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"Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic", Erik Hostens, Jeroen Dehaene, and Bart De Moor, 2005

Reviewed August 17, 2023

Citation: Hostens, Erik, Jeroen Dehaene, and Bart De Moor. "Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic." Physical Review A 71.4 (2005): 042315.

Web: https://arxiv.org/abs/quant-ph/0408190

Tags: Abelian-anyons, Computer-scientific, Pedagogical, Higher-dimensional

This is a great general reference for qudit Pauli matrices and the qudit Clifford group. It pulls together lots of work, and does everything very explicitly.

One of the biggest contributions of this paper is a proof that the Clifford group can be generated by generalizations of the standard Clifford group elements. Namely, there are no "exotic" Clifford group elements even over Zn. This was conjectured by Gottesman in

> Gottesman, Daniel. "Fault-tolerant quantum computation with higher-dimensional systems." NASA International Conference on Quantum Computing and Quantum Communications. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998.

but only proved for the case that d is prime. In fact, when d is prime, this result is entirely classical - Gottesman seems unaware of this, however. The Clifford group's action on the Pauli group is symplectic. Hence, when d is prime, this reduces to the statement that a special set of matrices generates the symplectic group over a finite field. The symplectic group (especially over finite fields) is very well understood, so it is not a surprise that this is classical. In fact, this is a re-casting of the classical fact that the symplectic group over a finite field is generated by its root subgroups. The standard reference for this is

> Steinberg, Robert, John Faulkner, and Robert Wilson. Lectures on Chevalley groups. New Haven: Yale University, 1967.

Though a more readable introduction is found on MathOverflow, here and here.