classical_jacobian(qnode, argnum=None, expand_fn=None, trainable_only=True)[source]

Returns a function to extract the Jacobian matrix of the classical part of a QNode.

This transform allows the classical dependence between the QNode arguments and the quantum gate arguments to be extracted.

  • qnode (pennylane.QNode) – QNode to compute the (classical) Jacobian of

  • argnum (int or Sequence[int]) – indices of QNode arguments with respect to which the (classical) Jacobian is computed

  • expand_fn (None or function) – an expansion function (if required) to be applied to the QNode quantum tape before the classical Jacobian is computed


Function which accepts the same arguments as the QNode. When called, this function will return the Jacobian of the QNode gate arguments with respect to the QNode arguments indexed by argnum.

Return type



Consider the following QNode:

>>> @qml.qnode(dev)
... def circuit(weights):
...     qml.RX(weights[0], wires=0)
...     qml.RY(0.2 * weights[0], wires=1)
...     qml.RY(2.5, wires=0)
...     qml.RZ(weights[1] ** 2, wires=1)
...     qml.RX(weights[2], wires=1)
...     return qml.expval(qml.Z(0) @ qml.Z(1))

We can use this transform to extract the relationship \(f: \mathbb{R}^n \rightarrow \mathbb{R}^m\) between the input QNode arguments \(w\) and the gate arguments \(g\), for a given value of the QNode arguments:

>>> cjac_fn = qml.gradients.classical_jacobian(circuit)
>>> weights = np.array([1., 1., 0.6], requires_grad=True)
>>> cjac = cjac_fn(weights)
>>> print(cjac)
[[1.  0.  0. ]
 [0.2 0.  0. ]
 [0.  2.  0. ]
 [0.  0.  1. ]]

The returned Jacobian has rows corresponding to gate arguments, and columns corresponding to QNode arguments; that is,

\[J_{ij} = \frac{\partial}{\partial g_i} f(w_j).\]

We can see that:

  • The zeroth element of weights is repeated on the first two gates generated by the QNode.

  • The third row consisting of all zeros indicates that the third gate RY(2.5) does not depend on the weights.

  • The quadratic dependence of the fourth gate argument yields \(2\cdot 0.6=1.2\).


The QNode is constructed during this operation.

For a QNode with multiple QNode arguments, the arguments with respect to which the Jacobian is computed can be controlled with the argnum keyword argument. The output and its format depend on the backend:

Output format of classical_jacobian




type(argnum) = Sequence[int]


tuple(array) [1]




array [2]











[1] If there only is one trainable QNode argument, the tuple is unpacked to a single array, as is the case for jacobian().

[2] For JAX, argnum=None defaults to argnum=0 in contrast to all other interfaces. This means that only the classical Jacobian with respect to the first QNode argument is computed if no argnum is provided.

Example with ``argnum``

>>> @qml.qnode(dev)
... def circuit(x, y, z):
...     qml.RX(qml.math.sin(x), wires=0)
...     qml.CNOT(wires=[0, 1])
...     qml.RY(y ** 2, wires=1)
...     qml.RZ(1 / z, wires=1)
...     return qml.expval(qml.Z(0) @ qml.Z(1))
>>> jac_fn = qml.gradients.classical_jacobian(circuit, argnum=[1, 2])
>>> x, y, z = np.array([0.1, -2.5, 0.71])
>>> jac_fn(x, y, z)
(array([-0., -5., -0.]), array([-0.        , -0.        , -1.98373339]))

Only the Jacobians with respect to the arguments x and y were computed, and returned as a tuple of arrays.