# Fast Reset Protocol for Superconducting Transmon Qubits
## 3 Results
### Quantization of the System  
As shown in the above, with a mixing microwave pulse, we can easily realize superconducting qubit reset with the assistant of two decays (\(\kappa_{c}\) and \(\kappa_{r}\)). To obtain the effective Hamiltonian Equation (8), we first use the black-boxing quantization method [31; 32] to quantize the coupled system, and the quantize Hamiltonian is Equation (5), as shown in the Method Section. Note that, when performing circuit analysis on a superconducting circuits with frequency-tunable couplers that couple qubits, the SQUID-loop [36] of the coupler can be equivalent to the single Josephson junction [37]. Therefore, the circuit in Figure 2b is equivalent to Figure 1a. In order to produce the desired coupling term, we add a mixing microwave pulse to the system, and the Hamiltonian of the pulse is Equation (7). The Hamiltonian of the system is  
\[H=H_{1}+H_{d}. \tag{9}\]  
Figure 2: (**a**) The circuit diagrams of system model. Two-dimensional arrangement structure of the qubits and coupler and (**b**) circuit structure of specific system model. (**c**) Energy-level diagram of the qubit reset process. The coupling strength between \(|0e0\rangle\) and \(|1g1\rangle\) states is \(G\); and \(\kappa_{r}\) and \(\kappa_{c}\) denote the decay rates of readout resonator and coupler, respectively. Under the action of \(\kappa_{r}\) and \(\kappa_{c}\), the \(|1g1\rangle\) state can rapidly decay to \(|0g0\rangle\) state. Then, we use the displacement operator and the picture transformation to simplify the Hamiltonian. Firstly, by displacement unitary, \(U_{1}=\exp(\alpha a_{q}^{\dagger}-\alpha^{*}a_{q})\) to perform the picture transform for the Hamiltonian of Equation (9). We have  
\[H_{2} = i\frac{\mathrm{d}U_{1}}{\mathrm{d}t}U_{1}^{\dagger}+U_{1}HU_{1}^{\dagger} \tag{10}\] \[= \omega_{c}a_{c}^{\dagger}a_{c}+\omega_{r}a_{r}^{\dagger}a_{r}+i \left(\dot{\alpha}a_{q}^{\dagger}-\dot{\alpha}^{*}a_{q}-\frac{1}{2}\dot{| \dot{\alpha}|}^{2}\right)+\omega_{q}\left[a_{q}^{\dagger}a_{q}-\left(\alpha a_{ q}^{\dagger}+\dot{\alpha}^{*}a_{q}\right)+|\alpha|^{2}\right]\] \[-E_{Iq}\left(\cos\varphi_{Iq1}+\frac{\varphi_{Iq1}^{2}}{2}\right) -E_{Ic}\left(\cos\varphi_{Ic1}+\frac{\varphi_{Ic1}^{2}}{2}\right)\] \[+\epsilon\Big{(}\dot{e}^{i\omega_{Q}t}a_{q}+e^{-i\omega_{Q}t}a_{ q}^{\dagger}\Big{)}-\epsilon\Big{(}e^{i\omega_{Q}t}a+e^{-i\omega_{Q}t}a^{*} \Big{)},\]  
where \(\varphi_{Iq1}=\varphi_{Iq}-\varphi_{0}^{q}(\alpha+\alpha^{*})\), and \(\varphi_{Ic1}=k\varphi_{Iq1}\). To get the similar form of the Equation (5), we can achieve this goal by selecting the appropriate \(\alpha\). Thus, this parameter \(\alpha\) satisfies two conditions; that is,  
\[i\left(\dot{\alpha}a_{q}^{\dagger}-\dot{\alpha}^{*}a_{q}\right)-\omega_{q} \left(\alpha a_{q}^{\dagger}+\dot{\alpha}^{*}a_{q}\right)+\epsilon\Big{(}e^{ i\omega_{Q}t}a_{q}+e^{-i\omega_{Q}t}a_{q}^{\dagger}\Big{)}=0, \tag{11}\]  
and \(\epsilon[\exp(i\omega_{Q}t)\alpha+\exp(-i\omega_{Q}t)\alpha^{*}]\) is time independent. By solving the differential equations, we have  
\[\alpha=-\frac{\epsilon}{\Delta}e^{-i\omega_{Q}t}, \tag{12}\]  
where \(\Delta=\omega_{q}-\omega_{d}\). Substitute Equation (12) into Equation (10): we get  
\[H_{2} = \omega_{c}a_{c}^{\dagger}a_{c}+\omega_{q}a_{q}^{\dagger}a_{q}+ \omega_{r}a_{r}^{\dagger}a_{r} \tag{13}\] \[-E_{Ic}\left(\cos\varphi_{Ic1}+\frac{\varphi_{Ic1}^{2}}{2}\right) -E_{Iq}\left(\cos\varphi_{Iq1}+\frac{\varphi_{Iq1}^{2}}{2}\right).\]  
Apply a transformation with respect to \(U_{2}=\exp[i(\omega_{q}a_{q}^{\dagger}a_{q}+\omega_{c}a_{c}^{\dagger}a_{c}+ \omega_{r}a_{r}^{\dagger}a_{r})t]\) to Equation (13); we get  
Figure 2:0 Figure 2:1  
where \(\varphi_{Iq}^{\prime}=\sum_{i=c,q,r}\{\varphi_{0}^{i}[a_{i}\exp(-i\omega_{i}t) +a_{i}^{\dagger}\exp(i\omega_{i}t)]\}+\varphi_{0}^{q}(\epsilon/\Delta)[\exp(i \omega_{d}t)+\exp(-i\omega_{d}t)]\) and \(\varphi_{Ic}^{\prime}=k\varphi_{Iq}^{\prime}\). Then, we expand the cosine term in Equation (14) to the fourth order and remove the constant term.