# Unsupported gradient configurations¶

## Device jacobian¶

In order to use diff_method="device" for a QNode, the device passed into the constructor of a QNode must have its "provides_jacobian" capability set to True and must contain a method jacobian(circuits, **kwargs) that returns the gradients for each quantum circuit. This is not implemented for the default.qubit device because the device doesn’t provide such a jacobian method (instead allows backpropagation to work).

See the custom plugins page for more detail.

An exception is raised if this configuration is used:

def print_grad():
dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, diff_method='device')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.expval(qml.PauliZ(wires=0))

x = np.array([0.1], requires_grad=True)

>>> print_grad()
Traceback (most recent call last):
...
File "C:\pennylane\pennylane\qnode.py", line 448, in _validate_device_method
raise qml.QuantumFunctionError(
pennylane.QuantumFunctionError: The default.qubit device does not provide a native method for computing the jacobian.


## Backpropagation¶

The backpropagation algorithm is analytic by nature, and hence passing shots=None is the only supported configuration when diff_method="backprop" is used. Though it is possible to always use the analytic gradient even when shots>0 (as is the case with adjoint differentiation, see next section), in the current state of the code this would break other things.

Currently an exception is raised if this invalid configuration is used:

def print_grad():
dev = qml.device('default.qubit', wires=1, shots=100)

@qml.qnode(dev, diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.expval(qml.PauliZ(wires=0))

x = np.array([0.1], requires_grad=True)

>>> print_grad()
Traceback (most recent call last):
...
File "C:\pennylane\pennylane\qnode.py", line 375, in _validate_backprop_method
raise qml.QuantumFunctionError("Backpropagation is only supported when shots=None.")
pennylane.QuantumFunctionError: Backpropagation is only supported when shots=None.


Changing to shots=None allows computing the analytic gradient:

def print_grad():
dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.expval(qml.PauliZ(wires=0))

x = np.array([0.1], requires_grad=True)

>>> print_grad()
[-0.09983342]


PennyLane implements the adjoint differentiation method from 2009.02823, which only discusses the gradient of expectation of observables. The implementation is specific to the paper, hence the return statement of the quantum function wrapped in qml.qnode can only contain expval() as a measurement.

In particular, the following code works as expected:

def print_grad():
dev = qml.device('default.qubit', wires=1, shots=None)

def circuit(x):
qml.RX(x[0], wires=0)
return qml.expval(qml.PauliZ(wires=0))

x = np.array([0.1], requires_grad=True)

>>> print_grad()
[-0.09983342]


But the following code raises an error:

def print_grad_bad():
dev = qml.device('default.qubit', wires=1, shots=None)

def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

def cost_fn(x):
out = circuit(x)
return np.abs(out[0])

x = np.array([0.1], requires_grad=True)

>>> print_grad_bad()
Traceback (most recent call last):
...
File "C:\pennylane\pennylane\_qubit_device.py", line 951, in adjoint_jacobian
raise qml.QuantumFunctionError(
pennylane.QuantumFunctionError: Adjoint differentiation method does not support measurement state


Furthermore, the adjoint differentiation algorithm is analytic by nature. If the user creates a device with shots>0, a warning is raised and gradients are computed analytically:

def print_grad_ok():
dev = qml.device('default.qubit', wires=1, shots=100)

def circuit(x):
qml.RX(x[0], wires=0)
return qml.expval(qml.PauliZ(wires=0))

x = np.array([0.1], requires_grad=True)

>>> print_grad_ok()
C:\pennylane\pennylane\qnode.py:434: UserWarning: Requested adjoint differentiation to be computed with finite shots. Adjoint differentiation always calculated exactly.
warnings.warn(
C:\pennylane\pennylane\_qubit_device.py:965: UserWarning: Requested adjoint differentiation to be computed with finite shots. The derivative is always exact when using the adjoint differentiation method.
warnings.warn(
[-0.09983342]


In general, the state of a quantum circuit will be complex-valued, so differentiating the state directly is not possible without the use of complex analysis. Though complex gradients can be implemented for most “simple” functions, this is not supported in Autograd but is done in the other three interfaces.

Instead, in Autograd, real scalar-valued post-processing should be performed on the output state to allow the auto-differentiation frameworks to backpropagate through them. For example, the following code uses a scalar cost function dependent on the output state:

def state_scalar_grad():
dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

def cost_fn(x):
out = circuit(x)
return np.abs(out[0])

x = np.array([0.1], requires_grad=True)

>>> state_scalar_grad()
[-0.02498958]


However, changing from differentiating the scalar cost to differentiating the state directly will fail with an error:

def state_vector_grad():
dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

x = np.array([0.1], requires_grad=True)
print(qml.jacobian(circuit)(x))

>>> state_vector_grad()
Traceback (most recent call last):
...
File "C:\Python38\lib\site-packages\numpy\core\fromnumeric.py", line 57, in _wrapfunc
return bound(*args, **kwds)
ValueError: cannot reshape array of size 4 into shape (2,1)


Using a different interface that supports complex differentiation will fix this error:

def state_vector_grad_jax():
dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, interface='jax', diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

x = jnp.array([0.1], dtype=np.complex64)
print(jax.jacrev(circuit, holomorphic=True)(x))

dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, interface='tf', diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

x = tf.Variable([0.1], trainable=True, dtype=np.complex64)
with tf.GradientTape() as tape:
out = circuit(x)

print(tape.jacobian(out, [x]))

dev = qml.device('default.qubit', wires=1, shots=None)

@qml.qnode(dev, interface='torch', diff_method='backprop')
def circuit(x):
qml.RX(x[0], wires=0)
return qml.state()

x = torch.tensor([0.1], requires_grad=True, dtype=torch.complex64)
print(F.jacobian(circuit, (x,)))

>>> state_vector_grad_jax()
[[-0.02498958+0.j        ]
[ 0.        -0.49937513j]]
[<tf.Tensor: shape=(2, 1), dtype=complex64, numpy=
array([[-0.02498958+0.j        ],
[-0.        +0.49937513j]], dtype=complex64)>]
(tensor([[-0.0250+0.0000j],
[ 0.0000+0.4994j]]),)


In PennyLane, samples are drawn from the eigenvalues of an observable, or from the computational basis states if no observable is provided. This process is not differentiable in general, so no gradient flow backwards through the sampling is allowed.

Currently, attempting to compute the gradient in this scenario will not raise an error, but the results will be incorrect:

def sample_backward():
dev = qml.device('default.qubit', wires=1, shots=20)

@qml.qnode(dev)
def circuit(x):
qml.RX(x[0], wires=0)
return qml.sample(wires=0)

x = np.array([np.pi / 2])
print(qml.jacobian(circuit)(x))

>>> sample_backward()
[[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]
[0.5]]


The forward pass is supported and will work as expected:

def sample_forward():
dev = qml.device('default.qubit', wires=1, shots=20)

@qml.qnode(dev)
def circuit(x):
qml.RX(x[0], wires=0)
return qml.sample(wires=0)

x = np.array([np.pi / 2])
print(circuit(x))

>>> sample_forward()
[0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0]