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In the paper Modelling and Simulating the Noisy Behaviour of Near-term Quantum Computers, when the authors discuss relaxation times ($T_1$), they define two probabilities, $P_{T_1}$ and $p_{\text{reset}}$, as follows:

Considering the thermal relaxation times $T_1(q)$, as well as the (known) gate execution times $T_g$, we can define the probability for each qubit $q$ to relax as $$p_{T_1}(q) = e^{-T_g/T_1(q)}.$$ We can then define the probability for a qubit to reset to an equilibrium state as $p_{\text{reset}}(q) = 1 - p_{T_1}(q)$.

Based on my understanding, if we consider relaxation errors with the operators $M_0$ and $M_1$ as given:

$$ M_0 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-p_{reset}} \end{bmatrix}, \quad M_1 = \begin{bmatrix} 0 & \sqrt{p_{\text{reset}}} \\ 0 & 0 \end{bmatrix}$$

and apply $M = M_0 + M_1$ to a single qubit state $|1\rangle$, $$ M |1\rangle = \sqrt{1-p_{\text{reset}}} |1\rangle + \sqrt{p_{\text{reset}}} |0\rangle = \sqrt{p_{T_1}}|1\rangle + \sqrt{p_{\text{reset}}} |0\rangle $$

it appears that $P_{T_1}$ is the probability that the qubit doesn't relax (doesn't decay from $|1\rangle$ to $|0\rangle$).

Could someone please clarify whether my interpretation is correct or if there's a misunderstanding in the definitions provided in the paper?

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$P_{T_1}$ is indeed the probability of measuring the qubit in its excited state $|1\rangle$, under the definition above - after time $T_G$ from the moment of excitation.

The operators $M_0$ and $M_1$ aren't necessary for understanding it. $T_1$ is characterized by a simple experiment that goes as follows: a qubit is prepared in its excited state $|1\rangle$. After time $t$ the qubit is measured. This process repeats multiple "shots" for each value of $t$ so it will be possible to assess the probability $P(|1\rangle)$, that is the probability to measure $|1\rangle$. The value of $t$ is gradually increased until $P(|1\rangle)$ sufficiently decays. Then, using numerical methods a curve is fitted to the data and the following function (for example, SciPy's curve_fit function uses the nonlinear least-squares method):

$$ P(|1\rangle) = e^{-\frac{t}{T_1}} \tag{1} $$

After the curve is fitted, the $T_1$ value is known, and by replacing $t$ in equation $(1)$ with any desired value, e.g. $T_G$ in the case above, one can obtain the probability to measure $|1\rangle$ after time $T_G$.

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