# Qutrit Control for Bucket Brigade RAM Using Transmon Systems
## 1 Introduction  
Although qubits logically represent two-level quantum systems, physical implementations are typically realized by systems that possess extra physical states. Good candidate architectures for implementing qubits are those that can efficiently isolate a two-level subsystem so that information does not leak to other states; fluxonium [1] and transmon [2] qubits are typical examples of such architectures. However, these extra states can be manipulated in many cases either to facilitate the computation in the two-level scheme or to expand the computational schema to include additional logical states. [PERSON] [3], for example, has demonstrated efficient pulses for single-qubit operations in a double donor structure; the pulses make use of the excited delocalized orbitals for achieving state transfer between the localized states used for encoding the logical qubit. Going beyond the two-level logical qubit, the concept of qudit is used to denote any quantum information system that contains an n-level logical state. Examples of these systems are qutrit (three-level system), ququart (four-level system), and so on. Similarly to qubit operations, it can be demonstrated that universal gate sets for qudits can also be derived. As shown by [PERSON] and [PERSON] [4], any two-qudit gate \(V\), combined with the collection of all 1-qudit gates \(A\), constitutes a universal gate set, provided that \(V\) is _imprimitive_. An imprimitive 2-qudit gate is one that when acting on a decomposablestate, its output is a state that, in general, cannot be decomposed. For example, if two qudits with states \(|x\rangle_{0}\) and \(|x\rangle_{1}\) are acted upon by the operator \(V\), then the resulting state \(V(|x\rangle_{0}|x\rangle_{1})\) cannot be decomposed into a product of the form \(|y\rangle_{0}|y\rangle_{1}\), with \(|y\rangle_{i}\) being single-qudit states. This situation is similar and can, in fact, be considered a generalization to universality requirements for qubit gates, where at least one entangling gate is required for a gate set to be universal. [PERSON] et al. [5] also studied and demonstrated the existence of universal gate sets for qudits, while also reviewing the extent to which alternative quantum computational models based on qubits, such as measurement-based computing, adiabatic quantum computing, and topological quantum computing, can also be adopted for qudit-based computation. One particularly interesting use case is the possibility of using hybrid qudit computation, in which qudits of different dimensions are used; such hybrid approaches may offer more efficient algorithms when the problem to be solved has some natural mappings that are better served by nonbinary mappings (e.g., simulation of massive spin 1 systems). An algorithm for factorization using a hybrid approach has already been presented in [6], where one spin 3/2 nuclei coupled with a spin 1 nuclei was used to encode qudits of dimensions four and three, respectively. One particular domain of the application of qudits that is also relevant to the scope of the present work is its usage to compress quantum information in a manner that minimizes the amount of operations and/or qubits needed to perform calculations. In this direction, [PERSON] et al. [7] have shown that by using qudits, it is possible to completely remove ancilla qubits when their number is \(O(n)\) and produce an equivalent quantum circuit based on qudits with the same order of depth. On the other hand, [PERSON] et al. [8] constructed a logarithmic depth decomposition scheme of the Toffoli gate based on qutrits, which uses no ancilla qubits or qutrits. Two of the authors of the present work have moreover shown that by using higher-energy states, quantum entanglement can be performed at the logical level by only using the spectrum of a single transmon [9]. Since each quantum hardware can support natively different types of pulses and two-qudit entanglement, the choice of single-qudit and two-qudit gates will naturally be different for each architecture.