Group Operations

Decoupling groups and group operations.

class gadd.group_operations.DecouplingGroup(elements: Dict[str, int], names: Dict[int, str], multiplication: List[List[int]], inverse_map: Dict[int, int] | None)

Bases: object

Mathematical structure defining a decoupling group for DD sequences.

This class represents the algebraic structure used in the GADD paper for constructing dynamical decoupling sequences. It encodes the group elements, their names, multiplication table, and inverse relationships according to the Pauli group algebra extended with phase information.

The default group used in the paper is G = {Ip, Im, Xp, Xm, Yp, Ym, Zp, Zm} where the subscripts p/m indicate positive/negative phase rotations on the Bloch sphere. This encoding allows for systematic exploration of DD sequences that maintain the group constraint of multiplying to the identity element.

elements

Mapping from element names to integer indices.

Type:

Dict[str, int]

names

Reverse mapping from indices to element names.

Type:

Dict[int, str]

multiplication

2D list encoding the group multiplication table where multiplication[i][j] gives the result of multiplying element i with element j.

Type:

List[List[int]]

inverse_map

Optional mapping from elements to their group inverses.

Type:

Dict[int, int] | None

Example

>>> group = DEFAULT_GROUP
>>> multiply("Xp", "Yp", group)  # Returns index for result
>>> group.element_name(2)  # Returns "Xp"

elements: Dict[str, int]

names: Dict[int, str]

multiplication: List[List[int]]

inverse_map: Dict[int, int] | None

property size: int

Size of the group.

element_name(index: int) → str

Get element name from index.

element_index(name: str) → int

Get element index from name.

__init__(elements: Dict[str, int], names: Dict[int, str], multiplication: List[List[int]], inverse_map: Dict[int, int] | None) → None

gadd.group_operations.multiply(p1: int | str, p2: int | str, group: DecouplingGroup | None = None) → int

Multiply two group elements.

Parameters:

• p1 – First element (index or name).

• p2 – Second element (index or name).

• group – Optional custom group structure. Defaults to the G = {Ip, Im, Xp, Xm, Yp, Ym, Zp, Zm} used in the paper.

Returns:

Index of the resulting element.

gadd.group_operations.invert(a: int | str, group: DecouplingGroup | None = None) → int

Find the inverse of a group element.

Parameters:

• a – Element to invert (index or name).

• group – Optional custom group structure. Defaults to the G = {Ip, Im, Xp, Xm, Yp, Ym, Zp, Zm} used in the paper.

Returns:

Index of the inverse element.

gadd.group_operations.complete_sequence_to_identity(partial_sequence: List[int | str], group: DecouplingGroup | None = None) → int

Find the element needed to complete a sequence to multiply to identity.

Parameters:

• partial_sequence – Incomplete sequence of group elements.

• group – Optional custom group structure.

Returns:

Index of element needed to complete sequence to identity.