Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic

2004/08/31 by Erik Hostens, Jeroen Dehaene, Bart De Moor · 1 voice · 12 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Abelian group #Algebra over a field #Algorithm #Arithmetic #Clifford algebra #Computer science #Cryptography #Cyclic group #Dimension (graph theory) #Discrete mathematics #Engineering #Group (periodic table) #Mathematical physics #Mathematics #Modular arithmetic #Modular design #Modular group #Modulo #Pauli matrices #Physics #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Stabilizer (aeronautics) #quant-ph

paper · pdf · doi:10.1103/physreva.71.042315

published as Phys. Rev. A 71, 042315 (2005) · 10 pages, RevTeX

arxiv created 2005/02/22 · openalex publication_date 2005/04/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We describe generalizations of the Pauli group, the Clifford group, and stabilizer states for qudits in a Hilbert space of arbitrary dimension d. We examine a link with modular arithmetic, which yields an efficient way of representing the Pauli group and the Clifford group with matrices over ℤ_d. We further show how a Clifford operation can be efficiently decomposed into one and two-qudit operations. We also focus in detail on standard basis expansions of stabilizer states.

Citations

• Universal simulation of Hamiltonian dynamics for quantum systems with finite-dimensional state spaces
• Measurement-based quantum computation with cluster states
• Stabilizer Codes and Quantum Error Correction

Cited by

• Stabilizer rank bounds for magic-state orbits
• Clifford disentanglers for entanglement reduction in molecular electronic structure simulations
• An ideal characterization of the Clifford operators
• Systems of Imprimitivity for the Clifford Group
• Multi-qubit stabilizer and cluster entanglement witnesses
• Quantum error correction with the toric Gottesman-Kitaev-Preskill code
• Improvement of stabilizer-based entanglement distillation protocols by encoding operators
• Perfect many-to-one teleportation with stabilizer states
• Qudit hypergraph states
• Optimizing quantum codes with an application to the loss channel with partial erasure information
• Efficient simulatability of continuous-variable circuits with large Wigner negativity
• A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems
• Qudit surface codes and gauge theory with finite cyclic groups

Discussions

• Fix the generalized Pauli (Heisenberg–Weyl) formalism first: with $\omega = e^{2\pi i/3}$ , the clock is $Z = \mathrm{diag}(1,\omega,\omega^2)$ — so $Z|j\rangle = \omega^j|j\rangle$ — and the shift is [stackexchange, 1 points]