# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 1 Introduction
For the regime of the transmon-based architecture, for example, [PERSON] et al. [10] have shown that qudit control can be efficiently implemented and have obtained excellent experimental results for the case of a cross-resonance gate in a two-ququart system. On the other hand, [PERSON] et al. [11] demonstrated an implementation scheme of a qudit processor that uses silicon-photonic integrated circuits. One of the main challenges of quantum computation is the design of efficient quantum circuits and finding optimal transpilations for decomposing the logical quantum circuits into a set of instructions that are supported by the native quantum architecture. The ADAPT-VQE and the qubit-ADAPT-VQE algorithms [12; 13], for example, provide techniques for constructing hardware efficient ansatze for the Variational Quantum Eigensolver problem; these techniques take into account the constraints of current NISQ processors, such as the limited sized universal gate set and the also limited qubit couplings. This situation is similar to the qudit-based computation; both the techniques employed and their efficiency depend upon the characteristics of the problem at hand and the underlying architecture. The present work investigates the process of designing optimal pulses for the transmon-based system and for the case of qutrits used in the implementation of the Bucket Brigade Algorithm (BBA), which is one of the candidate schemas for realizing a Quantum RAM (QRAM). Similarly to classical RAM, a QRAM is composed of a set of cells that contain quantum information (typically in the form of a quantum register). Each cell has an address, and the QRAM allows the selection of cells based on the given addresses. Various possible methodologies have been proposed to implement QRAM [14; 15]. Briefly, some of the most promising frameworks for designing a QRAM can be summarized as follows:  
* The Bucket Brigade QRAM was the first architecture proposed for quantum memory [16] and is the architecture considered in the present work. The Bucket Brigade QRAM uses qutrits for switches and requires the activation of \(O(n)\) switches to access an address. * The Fan-Out QRAM is a quantum extension of the classical Fan-Out RAM [14]. Although Fan-Out QRAM does not need qutrits to implement switches to address cells, it requires the activation of \(O(2^{n})\) switches to access and address a cell. * The Flip-Flop QRAM [17] uses a three-stage process to store data; while it achieves a linear circuit width in terms of the number of addresses, the depth of the circuit is exponential. * Parametric Quantum Circuit (PQC) QRAM is implemented by using a parametrized quantum circuit (a parametrized quantum circuit is one that contains gates that depend upon a set of parameters \(\theta_{i}\), which are typically computed by using an optimization process). This novel approach uses the principles of Machine Learning to train the parametrized circuit and produce a circuit that is efficient for addressing datasets that have similar statistical characteristics with those used for the training set. The Entangling Quantum Generative Adversarial Network (EQGAN) QRAM [18] is one such approach that has shown promise in efficiently storing quantum information, especially when this is used for tasks on the domain of Quantum Machine Learning (QML). As mentioned, our work is targeted towards the BBA QRAM model. Without going into much detail, the BBA uses three-level systems to build the tree that addresses the registers. Two states are used to denote the path (left-right) and one state is the \"wait\" state; the path to the target register is obtained by following nodes that have a state different from the \"wait\" state. Figure 1 depicts an example of an 8-qubit QRAM and how the switches are activated to route to a specific address (see also [19] for a more detailed analysis and how such trees can be simulated in quantum circuits). Figure 1: Example of addressing single-qubit data registers using the Bucket Brigade Algorithm.