Pauli (Encodings)

Pauli Type

Namespace: Encodings
Assembly: Encodings.dll
Parent Module: IndexedPauli
Base Type: obj
All Interfaces: IEquatable<Pauli> , IStructuralEquatable , IComparable<Pauli> , IComparable , IStructuralComparable

Single-qubit Pauli operators forming the Pauli group.

The four single-qubit Pauli matrices form a basis for all 2×2 Hermitian matrices:

I = identity matrix [[1,0],[0,1]]
X = bit-flip (NOT) [[0,1],[1,0]]
Y = bit+phase flip [[0,-i],[i,0]]
Z = phase-flip [[1,0],[0,-1]]

The multiplication table is cyclic: XY = iZ, YZ = iX, ZX = iY. Two distinct non-identity Paulis always anti-commute: XY = -YX. Each Pauli squares to identity: X² = Y² = Z² = I.

Union cases

Union case	Description
`I` Full Usage: `I`	Identity operator (no operation).
`X` Full Usage: `X`	Pauli-X (bit-flip, NOT gate, σₓ).
`Y` Full Usage: `Y`	Pauli-Y (bit and phase flip, σᵧ).
`Z` Full Usage: `Z`	Pauli-Z (phase-flip, σᵤ).

Instance members

Instance member	Description
`this.IsI` Full Usage: `this.IsI` Returns: `bool`	Returns: `bool`
`this.IsX` Full Usage: `this.IsX` Returns: `bool`	Returns: `bool`
`this.IsY` Full Usage: `this.IsY` Returns: `bool`	Returns: `bool`
`this.IsZ` Full Usage: `this.IsZ` Returns: `bool`	Returns: `bool`

Static members

Static member	Description
`l * r` Full Usage: `l * r` Parameters: l : `Pauli` - The left Pauli operator. r : `Pauli` - The right Pauli operator. Returns: `Pauli * Phase` A tuple (resultOp, phase) where resultOp is the resulting Pauli operator and phase is the phase factor from the multiplication.	Multiplies two Pauli operators, returning the result operator and phase. Pauli multiplication follows these rules: Any Pauli times I equals itself: P · I = I · P = P Any Pauli squared is identity: X² = Y² = Z² = I Cyclic products: XY = iZ, YZ = iX, ZX = iY Anti-commutation: YX = -iZ, ZY = -iX, XZ = -iY l : `Pauli` The left Pauli operator. r : `Pauli` The right Pauli operator. Returns: `Pauli * Phase` A tuple (resultOp, phase) where resultOp is the resulting Pauli operator and phase is the phase factor from the multiplication.
`Pauli.Apply` Full Usage: `Pauli.Apply` Returns: `string -> Pauli option` Some Pauli if valid; None otherwise.	Parses a Pauli operator from its string representation. Returns: `string -> Pauli option` Some Pauli if valid; None otherwise.
`Pauli.FromChar` Full Usage: `Pauli.FromChar` Returns: `char -> Pauli option` Some Pauli if valid; None otherwise.	Parses a Pauli operator from a single character. Returns: `char -> Pauli option` Some Pauli if valid; None otherwise.
`Pauli.FromString` Full Usage: `Pauli.FromString` Returns: `string -> IxOp<uint32, Pauli> option` A parser function for indexed Pauli strings.	Parses an indexed Pauli operator from string format "P_n" (e.g., "X_0", "Z_3"). Returns: `string -> IxOp<uint32, Pauli> option` A parser function for indexed Pauli strings.
`Pauli.Identity` Full Usage: `Pauli.Identity` Returns: `Pauli`	Gets the identity Pauli operator. Returns: `Pauli`