# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Qutrit Encoding
Such pulses are implemented in modern hardware in such a manner as to ensure that there is minimum leakage to higher excited states; indeed, the fact that the ground and excited states can be separated by the higher ones is one of the key factors that constitute a good quantum processor architecture, such as the one based on transmon systems. Ignoring the specifics of the pulse and assuming that it operates in the space spanned by the ground and first excited states, the transitions in the CNOT case for the transmon and logical states can be summarized in Table 1.  
As can be seen in Table 1, the logical state of the target qubit makes the transition \(|0\rangle_{L}\leftrightarrow|2\rangle_{L}\) when the logical state of the control qubit is \(|2\rangle\). This corresponds to the operation of \(CT_{0,2}\), which in explicit matrix notation can be described as  
\[CT_{0,2}= \begin{pmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0\\ \end{pmatrix}  \tag{7}\]  
The other controlled operations can be obtained by using the identities  
\[ \begin{split} CT_{0,1}&=(I_{3}\otimes P_{3})^{2}CT_{0,2}(I_{3} \otimes P_{3})\\ CT_{1,2}&=(I_{3}\otimes P_{3})CT_{0,2}(I_{3}\otimes P_{3})^{2}\end{split}  \tag{8}\]  
where \(I_{3}\) is the three-dimensional identity matrix and the symbol \(\otimes\) denotes tensor products. Similarly to the interpretation given for Equation (5), the identities can be logically deducted by observing that the \((I_{3}\otimes P_{3})\) operators act on the second qutrit subspace to alter its state  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Initial Physical** & **Final Physical** & **Initial Logical** & **Final Logical State** \\ \hline \(|00\rangle_{T}\) & \(|00\rangle_{T}\) & \(|00\rangle_{L}\) & \(|00\rangle_{L}\) \\ \(|01\rangle_{T}\) & \(|01\rangle_{T}\) & \(|02\rangle_{L}\) & \(|02\rangle_{L}\) \\ \(|02\rangle_{T}\) & \(|02\rangle_{T}\)  
accordingly, before and after the operation of the \(CT_{0,2}\) operator. As an example, consider the application of the \(CT_{1,2}\) operator to a target qutrit in state \(|2\rangle\). If the control qutrit is not in state \(|2\rangle\), then the \(CT_{0,2}\) operator will have no effect and, according to the second equation of Equation (8), the permutation operator will be applied three times to the state, performing the transitions \(|2\rangle\rightarrow|1\rangle\rightarrow|0\rangle\rightarrow|2\rangle\), effectively leaving the state invariant. If, however, the control qutrit is in state \(|2\rangle\), then after the second permutation, which sends the state of the target qutrit to \(|0\rangle\), the \(CT_{0,2}\) operator will alter its state to \(|2\rangle\). The final permutation will then send state \(|2\rangle\) to state \(|1\rangle\), thus performing the equivalent of a \(CT_{1,2}\) operation. We shall not exhaust all the other cases; the reader can refer to Appendix A for the computation that derives all the \(CT_{i,j}\) matrices using the identities of Equation (8). Under all of the assumptions above, we can see that the desired logical operations can be obtained by pulses that involve only single-qutrit \(T_{i,j}\) transitions and only one two-qutrit gate that can be implemented by applying existing approaches that realize the two-qubit CNOT gate.