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?


1 Answer 1


$P_{T_1}$ 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 $T_G$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.