# How are Rigetti and IBM QX device parameters related to Kraus operators?

Rigetti reports the following parameters: (https://www.rigetti.com/qpu)

• T1, T2* times
• 1-qubit gate fidelity (F1q)
• 2-qubit gate fidelity (F2q) and,

IBM QX reports the following: (https://quantumexperience.ng.bluemix.net/qx/devices)

• T1, T2 times
• (single) qubit gate error
• multi-qubit gate error and,

I understand that one can simulate the effect of noise on qubit state using operator-sum representation. According to Nielsen and Chuang, the operation elements are:

Amplitude damping $$E_0 = \begin{bmatrix} 1 & 0\\ 0 & \sqrt{1-\gamma}\end{bmatrix}$$ $$E_1 = \begin{bmatrix} 0 & \sqrt{\gamma}\\ 0 & 0\end{bmatrix}$$

Phase damping $$E_0 = \begin{bmatrix} 1 & 0\\ 0 & \sqrt{1-\gamma}\end{bmatrix}$$ $$E_1 = \begin{bmatrix} 0 & 0\\ 0 & \sqrt{\gamma}\end{bmatrix}$$

Phase flip $$E_0 = \sqrt{p}\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$ $$E_1 = \sqrt{1-p} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix}$$

Bit flip $$E_0 = \sqrt{p}\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$ $$E_1 = \sqrt{1-p} \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}$$

Bit-phase flip $$E_0 = \sqrt{p}\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$ $$E_1 = \sqrt{1-p} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix}$$

Depolarizing channel $$E_0 = \sqrt{1-3p/4}\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$ $$E_1 = \sqrt{p/4} \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}$$

$$E_2 = \sqrt{p/4} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix}$$ $$E_3 = \sqrt{p/4} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix}$$

How are the original device parameters related to the parameters in the operations elements i.e., $$p$$ and $$\gamma$$? (A first-order approximation of relation between these parameters are also welcomed.)

[P.S. are operations elements described in N&L and Kraus operators the same thing?]

• why do you say there should be a relation? The parameters in the operator-sum representations you report simply parametrise classes of operations of specific kinds. The parameters provided by the quantum devices you mention likely parametrise classes of gate operations. What is the connection between the two? – glS Nov 13 '18 at 16:10

For amplitude damping, $$\gamma$$ is something like $$e^{-\Delta t/T_1}$$ where $$\Delta t$$ is how long the Kraus operator is supposed to act. But be very careful, Kraus evolution assumes your system has no initial correlations, that every qubit interacts with identical baths and that every qubit is identical. All the assumptions are most likely violated and so there will be no simple relation between the actual evolution given by Kraus operators and the evolution on the devices. We may use Kraus evolution as an approximation. It's probably more appropriate to do a master equation simulation but that will not, in general, correspond to the Kraus operators one sees in Nielsen and Chuang.