qml.pauli.trace_inner_product

trace_inner_product(A, B)

Trace inner product $\langle A, B \rangle = \text{tr}\left(A^\dagger B\right)/\text{dim}(A)$ between two operators $A$ and $B$.

If the inputs are np.ndarray, leading [broadcasting](https://docs.pennylane.ai/en/stable/introduction/circuits.html#parameter-broadcasting-in-qnodes) axes are supported for either or both inputs.

Warning

Operator inputs are assumed to be Hermitian. In particular, sums of Pauli operators are assumed to have real-valued coefficients. We recommend to use matrix representations for non-Hermitian inputs. In case of non-Hermitian PauliSentence or Operator inputs, the Hermitian conjugation needs to be done manually by inputting $A^\dagger$.

Parameters

• A (Union[PauliSentence, Operator, TensorLike]) – First operator

• B (Union[PauliSentence, Operator, TensorLike]) – Second operator of the same type as A

Returns

Result is either a single float or an array of floats (in batches of the broadcasting dimension).

Return type

Union[float, TensorLike]

Example

>>> from pennylane.pauli import trace_inner_product
>>> trace_inner_product(qml.X(0) + qml.Y(0), qml.Y(0) + qml.Z(0))
1.0

If both operators are arrays, a leading batch dimension is broadcasted.

>>> batch = 10
>>> ops1 = np.random.rand(batch, 16, 16)
>>> op2 = np.random.rand(16, 16)
>>> trace_inner_product(ops1, op2).shape
(10,)
>>> trace_inner_product(op2, ops1).shape
(10,)

We can also have both arguments broadcasted.

>>> trace_inner_product(ops1, ops1).shape
(10, 10)

Usage Details

PauliSentence and Operator inputs are assumed to be Hermitian. In particular, the input A is not conjugated when operators are used. To get correct results, we can either use the matrix representation or manually conjugate the operator.

>>> A = qml.X(0) - 1j * qml.Y(0)
>>> Ad = qml.X(0) + 1j * qml.Y(0)
>>> B = qml.X(0) + 1j * qml.Y(0)
>>> trace_inner_product(Ad, B) == trace_inner_product(qml.matrix(A), qml.matrix(B))
True

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