrac I2$. In the circuit script a channel is a statement with the qubit first and then its parameter, for example $\texttt{amplitudeDamping(0, 0.1)}$, $\texttt{phaseDamping(0, 0.5)}$ or $\texttt{depolarizing(0, 0.05)}$; gates are written as $\texttt{H(0)}$ or $\texttt{RY(0, pi/3)}$, and statements are separated by semicolons or new lines. The grader requires the trace distance $\tfrac12\lVert\rho-\sigma\rVert_1\le10^{-6}$. Bit flip, phase flip and thermal relaxation are not available in this step.