Transition-Aware Decomposition of Single-Qudit Gates

Quantum computation with d-level quantum systems, also known as qudits, benefits from the possibility to use a richer computational space compared to qubits. However, for arbitrary qudit-based hardware platform the issue is that a generic qudit operation has to be decomposed into the sequence of native operations $-$ pulses that are adjusted to the transitions between two levels in a qudit. Typically, not all levels in a qudit are simply connected to each other due to specific selection rules. Moreover, the number of pulses plays a significant role, since each pulse takes a certain execution time and may introduce error. In this paper, we propose a resource-efficient algorithm to decompose single-qudit operations into the sequence of pulses that are allowed by qudit selection rules. Using the developed algorithm, the number of pulses is at most d(d-1)/2 for an arbitrary single-qudit operation. For specific operations the algorithm could produce even fewer pulses. We provide a comparison of qudit decompositions for several types of trapped ions, specifically ¹⁷¹Yb⁺, ¹³⁷Ba⁺ and ⁴⁰Ca⁺ with different selection rules, and also decomposition for superconducting qudits.