# Clarification on Definitions of Relaxation Probabilities

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?

$$P_{T_1}$$ 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 $$T_G$$.