Pauli group
E674930
The Pauli group is the set of all products of Pauli matrices (up to phase factors), forming a fundamental discrete group used to describe qubit operations in quantum mechanics and quantum computing.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Clifford group subgroup
ⓘ
discrete group ⓘ mathematical group ⓘ matrix group ⓘ |
| actsOn | single qubit Hilbert space ⓘ |
| basisRole | forms operator basis for single-qubit observables up to phase ⓘ |
| closureProperty |
closed under matrix multiplication
ⓘ
closed under taking inverses ⓘ |
| commutationStructure | elements either commute or anticommute ⓘ |
| containsElement |
-I
ⓘ
-iI ⓘ Pauli X matrix NERFINISHED ⓘ Pauli Y matrix NERFINISHED ⓘ Pauli Z matrix NERFINISHED ⓘ iI ⓘ identity matrix ⓘ |
| definedOver | 2-dimensional complex Hilbert space for single qubit ⓘ |
| elementType | 2×2 complex matrices for single-qubit case ⓘ |
| field |
quantum computing
ⓘ
quantum information theory ⓘ quantum mechanics ⓘ |
| generalizesTo | n-qubit Pauli group NERFINISHED ⓘ |
| generatedBy |
Pauli X matrix
ⓘ
Pauli Y matrix NERFINISHED ⓘ Pauli Z matrix NERFINISHED ⓘ global phase factors ⓘ |
| hasCenter | set of scalar phase multiples of identity ⓘ |
| hasProperty |
finite
ⓘ
non-abelian ⓘ unitary ⓘ |
| isAbelian | false ⓘ |
| measurementRole | generators correspond to spin-1/2 observables along orthogonal axes ⓘ |
| namedAfter | Wolfgang Pauli NERFINISHED ⓘ |
| nQubitVersionOrder | 4^(n+1) for n-qubit Pauli group including phases ±1, ±i ⓘ |
| order | 16 for single-qubit Pauli group including ±1, ±i phases ⓘ |
| relatedTo |
Clifford group
NERFINISHED
ⓘ
Heisenberg–Weyl group NERFINISHED ⓘ stabilizer group ⓘ |
| representation | projective representation of Z2×Z2 for single-qubit case ⓘ |
| usedFor |
defining stabilizer formalism
ⓘ
describing qubit errors ⓘ fault-tolerant quantum computation ⓘ modeling qubit operations ⓘ quantum error correction ⓘ stabilizer codes ⓘ |
| usedIn |
Pauli error channels
ⓘ
Pauli twirling protocols NERFINISHED ⓘ stabilizer measurement circuits ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.