# Transition-Aware Decomposition of Single-Qudit Gates

## TL;DR

This paper introduces a method to efficiently break down quantum operations on d-level systems into allowed pulses, improving quantum computing with qudits.

## Contribution

A resource-efficient algorithm is proposed for decomposing single-qudit operations into allowed pulses based on selection rules.

## Key findings

- The algorithm ensures the number of pulses is at most d(d−1)/2 for arbitrary single-qudit operations.
- The method is demonstrated for trapped ions like Yb+171, Ba+137, and Ca+40, and superconducting qudits.
- The approach is relevant for two-qudit operations via efficient single-qudit gate decomposition.

## Abstract

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 an 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 Yb+171, Ba+137 and Ca+40 with different selection rules, and also decomposition for superconducting qudits. Although our approach deals with single-qudit operations, the proposed approach is important for realizing two-qudit operations since they can be implemented as a standard two-qubit gate that is surrounded by efficiently implemented single-qudit gates.