Although only two quantum states of a physical system are often used to encode quantum information into qubits, in principle, qudits (a system with a number $d$ states) could be used to increase the information capacity of the system. In this work, three-state systems (qutrits) are studied using a parametrization of the group $U\left(3\right)$ that is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. The parametrization uses a decomposition of a general $3\times 3$ unitary matrix into the exponential of a diagonal matrix and the exponential of an off-diagonal matrix. This decomposition is numerically confirmed to allow the parametrization of any element in $U\left(3\right)$. However, many different sets of parameters can give the same element of $U\left(3\right)$. Furthermore, different elements could have different numbers of decompositions. Why are there different numbers? Which decompositions are better and why? These two questions are answered so that the parametrization can be better utilized and searches can be made more efficient. Other parametrizations are derived and discussed and the methods used are applicable to these other parametrizations as well.

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• Quantum control
• Quantum gates
• Quantum information theory