Phase (Encodings)

Phase Type

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

The global phase factor attached to a Pauli product.

When multiplying Pauli matrices, the result is another Pauli matrix times a phase in {+1, -1, +i, -i}. For example: X · Y = iZ (phase = +i).

Phases are tracked as a discriminated union enabling exact (no floating-point error) phase arithmetic. The multiplication table forms a cyclic group Z₄:

   P1 * P1 = P1    Pi * Pi = M1
   P1 * M1 = M1    Pi * Mi = P1
   M1 * M1 = P1    Mi * Mi = M1

Union cases

Union case	Description
`M1` Full Usage: `M1`	Phase factor -1.
`Mi` Full Usage: `Mi`	Phase factor -i (negative imaginary).
`P1` Full Usage: `P1`	Phase factor +1 (identity phase).
`Pi` Full Usage: `Pi`	Phase factor +i (positive imaginary).

Instance members

Instance member	Description
`this.FoldIntoGlobalPhase` Full Usage: `this.FoldIntoGlobalPhase` Parameters: globalPhase : `Complex` - The complex number to multiply by this phase. Returns: `Complex` The product of globalPhase and this phase factor.	Folds this phase into a complex global phase by multiplication. Converts the symbolic phase to its numeric effect: P1 → globalPhase, M1 → -globalPhase, Pi → i·globalPhase, Mi → -i·globalPhase. globalPhase : `Complex` The complex number to multiply by this phase. Returns: `Complex` The product of globalPhase and this phase factor.
`this.IsComplex` Full Usage: `this.IsComplex` Returns: `bool` `true` for Pi (+i) and Mi (-i); `false` for P1 (+1) and M1 (-1).	Gets whether this phase is purely imaginary. Returns: `bool` `true` for Pi (+i) and Mi (-i); `false` for P1 (+1) and M1 (-1).
`this.IsM1` Full Usage: `this.IsM1` Returns: `bool`	Returns: `bool`
`this.IsMi` Full Usage: `this.IsMi` Returns: `bool`	Returns: `bool`
`this.IsP1` Full Usage: `this.IsP1` Returns: `bool`	Returns: `bool`
`this.IsPi` Full Usage: `this.IsPi` Returns: `bool`	Returns: `bool`
`this.IsPositive` Full Usage: `this.IsPositive` Returns: `bool` `true` for P1 (+1) and Pi (+i); `false` for M1 (-1) and Mi (-i).	Gets whether this phase has a positive real or imaginary part. Returns: `bool` `true` for P1 (+1) and Pi (+i); `false` for M1 (-1) and Mi (-i).

Static members

Static member	Description
`l * r` Full Usage: `l * r` Parameters: l : `Phase` - The left phase factor. r : `Phase` - The right phase factor. Returns: `Phase` The product phase factor in {P1, M1, Pi, Mi}.	Multiplies two phase factors exactly. Phase multiplication follows the cyclic group Z₄ structure: i² = -1, i · (-i) = 1, (-1)² = 1, etc. l : `Phase` The left phase factor. r : `Phase` The right phase factor. Returns: `Phase` The product phase factor in {P1, M1, Pi, Mi}.