# expectation_value

Graph.expectation_value(state, operator, *, name=None)

Calculate the expectation value of an operator with respect to a state.

The last dimension of the state must be equal to the last two dimensions of the operator and their batch shapes must be broadcastable.

### Parameters

• state (np.ndarray or Tensor) – The state. It must be a vector of shape (..., D).
• operator (np.ndarray or Tensor) – The operator. It must be of shape (..., D, D).
• name (str or None , optional) – The name of the node.

### Returns

The expectation value with shape (...).

### Return type

Tensor

Graph.density_matrix_expectation_value : Expectation value of an operator with respect to a density matrix.

Graph.inner_product : Inner product of two vectors.

Graph.outer_product : Outer product of two vectors.

Graph.trace : Trace of an object.

## Notes

The expectation value of an operator $\mathbf{A}$ with respect to a vector $\mathbf{x}$

$\mathbf{x}^\dagger \mathbf{A} \mathbf{x} = \langle x \vert \mathbf{A} \vert x \rangle = \sum_{ij} x^\ast_{i} A_{ij} x_{j} .$

## Examples

>>> graph.expectation_value(np.array([1j, 1j]), np.eye(2), name="expectation")
<Tensor: name="expectation", operation_name="expectation_value", shape=()>
>>> result = bo.execute_graph(graph=graph, output_node_names="expectation")
>>> result["output"]["expectation"]["value"]
2.+0.j
>>> graph.expectation_value(np.ones([3,1,4]), np.ones([2,4,4]), name="expectation)
<Tensor: name="expectation", operation_name="expectation_value", shape=(3, 2)>
>>> result = bo.execute_graph(graph=graph, output_node_names="expectation")
>>> result["output"]["expectation"]["value"]
array([[16, 16], [16, 16], [16, 16]])