qml.grouping¶
Overview¶
This subpackage defines functions and classes for generating and manipulating elements of the Pauli group. It also contains Pauli-word partitioning functionality used in measurement optimization.
Functions¶
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Performs a check if two Pauli words have the same |
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Converts a binary vector of even dimension to an Observable instance. |
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Transforms the Pauli word to diagonal form in the computational basis. |
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Diagonalizes a list of qubit-wise commutative groupings of Pauli strings. |
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Diagonalizes a list of mutually qubit-wise commutative Pauli words. |
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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). |
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Checks if two Pauli words commute. |
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Checks if an observable instance is a Pauli word. |
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Checks if two Pauli words in the binary vector representation are qubit-wise commutative. |
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Converts a list of Pauli words to the binary vector representation and yields a row matrix of the binary vectors. |
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Partitions then diagonalizes a list of Pauli words, facilitating simultaneous measurement of all observables within a partition. |
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Partitions the \(n\)-qubit Pauli group into qubit-wise commuting terms. |
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Generate the \(n\)-qubit Pauli group. |
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Multiply two Pauli words together and return the product as a Pauli word. |
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Multiply two Pauli words together, and return both their product as a Pauli word and the global phase. |
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Converts a Pauli word to the binary vector representation. |
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Convert a Pauli word from a tensor to its matrix representation. |
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Convert a Pauli word from a tensor to a string. |
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Obtains the adjacency matrix for the complementary graph of the qubit-wise commutativity graph for a given set of observables in the binary representation. |
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Performs circuit implementation of diagonalizing unitary for a Pauli word. |
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Convert a string in terms of |
Classes¶
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Class for partitioning a list of Pauli words according to some binary symmetric relation. |
Graph colouring¶
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¶
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Performs graph-colouring using the Largest Degree First heuristic. |
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Performs graph-colouring using the Recursive Largest Degree First heuristic. |
Pauli group¶
The single-qubit Pauli group consists of the four single-qubit Pauli operations
Identity, PauliX,
PauliY , and PauliZ. The \(n\)-qubit
Pauli group is constructed by taking all possible \(N\)-fold tensor products
of these elements. Elements of the \(n\)-qubit Pauli group are often known
as Pauli words, and have the form \(P_J = \otimes_{i=1}^{n}\sigma_i^{(J)}\),
where \(\sigma_i^{(J)}\) is one of the Pauli operators
(PauliX, PauliY,
PauliZ) or identity (Identity) acting on
the \(i^{th}\) qubit. The full \(n\)-qubit Pauli group has size
\(4^n\) (neglecting the four possible global phases that may arise from
multiplication of its elements).
The Pauli group can be constructed using the pauli_group()
function. To construct the group, it is recommended to provide a wire map in
order to indicate the names and indices of the wires. (If none is provided, a
default mapping of integers will be used.)
>>> from pennylane.grouping import pauli_group
>>> pg_3 = list(pauli_group(3))
Multiplication of Pauli group elements can be performed using
pauli_mult() or
pauli_mult_with_phase():
>>> from pennylane.grouping import pauli_mult
>>> wire_map = {'a' : 0, 'b' : 1, 'c' : 2}
>>> pg = list(pauli_group(3, wire_map=wire_map))
>>> pg[3]
PauliZ(wires=['b']) @ PauliZ(wires=['c'])
>>> pg[55]
PauliY(wires=['a']) @ PauliY(wires=['b']) @ PauliZ(wires=['c'])
>>> pauli_mult(pg[3], pg[55], wire_map=wire_map)
PauliY(wires=['a']) @ PauliX(wires=['b'])
Pauli observables can be converted to strings (and vice versa):
>>> from pennylane.grouping import pauli_word_to_string, string_to_pauli_word
>>> pauli_word_to_string(pg[55], wire_map=wire_map)
'YYZ'
>>> string_to_pauli_word('ZXY', wire_map=wire_map)
PauliZ(wires=['a']) @ PauliX(wires=['b']) @ PauliY(wires=['c'])
The matrix representation for arbitrary Paulis and wire maps can also be performed.
>>> pennylane.grouping import pauli_word_to_matrix
>>> wire_map = {'a' : 0, 'b' : 1}
>>> pauli_word = qml.PauliZ('b') # corresponds to Pauli 'IZ'
>>> pauli_word_to_matrix(pauli_word, wire_map=wire_map)
array([[ 1., 0., 0., 0.],
[ 0., -1., 0., -0.],
[ 0., 0., 1., 0.],
[ 0., -0., 0., -1.]])
Grouping observables¶
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=[0]), PauliX(wires=[1]))],
[PauliY(wires=[0]), PauliZ(wires=[1])]]
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=[0]), PauliX(wires=[1]))],
[PauliY(wires=[0]), PauliZ(wires=[1])]]
>>> coeffs_groupings
[[4.21], [1.43, 0.97]]