NumPy interface¶
Note
This interface is the default interface supported by PennyLane’s QNode
.
Using the NumPy interface¶
Using the NumPy interface is easy in PennyLane, and the default approach — designed so it will feel like you are just using standard NumPy, with the added benefit of automatic differentiation.
All you have to do is make sure to import the wrapped version of NumPy provided by PennyLane alongside with the PennyLane library:
import pennylane as qml
from pennylane import numpy as np
This is powered via Autograd, and enables
automatic differentiation and backpropagation of classical computations using familiar
NumPy functions and modules (such as np.sin
, np.cos
, np.exp
, np.linalg
,
np.fft
), as well as standard Python constructs, such as if
statements, and for
and while
loops.
Via the QNode decorator¶
The QNode decorator is the recommended way for creating QNodes
in PennyLane. By default, all QNodes are constructed for the NumPy interface,
but this can also be specified explicitly by passing the interface='autograd'
keyword argument:
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev, interface='autograd')
def circuit1(phi, theta):
qml.RX(phi[0], wires=0)
qml.RY(phi[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.PhaseShift(theta, wires=0)
return qml.expval(qml.PauliZ(0)), qml.expval(qml.Hadamard(1))
The QNode circuit1()
is a NumPyinterfacing QNode, accepting standard Python
data types such as ints, floats, lists, and tuples, as well as NumPy arrays, and
returning NumPy arrays.
It can now be used like any other Python/NumPy function:
>>> phi = np.array([0.5, 0.1])
>>> theta = 0.2
>>> circuit1(phi, theta)
(tensor(0.87758256, requires_grad=True),
tensor(0.68803733, requires_grad=True))
The interface can also be automatically determined when the QNode
is called. You do not need to pass the interface
if you provide parameters.
Via the QNode constructor¶
In the introduction it was shown how to instantiate a QNode
object directly, for example, if you would like to reuse the same quantum function across
multiple devices, or even use different classical interfaces:
dev1 = qml.device('default.qubit', wires=2)
dev2 = qml.device('forest.wavefunction', wires=2)
def circuit2(phi, theta):
qml.RX(phi[0], wires=0)
qml.RY(phi[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.PhaseShift(theta, wires=0)
return qml.expval(qml.PauliZ(0)), qml.expval(qml.Hadamard(1))
qnode1 = qml.QNode(circuit2, dev1)
qnode2 = qml.QNode(circuit2, dev2)
By default, all QNodes created this way are NumPy interfacing QNodes.
Quantum gradients¶
To calculate the gradient of a NumPyQNode, we can simply use the provided
grad()
function.
For example, consider the following QNode:
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit3(phi, theta):
qml.RX(phi[0], wires=0)
qml.RY(phi[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.PhaseShift(theta, wires=0)
return qml.expval(qml.PauliZ(0))
We can now use grad()
to create a QNode gradient function,
with respect to both QNode parameters phi
and theta
:
phi = np.array([0.5, 0.1], requires_grad=True)
theta = np.array(0.2, requires_grad=True)
dcircuit = qml.grad(circuit3)
Evaluating this gradient function at specific parameter values:
>>> dcircuit(phi, theta)
(array([ 4.79425539e01, 1.11022302e16]), array(0.0))
Differentiable and nondifferentiable arguments¶
How does PennyLane know which arguments of a quantum function to differentiate, and which to ignore? For example, you may want to pass arguments to a QNode but not have PennyLane consider them when computing gradients.
Regular positional arguments provided to the QNode are not assumed to be differentiable
by default. This includes arguments in the form of builtin Python data types, and arrays from
the original NumPy module. Thus, arguments need to be explicitly marked as trainable or selected
using the argnum
keyword. To mark an argument as trainable, a special flag requires_grad
has been added to arrays from PennyLane’s NumPy module:
>>> from pennylane import numpy as np
>>> np.array([0.1, 0.2], requires_grad=True)
tensor([0.1, 0.2], requires_grad=True)
When omitted, the value for this flag is True
, so if you would like to provide a
nondifferentiable PennyLane NumPy array to the QNode or gradient function, make sure
to specify requires_grad=False
:
>>> from pennylane import numpy as np
>>> np.array([0.1, 0.2], requires_grad=False)
tensor([0.1, 0.2], requires_grad=False)
Note
The requires_grad
argument can be passed to any NumPy function provided by PennyLane,
including NumPy functions that create arrays like np.random.random
, np.zeros
, etc.
On the other hand, keyword arguments (whether they have a default value or not), are always
considered nontrainable, no matter their data type or flags they may have. For example, consider
the following QNode that accepts two arguments data
and weights
:
dev = qml.device('default.qubit', wires=5)
@qml.qnode(dev)
def circuit(data, weights):
qml.AmplitudeEmbedding(data, wires=[0, 1, 2], normalize=True)
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.RZ(weights[2], wires=2)
qml.CNOT(wires=[0, 1])
qml.CNOT(wires=[0, 2])
return qml.expval(qml.PauliZ(0))
rng = np.random.default_rng(seed=42) # make the results reproducable
data = rng.random([2 ** 3], requires_grad=False)
weights = np.array([0.1, 0.2, 0.3], requires_grad=True)
When we compute the derivative, arguments with requires_grad=False
as well as arguments
passed as keyword arguments are ignored by grad()
, which in this case means no gradient
is computed at all:
>>> qml.grad(circuit)(data, weights=weights)
UserWarning: Attempted to differentiate a function with no trainable parameters. If this is unintended, please add trainable parameters via the 'requires_grad' attribute or 'argnum' keyword.
()
Optimization¶
To optimize your hybrid classicalquantum model using the NumPy interface, use the provided optimizers:
Gradientdescent optimizer with pastgradientdependent learning rate in each dimension. 

Gradientdescent optimizer with adaptive learning rate, first and second moment. 

Basic gradientdescent optimizer. 

Lie algebra optimizer. 

Gradientdescent optimizer with momentum. 

Gradientdescent optimizer with Nesterov momentum. 

Optimizer with adaptive learning rate, via calculation of the diagonal or blockdiagonal approximation to the FubiniStudy metric tensor. 

Root mean squared propagation optimizer. 

Rotosolve gradientfree optimizer. 

Rotoselect gradientfree optimizer. 

Optimizer where the shot rate is adaptively calculated using the variances of the parametershift gradient. 
For example, we can optimize a NumPyinterfacing QNode (below) such that the weights x
lead to a final expectation value of 0.5:
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit4(x):
qml.RX(x[0], wires=0)
qml.RZ(x[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(x[2], wires=0)
return qml.expval(qml.PauliZ(0))
def cost(x):
return np.abs(circuit4(x)  0.5)**2
opt = qml.GradientDescentOptimizer(stepsize=0.4)
steps = 100
params = np.array([0.011, 0.012, 0.05], requires_grad=True)
for i in range(steps):
# update the circuit parameters
params = opt.step(cost, params)
The final weights and circuit value are:
>>> params
tensor([0.19846757, 0.012 , 1.03559806], requires_grad=True)
>>> circuit4(params)
tensor(0.5, requires_grad=True)
For more details on the NumPy optimizers, check out the tutorials, as well as the
pennylane.optimize
documentation.
Vectorvalued QNodes and the Jacobian¶
How does automatic differentiation work in the case where the QNode returns multiple expectation values?
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit5(params):
qml.Hadamard(wires=0)
qml.CNOT(wires=[0, 1])
qml.RX(params[0], wires=0)
qml.RY(params[1], wires=1)
qml.CNOT(wires=[0, 1])
return qml.expval(qml.PauliY(0)), qml.expval(qml.PauliZ(1))
If we were to naively try computing the gradient of circuit5
using the grad()
function,
>>> g1 = qml.grad(circuit5)
>>> params = np.array([np.pi/2, 0.2], requires_grad=True)
>>> g1(params)
TypeError: Grad only applies to real scalaroutput functions. Try jacobian, elementwise_grad or holomorphic_grad.
we would get an error message. This is because the gradient is
only defined for scalar functions, i.e., functions which return a single value. In the case where the QNode
returns multiple expectation values, the correct differential operator to use is
the Jacobian matrix.
This can be accessed in PennyLane as jacobian()
.
As the circuit5
returns a tuple of numpy arrays instead of a single numpy array, the results need
to be stacked into a single array before use with jacobian()
.
>>> j1 = qml.jacobian(lambda x: np.stack(circuit5(x)))
>>> j1(params)
array([[ 0. , 0.98006658],
[0.98006658, 0. ]])
The output of jacobian()
is a twodimensional vector, with the first/second element being
the partial derivative of the first/second expectation value with respect to the input parameter.
Advanced Autograd usage¶
The PennyLane NumPy interface leverages the Python library Autograd to enable automatic differentiation of NumPy code, and extends
it to provide gradients of quantum circuit functions encapsulated in QNodes. In order to make NumPy
code differentiable, Autograd provides a wrapped version of NumPy (exposed in PennyLane as
pennylane.numpy
).
Warning
As stated in other sections, using this interface, any hybrid computation should be coded using the wrapped version of NumPy provided by PennyLane. If you accidentally import the vanilla version of NumPy, your code will not be automatically differentiable.
Because of the way Autograd wraps NumPy, the PennyLane NumPy interface allows standard NumPy
functions and basic Python control statements (if
statements, loops, etc.) for declaring
differentiable classical computations.
That being said, Autograd’s coverage of NumPy is not complete. It is best to consult the Autograd docs for a more complete overview of supported and unsupported features. We highlight a few of the major ‘gotchas’ here.
Do not use:
Assignment to arrays, such as
A[0, 0] = x
.
Implicit casting of lists to arrays, for example
A = np.sum([x, y])
. Make sure to explicitly cast to a NumPy array first, i.e.,A = np.sum(np.array([x, y]))
instead.
A.dot(B)
notation. Usenp.dot(A, B)
orA @ B
instead.
Inplace operations such as
a += b
. Usea = a + b
instead.
Some
isinstance
checks, likeisinstance(x, np.ndarray)
orisinstance(x, tuple)
, without first doingfrom autograd.builtins import isinstance, tuple
.
SciPy Optimization¶
In addition to using autodifferentiation provided by Autograd, the NumPy interface also allows QNodes to be optimized directly using the SciPy optimize module.
Simply pass the QNode, or your hybrid cost function containing QNodes, directly
to the scipy.minimize
function:
from scipy.optimize import minimize
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit(x):
qml.RX(x[0], wires=0)
qml.RZ(x[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(x[2], wires=0)
return qml.expval(qml.PauliZ(0))
def cost(x):
return np.abs(circuit(x)  0.5) ** 2
params = np.array([0.011, 0.012, 0.05], requires_grad=True)
minimize(cost, params, method='BFGS')
Some of the SciPy minimization methods require information about the gradient
of the cost function via the jac
keyword argument. This is easy to include; we
can simply create a function that computes the gradient using qml.grad
. Since
minimize
does not use our wrapped version of numpy, we need to explicitly
specify which arguments are trainable via the argnum
keyword.
>>> minimize(cost, params, method='BFGS', jac=qml.grad(cost, argnum=0))
fun: 6.3491130264451484e18
hess_inv: array([[ 1.85642354e+00, 8.84954187e22, 3.89539943e+00],
[8.84954187e22, 1.00000000e+00, 4.02571211e21],
[ 3.89539943e+00, 4.02571211e21, 1.87180282e+01]])
jac: array([5.81636983e10, 3.23117427e27, 4.21456861e09])
message: 'Optimization terminated successfully.'
nfev: 8
nit: 2
njev: 8
status: 0
success: True
x: array([0.22685818, 0.012 , 1.03194789])