matrix(op, *, wire_order=None)[source]

The matrix representation of an operation or quantum circuit.

  • op (Operator, pennylane.QNode, QuantumTape, or Callable) – An operator, quantum node, tape, or function that applies quantum operations.

  • wire_order (Sequence[Any], optional) – Order of the wires in the quantum circuit. Defaults to the order in which the wires appear in the quantum function.


Function which accepts the same arguments as the QNode or quantum function. When called, this function will return the unitary matrix in the appropriate autodiff framework (Autograd, TensorFlow, PyTorch, JAX) given its parameters.

Return type

tensor_like or function


Given an instantiated operator, qml.matrix returns the matrix representation:

>>> op = qml.RX(0.54, wires=0)
>>> qml.matrix(op)
[[0.9637709+0.j         0.       -0.26673144j]
[0.       -0.26673144j 0.9637709+0.j        ]]

It can also be used in a functional form:

>>> x = torch.tensor(0.6, requires_grad=True)
>>> matrix_fn = qml.matrix(qml.RX)
>>> matrix_fn(x, wires=0)
tensor([[0.9553+0.0000j, 0.0000-0.2955j],
        [0.0000-0.2955j, 0.9553+0.0000j]], grad_fn=<AddBackward0>)

In its functional form, it is fully differentiable with respect to gate arguments:

>>> loss = torch.real(torch.trace(matrix_fn(x, wires=0)))
>>> loss.backward()
>>> x.grad

This operator transform can also be applied to QNodes, tapes, and quantum functions that contain multiple operations; see Usage Details below for more details.

qml.matrix can also be used with QNodes, tapes, or quantum functions that contain multiple operations.

Consider the following quantum function:

def circuit(theta):
    qml.RX(theta, wires=1)

We can use qml.matrix to generate a new function that returns the unitary matrix corresponding to the function circuit:

>>> matrix_fn = qml.matrix(circuit)
>>> theta = np.pi / 4
>>> matrix_fn(theta)
array([[ 0.92387953+0.j,  0.+0.j ,  0.-0.38268343j,  0.+0.j],
[ 0.+0.j,  -0.92387953+0.j,  0.+0.j,  0. +0.38268343j],
[ 0. -0.38268343j,  0.+0.j,  0.92387953+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.38268343j,  0.+0.j,  -0.92387953+0.j]])

Note that since wire_order was not specified, the default order [1, 0] for circuit was used, and the unitary matrix corresponds to the operation \(Z\otimes R_X(\theta)\). To obtain the matrix for \(R_X(\theta)\otimes Z\), specify wire_order=[0, 1] in the function call:

>>> matrix = qml.matrix(circuit, wire_order=[0, 1])

You can also get the unitary matrix for operations on a subspace of a larger Hilbert space. For example, with the same function circuit and wire_order=["a", 0, "b", 1] you obtain the \(16\times 16\) matrix for the operation \(I\otimes Z\otimes I\otimes R_X(\theta)\).

This unitary matrix can also be used in differentiable calculations. For example, consider the following cost function:

def circuit(theta):
    qml.RX(theta, wires=1) qml.PauliZ(wires=0)
    qml.CNOT(wires=[0, 1])

def cost(theta):
    matrix = qml.matrix(circuit)(theta)
    return np.real(np.trace(matrix))

Since this cost function returns a real scalar as a function of theta, we can differentiate it:

>>> theta = np.array(0.3, requires_grad=True)
>>> cost(theta)
>>> qml.grad(cost)(theta)