# qml.grouping¶

This subpackage defines functions and classes for Pauli-word partitioning functionality used in measurement optimization.

A Pauli word is defined as $$P_J = \prod_{i=1}^{N}\sigma_i^{(J)}$$, where $$\sigma_i^{(J)}$$ is one of the Pauli operators (PauliX, PauliY, PauliZ) or identity (Identity) acting on the $$i^{th}$$ qubit.

Pauli words can be used for expressing a qubit Hamiltonian. A qubit Hamiltonian has the form $$H_{q} = \sum_{J} C_J P_J$$ where $$C_{J}$$ are numerical coefficients, and $$P_J$$ are Pauli words.

A list of Pauli words can be partitioned according to certain grouping strategies. As an example, the group_observables() function partitions a list of observables (Pauli operations and tensor products thereof) into groupings according to a binary relation (e.g., qubit-wise commuting):

>>> observables = [qml.PauliY(0), qml.PauliX(0) @ qml.PauliX(1), qml.PauliZ(1)]
>>> obs_groupings = group_observables(observables)
>>> obs_groupings
[[Tensor(PauliX(wires=), PauliX(wires=))],
[PauliY(wires=), PauliZ(wires=)]]


The $$C_{J}$$ coefficients for each $$P_J$$ Pauli word making up a Hamiltonian can also be specified along with further options, such as the Pauli-word grouping method (e.g., qubit-wise commuting) and the underlying graph-colouring algorithm (e.g., recursive largest first) used for creating the groups of observables:

>>> obs = [qml.PauliY(0), qml.PauliX(0) @ qml.PauliX(1), qml.PauliZ(1)]
>>> coeffs = [1.43, 4.21, 0.97]
>>> obs_groupings, coeffs_groupings = group_observables(obs, coeffs, 'qwc', 'rlf')
>>> obs_groupings
[[Tensor(PauliX(wires=), PauliX(wires=))],
[PauliY(wires=), PauliZ(wires=)]]
>>> coeffs_groupings
[[4.21], [1.43, 0.97]]


## pennylane.grouping Package¶

This subpackage defines functions and classes for Pauli-word partitioning functionality used in measurement optimization.

### Functions¶

 are_identical_pauli_words(pauli_1, pauli_2) Performs a check if two Pauli words have the same wires and name attributes. binary_to_pauli(binary_vector[, wire_map]) Converts a binary vector of even dimension to an Observable instance. diagonalize_pauli_word(pauli_word) Transforms the Pauli word to diagonal form in the computational basis. diagonalize_qwc_groupings(qwc_groupings) Diagonalizes a list of qubit-wise commutative groupings of Pauli strings. diagonalize_qwc_pauli_words(qwc_grouping) Diagonalizes a list of mutually qubit-wise commutative Pauli words. group_observables(observables[, …]) Partitions a list of observables (Pauli operations and tensor products thereof) into groupings according to a binary relation (qubit-wise commuting, fully-commuting, or anticommuting). is_pauli_word(observable) Checks if an observable instance is a Pauli word. is_qwc(pauli_vec_1, pauli_vec_2) Checks if two Pauli words in the binary vector representation are qubit-wise commutative. observables_to_binary_matrix(observables[, …]) Converts a list of Pauli words to the binary vector representation and yields a row matrix of the binary vectors. optimize_measurements(observables[, …]) Partitions then diagonalizes a list of Pauli words, facilitating simultaneous measurement of all observables within a partition. pauli_to_binary(pauli_word[, n_qubits, wire_map]) Converts a Pauli word to the binary vector representation. qwc_complement_adj_matrix(binary_observables) Obtains the adjacency matrix for the complementary graph of the qubit-wise commutativity graph for a given set of observables in the binary representation. qwc_rotation(pauli_operators) Performs circuit implementation of diagonalizing unitary for a Pauli word.

### Classes¶

 PauliGroupingStrategy(observables[, …]) Class for partitioning a list of Pauli words according to some binary symmetric relation.

## pennylane.grouping.graph_colouring Module¶

A module for heuristic algorithms for colouring Pauli graphs.

A Pauli graph is a graph where vertices represent Pauli words and edges denote if a specified symmetric binary relation (e.g., commutation) is satisfied for the corresponding Pauli words. The graph-colouring problem is to assign a colour to each vertex such that no vertices of the same colour are connected, using the fewest number of colours (lowest “chromatic number”) as possible.

### Functions¶

 largest_first(binary_observables, adj) Performs graph-colouring using the Largest Degree First heuristic. recursive_largest_first(binary_observables, adj) Performs graph-colouring using the Recursive Largest Degree First heuristic.