"""
.. role:: html(raw)
:format: html
.. _quantum_neural_net:
Function fitting with a photonic quantum neural network
=======================================================
.. meta::
:property="og:description": Fit to noisy data with a variational quantum circuit.
:property="og:image": https://pennylane.ai/qml/_static/demonstration_assets/qnn_output_28_0.png
.. related::
qonn Optimizing a quantum optical neural network
pytorch_noise PyTorch and noisy devices
tutorial_noisy_circuit_optimization Optimizing noisy circuits with Cirq
*Author: Maria Schuld — Posted: 11 October 2019. Last updated: 20 March 2025.*
.. warning::
This demo is only compatible with PennyLane version ``0.29`` or below.
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) <https://arxiv.org/abs/1806.06871>`__.
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
from pennylane.optimize import AdamOptimizer
##############################################################################
# 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``
# (:html:`<a href="https://raw.githubusercontent.com/XanaduAI/pennylane/v0.3.0/examples/data/sine.txt"
# download="sine.txt" target="_blank">download the file here</a>`).
data = np.loadtxt("sine.txt")
X = np.array(data[:, 0], requires_grad=False)
Y = np.array(data[:, 1], requires_grad=False)
##############################################################################
# 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()
##############################################################################
# .. image:: ../_static/demonstration_assets/quantum_neural_net/qnn_output_20_0.png
#
# 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)
##############################################################################
# .. rst-class:: sphx-glr-script-out
#
#
# .. code-block:: none
#
# 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))
##############################################################################
# .. rst-class:: sphx-glr-script-out
#
#
# .. code-block:: none
#
# Iter: 0 | Cost: 0.3006065
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#
#
# Finally, we collect the predictions of the trained model for 50 values
# in the range :math:`[-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()
##############################################################################
# .. image:: ../_static/demonstration_assets/quantum_neural_net/qnn_output_28_0.png
#
# 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()
##############################################################################
# .. image:: ../_static/demonstration_assets/quantum_neural_net/qnn_output_30_0.png
##############################################################################
# About the author
# ----------------
#