Stabilizer code: Difference between revisions

Revision as of 14:45, 10 June 2023

Revision as of 20:22, 23 November 2023

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One of the fundamental notions in quantum error correction theory is that it

suffices to correct a [[Discrete set|discrete]] error set with [[Support (mathematics)|support]] in the [[Pauli group]]

<math>\Pi^{n}</math>. Suppose that the errors affecting an

encoded quantum state are a subset <math>\mathcal{E}</math> of the [[Pauli group]] <math>\Pi^{n}</math>:

:<math>\mathcal{E}\subset\Pi^{n}.</math>

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the syndrome via a [[parity measurement]] and applying a corrective operation.

== Relation between [[Pauli group]] and binary vectors ==

A simple but useful mapping exists between elements of <math>\Pi</math> and the binary

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</math>

The above binary representation and [[symplectic algebra]] are useful in making

the relation between classical linear [[error correction]] and [[quantum error correction]] more explicit.

By comparing quantum error correcting codes in this language to [[symplectic vector space]]s, we can see the following. A [[Symplectic vector space#Subspaces|symplectic]] subspace corresponds to a [[direct sum]] of Pauli algebras (i.e., encoded qubits), while an [[Symplectic vector space#Subspaces|isotropic]] subspace corresponds to a set of stabilizers.