# What is the connection between quantum thermal relaxation and phase and amplitude damping?

I have read that amplitude and phase damping of a quantum system describe the energy and quantum information loss of a quantum system due to the interaction with its environment. This is related to thermal relaxation of the system described by the relaxation times T1 and T2. Are amplitude and phase damping together fully analog to thermal relaxation? If yes what is the concrete connection between T1,T2 and the probability for amplitude and for phase damping?

(Longer explanation why I ask, although there seems to be literature about it: I found different sources on this topic but it seems to be inconsistent.

What I have read so far:

• incoherent quantum noise can in general be described by the operator-sum representation, where the effect of a noisy channel on a initial density matrix $$\rho$$ is given by $$N(\rho)=\sum _k K_k \rho K_k ^{\dagger}$$.

From qiskit I have the Kraus operators:

• for phase damping: $$P_0=\left[ \begin{array}{cc} 1&0 \\0 & \sqrt{1-p_p} \end{array}\right], P_1=\left[ \begin{array}{cc} 0&0 \\0 & \sqrt{p_p} \end{array}\right]$$
• for amplitude damping:$$A_0=\left[ \begin{array}{cc} 1&0 \\0 & \sqrt{1-p_a} \end{array}\right], A_1=\left[ \begin{array}{cc} 0& \sqrt{p_a} \\0 & 0 \end{array}\right]$$
• for the combined amplitude and phase damping: $$AP_0=\left[ \begin{array}{cc} 1&0 \\0 & \sqrt{1-p_p-p_a} \end{array}\right], AP_1=\left[ \begin{array}{cc} 0&0 \\0 & \sqrt{p_p} \end{array}\right], AP_2=\left[ \begin{array}{cc} 0& \sqrt{p_a} \\0 & 0 \end{array}\right]$$ (I)

Nielsen&Chuang chapter 8 gives:

• same kraus operators for phase and for amplitude damping as used in qiskit
• an alternative formulation as phase damping: $$\hat {P_0}= \sqrt{\alpha} I, \hat{P_1}=\sqrt{1-\alpha}Z$$, where I is the identity and Z the Pauli-Z-matrix. The connection between the formulations is given by $$\alpha=(1+\sqrt{1-p_p})/2$$.
• $$p_a=1-exp(-t/T1), \sqrt{1-p_p}=exp(-t/2*T2)$$ (II)

This post explains:

• Amplitude damping changes the diagonal and off-diagonal elements of the density matrix. Phase damping changes the off-diagonal elements only. Thus, the changes of the off-diagonal elements are put together like $$\frac{1}{T2}=\frac{1}{2T1}+\frac{1}{T_{pure dephasing}}$$
• For amplitude damping the same kraus operators as with qiskit are given and the definition of the probability of amplitude damping is given as in Nielsen&Chuang.

Tilly et al. writes:

• Kraus operators for amplitude damping defined as in qiskit
• Phase damping defined like the alternative in Nielsen&Chuang
• The effect thermal relaxation has on a initial density matrix $$\rho=\left[ \begin{array}{cc} a & b \\b^* & 1-a \end{array}\right]$$ is (for temperature=0 K) $$N(\rho)=\left[ \begin{array}{cc}(a-1) \times exp(-t/T1)+1 & b \times exp(-t/T2)\\ b^*\times exp(-t/T2) & (1-a)\times exp(-t/T1)\end{array}\right]$$ (III)

My problem: I would expect that if thermal relaxation was just amplitude and phase damping together, I could take the joint amplitude and phase damping channel as defined by (I), plug in the definitions for the error probabilities in terms of the relaxation times (II) and get the thermal relaxation error as defined at (III). Instead it gives me (with the initial density matrixv$$\rho$$ defined as above): $$N(\rho)=\left[ \begin{array}{cc}(a-1) \times exp(-t/T1)+1 & b \times \sqrt{exp(-t/T2)-exp(-t/T1)-1}\\ b^*\times \sqrt{exp(-t/T2)-exp(-t/T1)-1} & (1-a)\times exp(-t/T1)\end{array}\right]$$)

Edit: The definition of the joint amplitude and phase damping of qiskit is an approximation according to this post, where the combination of error channels is explained. Using the Kraus operators of this post, the combined amplitude and phase damping is equal to the thermal relaxation (III) if $$\sqrt{1-p_p}=exp(-t/T_{pure dephasing})$$.This relation of $$p_p$$ seems to make sense, because $$p_p$$ describes phase damping only. But it is not consistent with Nielsen&Chuang (II). Is this a matter of convention?

• Your question is totally appropriate on this forum, but I would like to draw your attention on quantumcomputing.stackexchange.com in which your question could also fit (and there is a large quantum info/computing community there). Cheers! Commented Aug 17, 2022 at 10:11

All definitions above define a combined phase and amplitude damping channel and are connected via parameter transformations.

For the definition of the combined amplitude and phase damping in qiskit with
$$AP_0=\left[ \begin{array}{cc} 1&0 \\0 & \sqrt{1-p_p-p_a} \end{array}\right], AP_1=\left[ \begin{array}{cc} 0&0 \\0 & \sqrt{p_p} \end{array}\right], AP_2=\left[ \begin{array}{cc} 0& \sqrt{p_a} \\0 & 0 \end{array}\right]$$ we get \begin{align} &p_a=1-\exp(-t/T_1), \sqrt{1-p_p-p_a} = \exp(-t/T_2) \\ &p_p=\exp(-t/T_1) - \exp(-2t/T_2).\end{align} With these probabilities defined and $$T_2\le 2 T_1$$ in qiskit thermal_relaxation_error(t1,t2,t) corresponds to phase_amplitude_damping_error(p_a,p_p).

If you calculate the Kraus matrices from the combination of the Kraus matrices for an amplitude damping channel and a phase damping channel, you get, for example (Kraus matrices from an operator sum representation of a quantum channel are not unique) $$AP_0=\left[ \begin{array}{cc} 1&0 \\0 & \sqrt{1-p_p}\sqrt{1-p_a} \end{array}\right], AP_1=\left[ \begin{array}{cc} 0&0 \\0 & \sqrt{1-p_a}\sqrt{p_p} \end{array}\right], AP_2=\left[ \begin{array}{cc} 0& \sqrt{p_a} \\0 & 0 \end{array}\right]$$ and the relation \begin{align} &p_a=1-\exp(-t/T_1), \sqrt{1-p_p}\sqrt{1-p_a} = \exp(-t/T_2) \\ &p_p=1-\exp(-t/T_\phi) \;\mathrm{with}\; T_\phi= \frac{T_1 T_2}{2 T_1 - T_2}.\end{align} With these probabilities defined and $$T_2\le 2 T_1$$ in qiskit thermal_relaxation_error(t1,t2,t) corresponds to phase_damping_error(p_p).compose(amplitude_damping_error(p_a)) or eqivalent to amplitude_damping_error(p_a).compose(phase_damping_error(p_p)).

You can check the equivalence by calculating the Choi matrix representation $$J(\mathcal{E}) = \sum_k \text{vec}(AP_k) \text{vec}(AP_k)^{\dagger}$$ of the quantum channel $$\mathcal{E}$$ or simply the average fidelity $$\overline{F} = \int\langle\psi|\mathcal{E}(|\psi\rangle\langle\psi|)|\psi\rangle d\psi$$, which all give the same result, in particular $$\overline{F_{AP}} = \frac{1}{2} + \frac{1}{6}\exp(-t/T_1) + \frac{1}{3}\exp(-t/T_2)$$ (in qiskit you can calculate the average fidelity via average_gate_fidelity(error)).

Perhaps the question remains why in the qiskit source code for the thermal relaxation error the spectral decomposition of the Choi matrix is applied and not simply one of the above parameter transformations, which I cannot answer.

Another open question for me is where the factor 2 in the calculation of the exited state population (default = 0) in the qiskit source code has its origin (excited state population = 1/(1+exp((2 h f)/(kB T))) where f and T are the frequency and temperature of the qubit, h and kB are Planck and Boltzman constants, repectively). Instead, the equilibrium Boltzmann distribution with $$p_1=1-p_0=1/(1+\exp(h f/k_BT))$$ should be used in the calculation of the thermal relaxation error, where $$\rho_\infty = p_0 |0\rangle \langle 0| + p_1 |1\rangle \langle 1|$$ is the steady state of the general amplitude damping channel $$\mathcal{E}_{GAD}(\rho_\infty)=\rho_\infty$$.

Edit: I asked these questions in another post.

• Thank you for your detailed answer, this was helpful. Unfortunately, I'm relative new on Stack exchange and can not yet upvote your answer. Commented Aug 14, 2022 at 8:37
• Some fundamental questions, however, remain for me: 1) Why do you use the Choi matrix representation or the average fidelity for checking the equivalence? I naively applied the channels to an arbitrary density matrix to see whether it gives the same. What are possible disadvantages of this? For the average fidelity: We take the average of what? All possible states, I guess, but what are these? Commented Aug 14, 2022 at 8:44
• 2) Maybe connected to the first question on the equvalence: I understand that the representation in terms of Kraus operators are not unique, as in the case of phase damping shown in Nielsen&Chuang. But how can the first set of Kraus operators of combined amplitude and phase damping as defined in qiskit be equivalent to the second set defined via the composition of the two? If I apply both representations on a general 2x2 matrix I get different results which I cannot translate into eachother by finding a relation between the probabilities defined in the one case and the other case. Commented Aug 14, 2022 at 9:01
• The Choi matrix is unique because the unidal transformations of Kraus matrices cancel out. Of course, you can also calculate the transformation of a general density matrix with the Kraus matrices, which is a little more tedious. In both cases you can see that only the off-diagonal elements differ. Therefore, for both Kraus sets to define the same error channel, $1-p_p^q-p_a = (1-p_p^c)(1-p_a)$ must apply (see above). Commented Aug 15, 2022 at 10:29
• The average fidelity, which in qiskit defines the gate error of the simulated backends, provides only one number and must of course give the same value for equal error channels. The average is formed over all possible states, e.g. for a qubit the integral over the entire Bloch Sphere. To calculate the average fidelity, however, the integral does not have to be performed explicitly (see equation 9.135 in Nielsen & Chuang and Horodecki et al. [Phys. Rev. A 60 (1999) 1888], connecting average fidelity to entanglement fidelity). Commented Aug 15, 2022 at 11:14