# Function fitting with a photonic quantum neural network¶

Author: Maria Schuld — Posted: 11 October 2019. Last updated: 25 January 2021.

In this example we show how a variational circuit can be used to learn a fit for a one-dimensional function when being trained with noisy samples from that function.

The variational circuit we use is the continuous-variable quantum neural network model described in Killoran et al. (2018).

## Imports¶

We import PennyLane, the wrapped version of NumPy provided by PennyLane, and an optimizer.

import pennylane as qml
from pennylane import numpy as np


The device we use is the Strawberry Fields simulator, this time with only one quantum mode (or wire). You will need to have the Strawberry Fields plugin for PennyLane installed.

dev = qml.device("strawberryfields.fock", wires=1, cutoff_dim=10)


## Quantum node¶

For a single quantum mode, each layer of the variational circuit is defined as:

def layer(v):
# Matrix multiplication of input layer
qml.Rotation(v[0], wires=0)
qml.Squeezing(v[1], 0.0, wires=0)
qml.Rotation(v[2], wires=0)

# Bias
qml.Displacement(v[3], 0.0, wires=0)

# Element-wise nonlinear transformation
qml.Kerr(v[4], wires=0)


The variational circuit in the quantum node first encodes the input into the displacement of the mode, and then executes the layers. The output is the expectation of the x-quadrature.

@qml.qnode(dev)
def quantum_neural_net(var, x):
# Encode input x into quantum state
qml.Displacement(x, 0.0, wires=0)

# "layer" subcircuits
for v in var:
layer(v)

return qml.expval(qml.X(0))


## Objective¶

As an objective we take the square loss between target labels and model predictions.

def square_loss(labels, predictions):
loss = 0
for l, p in zip(labels, predictions):
loss = loss + (l - p) ** 2

loss = loss / len(labels)
return loss


In the cost function, we compute the outputs from the variational circuit. Function fitting is a regression problem, and we interpret the expectations from the quantum node as predictions (i.e., without applying postprocessing such as thresholding).

def cost(var, features, labels):
preds = [quantum_neural_net(var, x) for x in features]
return square_loss(labels, preds)


## Optimization¶

We load noisy data samples of a sine function from the external file sine.txt (download the file here).

data = np.loadtxt("sine.txt")


Before training a model, let’s examine the data.

Note: For the next cell to work you need the matplotlib library.

import matplotlib.pyplot as plt

plt.figure()
plt.scatter(X, Y)
plt.xlabel("x", fontsize=18)
plt.ylabel("f(x)", fontsize=18)
plt.tick_params(axis="both", which="major", labelsize=16)
plt.tick_params(axis="both", which="minor", labelsize=16)
plt.show()


The network’s weights (called var here) are initialized with values sampled from a normal distribution. We use 4 layers; performance has been found to plateau at around 6 layers.

np.random.seed(0)
num_layers = 4
var_init = 0.05 * np.random.randn(num_layers, 5, requires_grad=True)
print(var_init)


Out:

array([[ 0.08820262,  0.02000786,  0.0489369 ,  0.11204466,  0.0933779 ],
[-0.04886389,  0.04750442, -0.00756786, -0.00516094,  0.02052993],
[ 0.00720218,  0.07271368,  0.03805189,  0.00608375,  0.02219316],
[ 0.01668372,  0.07470395, -0.01025791,  0.01565339, -0.04270479]])


Using the Adam optimizer, we update the weights for 500 steps (this takes some time). More steps will lead to a better fit.

opt = AdamOptimizer(0.01, beta1=0.9, beta2=0.999)

var = var_init
for it in range(500):
(var, _, _), _cost = opt.step_and_cost(cost, var, X, Y)
print("Iter: {:5d} | Cost: {:0.7f} ".format(it, _cost))


Out:

Iter:     0 | Cost: 0.3006065
Iter:     1 | Cost: 0.2689702
Iter:     2 | Cost: 0.2472125
Iter:     3 | Cost: 0.2300139
Iter:     4 | Cost: 0.2157100
Iter:     5 | Cost: 0.2035455
Iter:     6 | Cost: 0.1931103
Iter:     7 | Cost: 0.1841536
Iter:     8 | Cost: 0.1765061
Iter:     9 | Cost: 0.1700410
Iter:    10 | Cost: 0.1646527
Iter:    11 | Cost: 0.1602444
Iter:    12 | Cost: 0.1567201
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Iter:    15 | Cost: 0.1504356
Iter:    16 | Cost: 0.1494099
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Iter:   496 | Cost: 0.0119462
Iter:   497 | Cost: 0.0119405
Iter:   498 | Cost: 0.0119349
Iter:   499 | Cost: 0.0119293


Finally, we collect the predictions of the trained model for 50 values in the range $$[-1,1]$$:

x_pred = np.linspace(-1, 1, 50)
predictions = [quantum_neural_net(var, x_) for x_ in x_pred]


and plot the shape of the function that the model has “learned” from the noisy data (green dots).

plt.figure()
plt.scatter(X, Y)
plt.scatter(x_pred, predictions, color="green")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.tick_params(axis="both", which="major")
plt.tick_params(axis="both", which="minor")
plt.show()


The model has learned to smooth the noisy data.

In fact, we can use PennyLane to look at typical functions that the model produces without being trained at all. The shape of these functions varies significantly with the variance hyperparameter for the weight initialization.

Setting this hyperparameter to a small value produces almost linear functions, since all quantum gates in the variational circuit approximately perform the identity transformation in that case. Larger values produce smoothly oscillating functions with a period that depends on the number of layers used (generically, the more layers, the smaller the period).

variance = 1.0

plt.figure()
x_pred = np.linspace(-2, 2, 50)
for i in range(7):
rnd_var = variance * np.random.randn(num_layers, 7)
predictions = [quantum_neural_net(rnd_var, x_) for x_ in x_pred]
plt.plot(x_pred, predictions, color="black")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.tick_params(axis="both", which="major")
plt.tick_params(axis="both", which="minor")
plt.show()