# 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)

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)


The QNode circuit1() is a NumPy-interfacing 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)
array([ 0.87758256,  0.68803733])


### 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)

qnode1 = qml.QNode(circuit2, dev1)
qnode2 = qml.QNode(circuit2, dev2)


By default, all QNodes created this way are NumPy interfacing QNodes.

To calculate the gradient of a NumPy-QNode, 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)


Evaluating this gradient function at specific parameter values:

>>> dcircuit(phi, theta)
(array([ -4.79425539e-01,   1.11022302e-16]), array(0.0))


### Differentiable and non-differentiable 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 built-in 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)


When omitted, the value for this flag is True, so if you would like to provide a non-differentiable 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)


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 non-trainable, 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 classical-quantum model using the NumPy interface, use the provided optimizers:

 AdagradOptimizer Gradient-descent optimizer with past-gradient-dependent learning rate in each dimension. AdamOptimizer Gradient-descent optimizer with adaptive learning rate, first and second moment. GradientDescentOptimizer Basic gradient-descent optimizer. LieAlgebraOptimizer Lie algebra optimizer. MomentumOptimizer Gradient-descent optimizer with momentum. NesterovMomentumOptimizer Gradient-descent optimizer with Nesterov momentum. QNGOptimizer Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. RMSPropOptimizer Root mean squared propagation optimizer. RotosolveOptimizer Rotosolve gradient-free optimizer. RotoselectOptimizer Rotoselect gradient-free optimizer. ShotAdaptiveOptimizer Optimizer where the shot rate is adaptively calculated using the variances of the parameter-shift gradient.

For example, we can optimize a NumPy-interfacing 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

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)


For more details on the NumPy optimizers, check out the tutorials, as well as the pennylane.optimize documentation.

## Vector-valued 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.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 scalar-output 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():

>>> j1 = qml.jacobian(circuit5)
>>> j1(params)
array([[ 0.        , -0.98006658],
[-0.98006658,  0.        ]])


The output of jacobian() is a two-dimensional vector, with the first/second element being the partial derivative of the first/second expectation value with respect to the input parameter.

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).

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. Use np.dot(A, B) or A @ B instead.

• In-place operations such as a += b. Use a = a + b instead.

• Some isinstance checks, like isinstance(x, np.ndarray) or isinstance(x, tuple), without first doing from 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: 6.3491130264451484e-18
hess_inv: array([[ 1.85642354e+00, -8.84954187e-22,  3.89539943e+00],
[-8.84954187e-22,  1.00000000e+00, -4.02571211e-21],
[ 3.89539943e+00, -4.02571211e-21,  1.87180282e+01]])
jac: array([5.81636983e-10, 3.23117427e-27, 4.21456861e-09])
message: 'Optimization terminated successfully.'
nfev: 8
nit: 2
njev: 8
status: 0
success: True
x: array([0.22685818, 0.012     , 1.03194789])