Physically motivated decompositions of qutrit gates

Here we find sets of unitary transformations for possible use as gates for transmon qutrits. Unitary 3X3 matrices (matrices in U(3)) can be decomposed in numerous ways. One way is to decompose it into a product of an exponential of diagonal terms and an exponential of off-diagonal terms. (Shown in PRL. 125, 180504 (2020)) We showed that many randomly generated U(3) matrices can be represented using this decomposition, providing evidence for the validity of this decomposition. Here we show that there are many different choices for the sets of parameters that can give the Walsh-Hadamard decomposition and how they are related. Furthermore, we find a Hamiltonian that will generate the Walsh-Hadamard matrix from a one-parameter subgroup using a general method. We also show that it is also possible to parameterize the group using various different Cartan decompositions.