# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Pulse Optimization
The exact strength and units of the signal that a qubit would 'feel', is dependent on the physical realization of the qubit and of any transformations that the signal converter would do. For a flux qubit, for example, the signal would have to be converted to currents. Qiskit hides these details by exposing a dimensionless amplitude parameters and performing the underlying signal conversions.) The dimensionless coupling parameter is taken to be equal to \(J=2\times 10^{-2}\). In Figure 3, some indicative examples of how states are evolved are depicted for the case of the first qubit. As shown, \(T_{0,1}\) swaps the populations of \(|0\rangle\) and \(|1\rangle\) states, while leaving state \(|2\rangle\) invariant. The \(T_{0,2}\) pulse, on the other hand, performs the \(|0\rangle\rightarrow|2\rangle\) transition as expected. Figure 2: Mixer output for the \(T_{0,2}\) signal which consist of a \(T_{0,1}\) and \(T_{1,2}\) signal in sequence. The first part (blue) is the modulated signal of the first pulse \(T_{0,1}\) and the second one (orange) of the second pulse \(T_{1,2}\). The two pulses are constant and are modulated with frequencies \(v\) and \(v+anharm\), respectively. Regarding the pulses' accuracies, these are estimated by calculating the corresponding infidelities, which are obtained by comparing the relative populations between the target and the computed states. More specifically, the infidelities \(inf_{T_{i,j}}\) are computed using the type \(inf_{T_{i,j}}=1-f_{T_{i,j}}\), where \(f_{T_{i,j}}\) are the Hellinger fidelities of the count distributions. Removing the operator subscripts for clarity, for each operator, the Hellinger fidelities are computed by  
\[f=(\sum_{i=0}^{5}\sqrt{q_{i}p_{i}})^{2} \tag{10}\]  
where \(q_{i}\) and \(p_{i}\) denote the final populations of the simulated and ideal states, respectively, for all possible outcomes. The possible outcomes are \(i=\{0,1,2\}\) for the possible qutrit states and \(i=\{4,5\}\) to include the possibility of unwanted leakage to even higher-energy states. (It can be seen that, when comparing to ideal distribution, the sum degenerates to a single term since \(p_{i}=0\) for all populations not belonging to the target state. However, the Hellinger fidelity thus defined is used since it can be directly computed by using [PERSON]'s  
Figure 3: Indicative results for a combination of simulated pulses and starting states. (**a–c**) show how the states evolve when a \(T_{0,1}\) pulse is performed, with starting states equal to \(|0\rangle\) (**a**), \(|1\rangle\) (**b**) and \(|2\rangle\) (**c**). As expected, populations in states \(|0\rangle\) and \(|1\rangle\) are exchanged, whereas there is minimum leakage to other states. When the system starts in state \(|2\rangle\), all populations remain unaffected to a large degree. (**d**) shows the full evolution of state \(|0\rangle\) under a \(T_{0,2}\) pulse. The system firstly goes to the excited state \(|1\rangle\); a resonant pulse with frequency equal to \(r_{0}+a_{0}\) then drives the \(|1\rangle\rightarrow|2\rangle\) transition. As the \(T_{0,2}\) pulse consists of two sequential pulses, its duration is the sum of the duration of the two constituent pulses. library and can be easily extended to more general cases. )For a description of the basic setup and simulation parameters of the experiment, the reader can refer to Appendix B. Table 2 shows the calculated infidelities for the three pulses. Note that the operations are the _physical_ ones that correspond to the energy levels of the transmon. It is to be noted that even when the second qutrit is taken into account, noises due to crosstalk can be minimal when the frequencies, anharmonicities, and couplings are appropriately defined. Figure 4 for example, shows the evolution of the first qutrit under a \(T_{0,2}\) gate, where the second qutrit involved has a frequency equal to \(v_{1}=4.97\) GHz, and an anharmonicity of \(\alpha_{1}=-320\) MHz, in the presence of crosstalk due to the coupling.