# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Qutrit Encoding  
Transitions of single-qutrit states can be described by \(3\times 3\) unitary operators. In the case of qutrits used for BBA, the most common operations are state preparations and state-based transitions. We identify two kinds of operators:  
* The translation operators \(T_{i,j}\) that perform a flip between the \(|i\rangle\) and \(|j\rangle\) basis states of the qutrit; for example, the operator \(T_{0,2}\) alters the state \(|0\rangle(|2\rangle)\) to \(|2\rangle(|0\rangle)\), while leaving the state \(|1\rangle\) invariant. * The permutation operator \(P\), which permutes the states according to \(|i\rangle\rightarrow|(i+2)\mod 3\rangle\). In matrix form, the operators can be given by \[T_{0,1}= \begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix} ,T_{0,2}= \begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix} ,T_{1,2}= \begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix} \] (3) for the translation operators, and \[P_{3}= \begin{pmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{pmatrix} \] (4) for the permutation operator. Similarly to qubits, controlled operations involve the change of the state of a target qutrit based on the condition of the control qutrit. In direct extrapolation, we can identify the control operation \(CT_{i,j}\) that performs the operation \(T_{i,j}\) on the target qutrit if the control qutrit is in state \(|2\rangle\). The implementation of pulses that perform the \(T_{i,j}\) transitions can be performed by sending signals that are resonant with the \(i\leftrightarrow j\) transition frequency. The permutation operators can be more difficult to implement directly; however, they can be deconstructed by a set of \(T_{i,j}\) pulses by using the identity \[P_{3}=T_{0,1}T_{0,2}\] (5)  
Equation (5) can be validated by direct matrix multiplication; however, it can be instructive to see how it works to cycle qutrit states using qutrit translations. Reading from right to left (to simulate how, as a quantum operator, it would act on a qutrit state) and taking as an example the state \(|2\rangle\), the state is first acted on by \(T_{0,2}\). Since we consider the state with index equal to 2 (we use a zero-indexed scheme, so the column indexed at position 2 is actually the third column), the relevant column of the \(T_{0,2}\) operator is \(T_{0,2}^{2}= \begin{pmatrix}1&0&0\end{pmatrix} ^{T}\) which results in the transition to state \(|0\rangle\). We then take operator \(T_{0,1}\) and specifically its zero indexed column \(T_{0,1}^{0}= \begin{pmatrix}0&1&0\end{pmatrix} ^{T}\)(since the state is now \(|0\rangle\)) and observe that it changes the state to \(|1\rangle\). This is exactly the same as the operation performed by \(P_{3}\) alone as can be seen by taking Equation (4) and isolating the second-indexed column. For a proof using explicit matrix multiplication, the reader can refer to Appendix A. Considering the \(CT_{i,j}\) gates, controlling an operation conditional on the control qutrit being in the first hyper-excited state may involve the design of very complex pulses. The operation itself may be prone to additional error since, generally, coupling operations involve pulses that have a longer duration, and the first hyper-excited state has a lower relaxation time than the two lower ones. Various approaches to surpass this problem can be followed, such as redefining the control gates to involve other control states. For ourpurposes, however, we follow the simple approach of re-mapping the physical and logical states of the system. For a single transmon, we adopt the following mapping:  
\[ \begin{split}|0\rangle_{T}&\rightarrow|0\rangle_{L}\\ |1\rangle_{T}&\rightarrow|2\rangle_{L}\\ |2\rangle_{T}&\rightarrow|1\rangle_{L}\end{split}  \tag{6}\]  
where \(|\rangle_{T}\) denotes the physical state of the transmon and \(|\rangle_{L}\) the logical ones. Consider now a pulse sequence that performs the usual two-qubit CNOT pulse.