exp(op, coeff=1, num_steps=None, id=None)[source]

Take the exponential of an Operator times a coefficient.

  • base (Operator) – The Operator to be exponentiated

  • coeff (float) – a scalar coefficient of the operator

  • num_steps (int) – The number of steps used in the decomposition of the exponential operator, also known as the Trotter number. If this value is None and the Suzuki-Trotter decomposition is needed, an error will be raised.

  • id (str) – id for the Exp operator. Default is None.


An Operator representing an operator exponential.

Return type



This operator supports a batched base, a batched coefficient and a combination of both:

>>> op = qml.exp(qml.RX([1, 2, 3], wires=0), coeff=4)
>>> qml.matrix(op).shape
(3, 2, 2)
>>> op = qml.exp(qml.RX(1, wires=0), coeff=[1, 2, 3])
>>> qml.matrix(op).shape
(3, 2, 2)
>>> op = qml.exp(qml.RX([1, 2, 3], wires=0), coeff=[4, 5, 6])
>>> qml.matrix(op).shape
(3, 2, 2)

But it doesn’t support batching of operators:

>>> op = qml.exp([qml.RX(1, wires=0), qml.RX(2, wires=0)], coeff=4)
AttributeError: 'list' object has no attribute 'batch_size'


This symbolic operator can be used to make general rotation operators:

>>> x = np.array(1.23)
>>> op = qml.exp( qml.PauliX(0), -0.5j * x)
>>> qml.math.allclose(op.matrix(), qml.RX(x, wires=0).matrix())

This can even be used for more complicated generators:

>>> t = qml.PauliX(0) @ qml.PauliX(1) + qml.PauliY(0) @ qml.PauliY(1)
>>> isingxy = qml.exp(t, 0.25j * x)
>>> qml.math.allclose(isingxy.matrix(), qml.IsingXY(x, wires=(0,1)).matrix())

If the coefficient is purely imaginary and the base operator is Hermitian, then the gate can be used in a circuit, though it may not be supported by the device and may not be differentiable.

>>> @qml.qnode(qml.device('default.qubit', wires=1))
... def circuit(x):
...     qml.exp(qml.PauliX(0), -0.5j * x)
...     return qml.expval(qml.PauliZ(0))
>>> print(qml.draw(circuit)(1.23))
0: ──Exp─┤  <Z>

If the base operator is Hermitian and the coefficient is real, then the Exp operator can be measured as an observable:

>>> obs = qml.exp(qml.PauliZ(0), 3)
>>> @qml.qnode(qml.device('default.qubit', wires=1))
... def circuit():
...     return qml.expval(obs)
>>> circuit()
tensor(20.08553692, requires_grad=True)