pauli_decompose(H, hide_identity=False, wire_order=None, pauli=False, check_hermitian=True)¶
Decomposes a Hermitian matrix into a linear combination of Pauli operators.
H (tensor_like[complex]) – a Hermitian matrix of dimension \(2^n\times 2^n\).
hide_identity (bool) – does not include the Identity observable within the tensor products of the decomposition if
wire_order (list[Union[int, str]]) – the ordered list of wires with respect to which the operator is represented as a matrix.
pauli (bool) – return a
check_hermitian (bool) – check if the provided matrix is Hermitian if
- Return type
We can use this function to compute the Pauli operator decomposition of an arbitrary Hermitian matrix:
>>> A = np.array( ... [[-2, -2+1j, -2, -2], [-2-1j, 0, 0, -1], [-2, 0, -2, -1], [-2, -1, -1, 0]]) >>> H = qml.pauli_decompose(A) >>> print(H) (-1.5) [I0 X1] + (-1.5) [X0 I1] + (-1.0) [I0 I1] + (-1.0) [I0 Z1] + (-1.0) [X0 X1] + (-0.5) [I0 Y1] + (-0.5) [X0 Z1] + (-0.5) [Z0 X1] + (-0.5) [Z0 Y1] + (1.0) [Y0 Y1]
We can return a
PauliSentenceinstance by using the keyword argument
>>> ps = qml.pauli_decompose(A, pauli=True) >>> print(ps) -1.0 * I + -1.5 * X(1) + -0.5 * Y(1) + -1.0 * Z(1) + -1.5 * X(0) + -1.0 * X(0) @ X(1) + -0.5 * X(0) @ Z(1) + 1.0 * Y(0) @ Y(1) + -0.5 * Z(0) @ X(1) + -0.5 * Z(0) @ Y(1)
By default the wires are numbered [0, 1, …, n], but we can also set custom wires using the
>>> ps = qml.pauli_decompose(A, pauli=True, wire_order=['a', 'b']) >>> print(ps) -1.0 * I + -1.5 * X(b) + -0.5 * Y(b) + -1.0 * Z(b) + -1.5 * X(a) + -1.0 * X(a) @ X(b) + -0.5 * X(a) @ Z(b) + 1.0 * Y(a) @ Y(b) + -0.5 * Z(a) @ X(b) + -0.5 * Z(a) @ Y(b)
This method internally uses a generalized decomposition routine to convert the matrix to a weighted sum of Pauli words acting on \(n\) qubits in time \(O(n 4^n)\). The input matrix is written as a quantum state in the computational basis following the channel-state duality. A Bell basis transformation is then performed using the Walsh-Hadamard transform, after which coefficients for each of the \(4^n\) Pauli words are computed while accounting for the phase from each
PauliYterm occurring in the word.
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