The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems

Rahul Sarkar1 and Theodore J. Yoder2

1Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305
2IBM T.J. Watson Research Center, Yorktown Heights, NY

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Qudits with local dimension $d \gt 2$ 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. We study the structure of the qudit Pauli group for any, including composite, $d$ in several ways. To cover composite values of $d$, we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give 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. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.

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Cited by

[1] Samo Novák, "Homological Quantum Error Correction with Torsion", arXiv:2405.03559, (2024).

[2] Lane G. Gunderman, "Stabilizer Codes with Exotic Local-dimensions", Quantum 8, 1249 (2024).

[3] Lane G. Gunderman, Andrew Jena, and Luca Dellantonio, "Minimal qubit representations of Hamiltonians via conserved charges", Physical Review A 109 2, 022618 (2024).

[4] Ben DalFavero, Rahul Sarkar, Daan Camps, Nicolas Sawaya, and Ryan LaRose, "$k$-commutativity and measurement reduction for expectation values", arXiv:2312.11840, (2023).

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