# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Two-Qutrit Gates  
As we have seen in Section 2.1, controlled qutrit operations can be reduced to a series of \(T_{i,j}\) and 2-qubit \(CNOT\) operations with the appropriate encodings. Designing a pulse for performing a CNOT in transmon can then be seen as a trivial problem since we can calibrate existing pulses that modern quantum architectures offer. Figure 5, for example, shows a schedule that performs a CNOT gate on the Valencia processor of the IBM Quantum Platform. The main issues of obtaining such pulses out of the box, calibrating them, if needed, and applying them for qutrit manipulation according to the schematics presented are the following:  
* CNOT gates are rarely truly native gates in transmon architectures. For transmons with fixed frequencies, they are usually based on the cross-resonance gates, which are implemented by driving the control qubit with the target's qubit frequency. Calibrating such a CNOT may involve processes that are not directly applicable to processors that expose different gate-sets (e.g., an ECR gate) and additional calibration processes may need to be applied. * As is usual with many transmon architectures, arbitrary rotations around the \(Z\) axis by and angle \(\theta\) (realized \(Rz(\theta)\) gates) are implemented with virtual pulses (i.e., changes of reference frames) that must be applied across channels in the case of a CNOT gate. When higher-frequency states are involved, these rotations may need to be recalculated to accommodate for the additional wiggle factors of the expanded basis set. Figure 4: State population of the 2-qutrit system under a \(T_{0,2}\) gate on the first qutrit. \begin{table}
\begin{tabular}{c c} \hline \hline
**Pulse** & **Infidelity** \\ \hline \(T_{0,1}\) & \(9.81\times 10^{-7}\) \\ \(T_{1,2}\) & \(9.86\times 10^{-7}\) \\ \(T_{0,2}\) & \(1.90\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pulse infidelities for the \(T_{ij}\) pulses. To this end, we opt to adopt a more general approach in which a cross-resonance gate can be defined using the characteristics of the underlying Hamiltonian. If such a schedule is found, then a CNOT gate can be obtained by  
\[CNOT=(I\otimes Z)^{-1/2}(X\otimes Z)^{-1/2}(X\otimes I)^{-1/2} \tag{11}\]  
where \((X\otimes Z)^{-1/2}\) is the maximally entangling cross-resonance gate obtained from the general \(R_{xz}(\theta)=\exp{(-i\frac{\theta}{2}X\otimes Z)}\) gate. The right term corresponds to a one-qubit \(\sqrt{X}\); in the gate set used in our model, this corresponds to a \(\sqrt{T_{0,1}}\) gate, which can be performed by the same pulse used for the \(T_{0,1}\) if it is applied for half a duration of the pulse. The leftmost term corresponds to a one-qubit \(\sqrt{Z}\) gate. \(R_{z}\) rotations, of which \(\sqrt{Z}\) is a special case, can be performed by different methods in transmon systems. If an external pulse \(V(t)\) of the form  
\[V(t)=\Omega\cos{(\omega_{d}t-\phi)} \tag{12}\]  
is applied to a transmon qubit, then the total Hamiltonian of the qubit can be written as follows (we use the Pauli matrices for simplicity since we focus for now to two-level systems; in addition, we remove all \(\hbar\) factors for convenience):  
\[H=-\frac{\omega_{0}}{2}\sigma_{z}+V(t)\sigma_{x} \tag{13}\]  
It can be shown that by changing to the frame that rotates with the qubit's frequency and under the rotating wave approximation, the rotating Hamiltonian can be expressed as  
\[H_{R}=\frac{\Omega}{2} \begin{pmatrix}0&e^{i\Delta t-\phi}\\ e^{-i\Delta t-\phi}&0\end{pmatrix}  \tag{14}\]  
Figure 5: Pulse schedule for the Valencia backend of the IBM Quantum Platform for the CNOT gate. The target qubit is controlled by signals sent in the control’s qubit control channel that is driven by the target qubit’s frequency. The figure shows the unmodulated Gaussian pulses used for realizing the gate. \(D_{i}\) channels are used to drive qubit \(i\).