Alphabeta MathHow statement and proof provenance work
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Binary Pauli label and quotient
Definition
Let be the scalar center of . Define when , and let be the set of equivalence classes . The proposed coset product is . The quotient theorem verifies this product is well-defined and proves that the assignment
is a group isomorphism to the additive group of . A binary label deliberately forgets the scalar phase.
Depends on
- Phase aware n qubit pauli group
- Pauli product and adjoint
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- Finite fields and their order
- Vector space over a field
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
Used by
- Stabilizer encoder by symplectic elimination Algorithm
- Binary symplectic form Definition
- Phase-aware binary Clifford tableau Definition
- Phase-aware stabilizer tableau Definition
- Stabilizer code space Definition
- Stabilizer subgroup Definition
- Stabilizer syndromes and logical Pauli operators Definition
- Computational basis states have an extended stabilizer tableau Lemma
- Random Pauli measurement updates stabilizer generators Lemma
- Stabilizer labels form an isotropic space of the same rank Lemma
- Tableau row reduction returns the deterministic Pauli sign Lemma
- Binary Pauli quotient isomorphism Theorem
- Clifford induces symplectic map Theorem
- CSS encoding dimension and commutation Theorem
- Dimension of the logical Pauli quotient Theorem
Dependency tree · two levels
42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John Preskill, Quantum Error Correction, Physics 229 Lecture Notes, Chapter 7 (standard reference, not scraped)
- M. Hein et al., Entanglement in Graph States and Its Applications (standard reference, not scraped)