qml.pauli.PauliGroupingStrategy

class PauliGroupingStrategy(observables, grouping_type='qwc', graph_colourer='lf')

Bases: object

Class for partitioning a list of Pauli words according to some binary symmetric relation.

Partitions are defined by the binary symmetric relation of interest, e.g., all Pauli words in a partition being mutually commuting. The partitioning is accomplished by formulating the list of Pauli words as a graph where nodes represent Pauli words and edges between nodes denotes that the two corresponding Pauli words satisfy the symmetric binary relation.

Obtaining the fewest number of partitions such that all Pauli terms within a partition mutually satisfy the binary relation can then be accomplished by finding a partition of the graph nodes such that each partition is a fully connected subgraph (a “clique”). The problem of finding the optimal partitioning, i.e., the fewest number of cliques, is the minimum clique cover (MCC) problem. The solution of MCC may be found by graph colouring on the complementary graph. Both MCC and graph colouring are NP-Hard, so heuristic graph colouring algorithms are employed to find approximate solutions in polynomial time.

Parameters

• observables (list[Operator]) – A list of Pauli words to be partitioned according to a grouping strategy.

• grouping_type (str) – The binary relation used to define partitions of the Pauli words, can be 'qwc' (qubit-wise commuting), 'commuting', or 'anticommuting'.

• graph_colourer (str) – The heuristic algorithm to employ for graph colouring, can be 'lf' (Largest First), 'rlf' (Recursive Largest First), 'dsatur' (Degree of Saturation), or 'gis' (IndependentSet). Defaults to 'lf'.

Raises

ValueError – If arguments specified for grouping_type or graph_colourer are not recognized.

See also

rustworkx.ColoringStrategy for more information on the ('lf', 'dsatur', 'gis') strategies.

Attributes

adj_matrix	Adjacency matrix for the complement of the Pauli graph determined by the grouping_type.
binary_observables	Binary Matrix corresponding to the symplectic representation of self.observables.
complement_graph	Complement graph of the (anti-)commutation graph constructed from the Pauli observables.

adj_matrix

Adjacency matrix for the complement of the Pauli graph determined by the grouping_type.

The adjacency matrix for an undirected graph of N nodes is an N x N symmetric binary matrix, where matrix elements of 1 denote an edge (grouping strategy is not satisfied), and matrix elements of 0 denote no edge (grouping strategy is satisfied).

binary_observables

Binary Matrix corresponding to the symplectic representation of self.observables.

It is an m x n matrix where each row is the symplectic (binary) representation of self.observables, with m = len(self.observables) and n the number of qubits acted on by the observables.

complement_graph

Complement graph of the (anti-)commutation graph constructed from the Pauli observables.

Edge (i,j) is present in the graph if observable[i] and observable[j] do not satisfy the grouping_type strategy.

The nodes are the observables (can only be accessed through their integer index).

Methods

binary_repr([n_qubits, wire_map])	Converts the list of Pauli words to a binary matrix, i.e. a matrix where row m is the symplectic representation of self.observables[m].
idx_partitions_from_graph([observables_indices])	Use Rustworkx graph colouring algorithms to partition the indices of the Pauli observables into tuples containing the indices of observables satisying the binary relation determined by self.grouping_type.
partition_observables()	Partition the observables into groups of observables mutually satisfying the binary relation determined by self.grouping_type.
pauli_partitions_from_graph()	Partition Pauli observables into lists of (anti-)commuting observables using Rustworkx graph colouring algorithms based on binary relation determined by self.grouping_type.

binary_repr(n_qubits=None, wire_map=None)

Converts the list of Pauli words to a binary matrix, i.e. a matrix where row m is the symplectic representation of self.observables[m].

Parameters

• n_qubits (int) – number of qubits to specify dimension of binary vector representation

• wire_map (dict) – dictionary containing all wire labels used in the Pauli word as keys, and unique integer labels as their values

Returns

a column matrix of the Pauli words in binary vector representation

Return type

array[int]

idx_partitions_from_graph(observables_indices=None)

Use Rustworkx graph colouring algorithms to partition the indices of the Pauli observables into tuples containing the indices of observables satisying the binary relation determined by self.grouping_type.

Parameters

observables_indices (Optional[TensorLike]) – A tensor or list of indices associated to each observable. This argument is helpful when the observables used in the graph colouring are part of a bigger set of observables. Defaults to None. If None, the partitions are made up of the relative indices, i.e. assuming self.observables have indices in [0, len(observables)-1].

Raises

IndexError – When observables_indices is not of the same length as the observables.

Returns

Tuple of tuples containing the indices of the partitioned observables.

Return type

tuple[tuple[int]]

partition_observables()

Partition the observables into groups of observables mutually satisfying the binary relation determined by self.grouping_type.

Returns

List of partitions of the Pauli observables made up of mutually (anti-)commuting observables.

Return type

list[list[Operator]]

pauli_partitions_from_graph()

Partition Pauli observables into lists of (anti-)commuting observables using Rustworkx graph colouring algorithms based on binary relation determined by self.grouping_type.

Returns

List of partitions of the Pauli observables made up of mutually (anti-)commuting terms.

Return type

list[list[Observable]]

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