# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 2 Materials and Methods
### Pulse Optimization  
The methodology for designing efficient pulses for state preparations and controlled state transitions relies on leveraging existing techniques for accessing the higher-energy states of transmon systems, while also avoiding performing controlled operations that involve higher excited states both as controls and as targets to reduce the computational error. For estimating pulse parameters, we adopt the following Hamiltonian (see, for example, the overview of [PERSON] et al. [23] for a complete description of how to describe the static and interacting terms of transmon Hamiltonians):  
\[ \begin{split} H&=\sum_{i=\{0,1\}}v_{i}a_{i}^{ \dagger}a_{i}\\ &+\frac{1}{2}\sum_{i=\{0,1\}}\alpha_{i}a_{i}^{\dagger}a_{i}(a_{i} ^{\dagger}a_{i}-I)\\ &+\sum_{i=\{0,1\}}d_{i}(t)(a_{i}+a_{i}^{\dagger})\\ &+J\prod_{i=\{0,1\}}(a_{i}+a_{i}^{\dagger})\end{split}  \tag{9}\]  
where the first two rows refer to the free Hamiltonians of the two transmons with frequencies \(v_{0}\) and \(v_{1}\) and with anharmonicities equal to \(\alpha_{0}\) and \(\alpha_{1}\). \(a_{i}^{\dagger}(a_{i})\) are the creation(annihilation) operator for qutrit \(i\). \(d_{i}(t)\) refer to the drive signals that can be applied to the transmons to achieve control; these can be further decomposed to the Rabi strengths and the time signals themselves, but we keep these two terms together for simplicity. Factors of \(2\pi\) are omitted for simplicity. Finally, the coupling of the systems with coupling strength \(J\) is included in the final row. While a perfect transmon could be described by the Pauli operators, we introduce the more generic annihilation and creation operators both to access the higher-energy states and to ensure that unwanted leakage to other states is modeled appropriately. To this end, we adopt a 5-level oscillator to accommodate both cases. For the one-qutrit pulses, the following assumptions are made:  
* Relaxation and dephasing times have been ignored as a first approximation. Our main focus is to establish errors due to the leakage to unwanted states, which are typically introduced due to imperfect calibrations or crosstalk terms. These are modeled in our Hamlitonian. * A maximum power of 1.0 is imposed on our simulations. Although driving a transmon from its ground state directly to the second excited state is, in theory, possible, it may involve the realization of strong pulses so that current architectures may not possess (the IBM Quantum Platform for example, has an upper limit of a pulse amplitude equal to one; the value can be overridden but only for simulations). Under those assumptions, the process for constructing the \(T_{i,j}\) pulse is the following (for this section, the indices of the \(T\) pulses refer to physical states and not logical ones):  
1. Perform a Rabi experiment to estimate the amplitude of the pulse. 2. Drive the qutrit and obtain the fidelity. 3. For the \(T_{0,2}\) pulse especially, use the two calibrated \(T_{0,1}\) and \(T_{1,2}\) pulses in sequence (here, and for all single-qutrit operations described in the present section, we refer to the _physical_ operations that alter the \(|\rangle_{T}\) states of Equation (6)). Using a frequency of 4.86 GHz for the first transmon and an anharmonicity of \(-320\) MHz (these, and all the other system parameters, are taken from the existing Almaden backend from IBM Quantum Platform to make the experiments more realistic), the system is simulated using the Qiskit Dynamics library. A five-level oscillator is used to model the possibility of leakage to unwanted higher-energy states. For simplicity, constant pulses are used. For modulating the pulses to signals, we use the \(v_{0}+\alpha_{0}\) frequency as a carrier frequency for the transition \(T_{0,2}\). Figure 2 shows the mixer output of the two combined signals that performs a \(T_{0,2}\) transition. A Rabi strength of 0.21 is used. (Here, we use the normalized dimensionless units that Qiskit uses.