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qp.pauli.trace_inner_product

trace_inner_product(A, B)[source]

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 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 \(A^\dagger\).

Args:

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

Returns:

Union[float, TensorLike]: Result is either a single float or an array of floats (in batches of the broadcasting dimension).

Example

>>> from pennylane.pauli import trace_inner_product
>>> trace_inner_product(qp.X(0) + qp.Y(0), qp.Y(0) + qp.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)

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