Qudits with local dimension d > 2 can have unique structure and uses that qubits (d=2) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. In this talk, I'll present some recently obtained structural results about the qudit Pauli group for any dimension d, including composite ones. I'll briefly discuss the central question that started this study and present a complete solution: for any specified set of commutation relations, can we construct a set of qudit Paulis satisfying those relations using the minimum number of qudits? Next, I'll present some combinatorial results quantifying the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, I'll talk about methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Some applications will be discussed, for example, we will see that this leads to a polynomial time algorithm to calculate the dimension of the codespace for a qudit stabilizer code. Useful tools in this study are normal forms from linear algebra over commutative rings (principal rings), including the Smith normal form, and alternating Smith normal form.
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