# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Two-Qutrit Gates
The \(U\) control channel is used to perform controlled operations in the cross-resonance scheme employed by the Valencia processor; pulses in the \(U\) channel use the frequency of the target qubit to realize controlled rotations. Virtual \(Z\) rotations are depicted as cyclic arrows with the notation \(VZ(\phi)\), where \(\phi\) is the angle of the virtual rotation. where \(\Delta\), assuming that we are studying the 0-indexed qubit, is equal to \(\Delta=\omega_{d}-\omega_{0}\). For \(\Delta=0\), we have resonant pulses that perform rotations around the \(X\) axis; all single-gate pulses that we have studied thus far fall under this category, with the main difference being that they are performed on systems that have more than two states. An \(R_{z}\) rotation can be performed via the following:  
* Use a slightly off-resonant pulse to introduce a phase difference between the qubit's state vector and the rotating frame. * Use the \(\phi\) angle in Equation (14) to place the rotation axis parallel to the \(z\) axis. Many platforms, such as the IBM Quantum Platform, use the second approach (please also refer to [[https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RZGate](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RZGate)]([https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RZGate](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RZGate)) (accessed on 6 February 2025)), as this phase change can be performed by simply adjusting the phase of the pulses; such frame changes, also referred to as virtual gates, can be performed classically via the signal generators by introducing the appropriate delays. At the cost of having to track the delays and changes across the channels via the program that implements the pulse schedule of the whole quantum program, this approach implements \(R_{z}\) pulses with effectively zero duration and error (for a complete treatise of how to implement virtual \(R_{z}\) gates and how update the phases to take into account subsequent multi-qubit interactions, see [24]). In our implementation, arbitrary \(R_{z}\) rotations are seldom needed for qutrit control that is typically encountered in BBA circuits, while, due to the extra computational states used, the extra wiggle factors of these states may increase the complexity of bookkeeping the phase delays. We are therefore using a simple calibration process for identifying optimum pulse parameters for implementing the  
\[\sqrt{Z}= \begin{pmatrix}1&0\\ 0&i\end{pmatrix}  \tag{15}\]  
gate by using the Boulder Opal framework. The Boulder Opal framework offers a vast array of functionalities for computing optimal parameters for quantum control and for simulating systems, including, among others, superconducting qubits [25]. In similar fashion to the one-qutrit gates and the simulations based on Qiskit Dynamics, we define a two fixed-frequency transmon system and use the same frequency, anharmonicity, and coupling strengths. We then define the target operation and run the Boulder Opal optimizer to generate a pulse that is being composed by piecewise-constant segments that minimize the gate infidelity. Boulder Opal calculates the operational infidelity of a gate by using the formula:  
\[Inf=1-\left|\frac{Tr(U_{T}^{\dagger}U)}{Tr(U_{T}^{\dagger}U_{T})}\right|^{2} \tag{16}\]  
where \(U_{T}\) is the target operator (in our case, these operators are the \((XZ)^{-1/2}\) two-qutrit operator operating in the subspace of the two lowest energy states and \(\sqrt{Z}\) operating in the subspace spanned by the two lowest energy states of the first qutrit) and the \(U\) is the operator that corresponds to the pulses computed by Boulder Opal using gradient optimization (the reader can refer to Appendix C for a brief description of how Boulder Opal can be used to instantiate problems and compute infidelities).