# 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 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)
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 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])
theta = 0.2
dcircuit = qml.grad(circuit3)
```

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.

As a basic rule, **all positional arguments provided to the QNode are assumed to be differentiable
by default**. To accomplish this, arguments are internally turned into arrays of the PennyLane NumPy module,
which have a special flag `requires_grad`

specifying whether they are trainable or not:

```
>>> from pennylane import numpy as np
>>> np.array([0.1, 0.2])
tensor([0.1, 0.2], requires_grad=True)
```

If you would like to provide explicit non-differentiable arguments to the
QNode or gradient function, make sure to use a NumPy array that specifies
`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.

An alternative way to avoid having positional arguments turned into differentiable PennyLane NumPy arrays is to use a keyword argument syntax when the QNode is evaluated or when its gradient is computed.

For example, consider the following QNode that accepts one trainable argument `weights`

,
and two non-differentiable arguments `data`

and `wires`

:

```
dev = qml.device('default.qubit', wires=5)
@qml.qnode(dev)
def circuit(weights, data, wires):
qml.templates.AmplitudeEmbedding(data, wires=wires, normalize=True)
qml.RX(weights[0], wires=wires[0])
qml.RY(weights[1], wires=wires[1])
qml.RZ(weights[2], wires=wires[2])
qml.CNOT(wires=[wires[0], wires[1]])
qml.CNOT(wires=[wires[0], wires[2]])
return qml.expval(qml.PauliZ(wires[0]))
```

For `data`

, which is a PennyLane NumPy array, we can simply specify `requires_grad=False`

:

```
>>> np.random.seed(42) # make the results reproducable
>>> data = np.random.random([2**3], requires_grad=False)
```

But `wires`

is a list in this example, and if we turn it into a PennyLane NumPy array we would have to
create a device that understands custom wire labels of this type.
It is much easier to use the second option laid out above, and pass `wires`

to the
QNode using keyword argument syntax:

```
>>> wires = [2, 0, 1]
>>> weights = np.array([0.1, 0.2, 0.3])
>>> circuit(weights, data, wires=wires)
0.4124409353413991
```

When we compute the derivative, arguments with `requires_grad=False`

as well as arguments passed as
keyword arguments are ignored by `grad()`

:

```
>>> grad_fn = qml.grad(circuit)
>>> grad_fn(weights, data, wires=wires)
[-4.1382126e-02 0.0000000e+00 -6.9388939e-18]
```

Note

**Keyword arguments**

The `grad()`

function does not differentiate keyword arguments. A QNode may be defined
using arguments with default values; for example

```
@qml.qnode(dev)
def circuit(weights, data=None):
```

These arguments must always be passed using `keyword=value`

syntax:

```
>>> circuit(weights, data=[0.34, 0.1])
```

These arguments will always be treated as non-differentiable by the QNode and `grad()`

function.

## Optimization¶

To optimize your hybrid classical-quantum model using the NumPy interface, use the provided PennyLane optimizers.

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
opt = qml.GradientDescentOptimizer(stepsize=0.4)
steps = 100
params = np.array([0.011, 0.012, 0.05])
for i in range(steps):
# update the circuit parameters
params = opt.step(cost, params)
```

The final weights and circuit value are:

```
>>> params
array([ 0.19846757, 0.012 , 1.03559806])
>>> circuit4(params)
0.5
```

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.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])
>>> g1(params)
```

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.

## 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`

).

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])
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`

:

```
>>> minimize(cost, params, method='BFGS', jac=qml.grad(cost))
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])
```

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