ARTIFICIAL NEURAL NETWORKS AND
THEIR APPLICATIONS TO QUANTUM
COMPUTING
SEMINAR COURSE IDC601
COURSE INSTRUCTOR: PROF. KAVITA DORAI
PRESENTED BY: MATREYEE KANDPAL (PH23022)
OUTLINE:
• INTRODUCTION
• BIOLOGICAL NEURAL NETWORKS
• ARTIFICIAL NEURAL NETWORKS




HISTORY
MODEL OF AN ANN
MODEL OF SINGLE NEURON
ACTIVATION FUNCTIONS
• NEURAL NETWORKS ARCHITECTURE TYPES
• LEARNING IN NEURAL NETS
• FEEDFORWARDED NEURAL NETWORKS
 TRAINING FFNN’s
 BACK PROPAGATION ALGORITHM
•
•
•
•
FUNDAMENTALS OF QUANTUM COMPUTING
APPLICATIONS OF ANN’s TO QC
EXAMPLE: QUANTUM STATE TOMOGRAPHY USING NEURAL NETS
ADVANTAGES OF COMBINING QC AND ANN’s
INTRODUCTION:
• Computational models called Artificial
Neural Networks (ANNs) are a key
element in machine learning's branch
known as deep learning and they are
modelled after the neurons in the human
brain.
• Artificial Neural Network (ANN) or
Neural Network(NN) has provide an
exciting alternative method for solving a
variety of problems in different fields of
science and engineering.
BIOLOGICAL NEURAL NETWORKS:
Four parts of a typical nerve cell : • DENDRITES : Accepts the inputs
• SOMA: Process the inputs
• AXON : Turns the processed inputs into
outputs.
• SYNAPSES : The electrochemical
contact between the neurons.
ARTIFICIAL NEURAL NETWORKS:
HISTORY:
• Human brain has many incredible characteristics such as massive parallelism, computation, learning
ability, generalization ability, adaptivity, which seems simple but is really complicated.
• ANN models was an effort to apply the same method as human brain uses to solve perceptual
problems.
• Three periods of development for ANN:-
• 1940: Mcculloch and Pitts: Initial works
• 1960: Rosenblatt: perceptron convergence theorem. Minsky and Papert : work showing the
limitations of a simple perceptron
• 1980: Hopfield/Werbos and Rumelhart : Hopfield's energy approach/back-propagation learning
algorithm.
MODEL OF AN ARTIFICIAL NEURAL NETWORK:
(dendrite)
X1
X2
X3
Input units
(synapse)
(axon)
(soma)
Wa
Wb

f()
Wc
Connection
weights
Summing
function
computation
Y
MODEL OF A SINGLE NEURON (PERCEPTRON):
• McCulloch and Pitts (1943) proposed the 'integrate and fire' model:
Output is given by:
φ
𝑛
𝑖=1 𝑤𝑘𝑖
𝑥𝑖 +𝑏𝑘
where, 𝜑 =activation function
𝑤𝑘𝑖 = weights
𝑥𝑖 = inputs
𝑏𝑘 = bias
ACTIVATION FUNCTIONS:
The activation function decides whether a neuron should be activated or not by
calculating the weighted sum and further adding bias to it.
• Identity
f(x) = x
• Binary step
f(x) = 1 if x >= q
f(x) = 0 otherwise
• Binary sigmoid
f(x) = 1 / (1 + e-s𝑥)
• Bipolar sigmoid
f(x) = -1 + 2 / (1 + e-s𝑥)
• Hyperbolic tangent
f(x) = (ex – e-x) / (ex + e-x)
• RELU (Rectified linear unit)
f(x) = max(0,x)
• Leaky RELU
f (x) = max (ax, x),
where a is a small constant.
NEURAL NETWORK ARCHITECTURE TYPES:
Simple perceptron
Feedforward NN
Deep Feedforward NN
Recurrent NN
LSTM NN
LEARNING IN NEURAL NETWORKS:
• Training: It is the process
in which the network is
taught to change its weight
and bias.
• Learning: It is the internal
process of training where
the artificial neural system
learns to update/adapt the
weights and biases.
Supervised Learning:
• Learning is performed by
presenting pattern with target.
• During learning, produced
output is compared with the
desired output.
• The difference between both
output is used to modify
learning weights according to
the learning algorithm.
• Recognizing
hand-written
digits, pattern recognition and
etc.
• Neural Network models:
perceptron,
feed-forward,
radial basis function, support
vector machine.
Unsupervised Learning:
• Targets are not provided
• Appropriate for clustering
task.
• Find similar groups of
documents in the web,
content
addressable
memory, clustering.
• Neural Network models:
Kohonen, self organizing
maps, Hopfield networks.
Reinforcement Learning:
• Target is provided, but
the desired output is
absent.
• The net is only
provided with guidance
to
determine
the
produced output is
correct or vise versa.
• Weights are modified
in the units that have
errors
FEEDFORWARDED NEURAL NETWORKS:
• Networks without cycles (feedback loops) are called a feed-forward networks (or perceptron).
Their architecture has following layers:
• Input layer: Number of neurons in this layer corresponds to the number of inputs to the neuronal
network. This layer consists of passive nodes, i.e., which do not take part in the actual signal
modification, but only transmits the signal to the following layer.
• Hidden layer: This layer has arbitrary number of layers with arbitrary number of neurons. The
nodes in this layer take part in the signal modification, hence, they are active.
• Output layer: The number of neurons in the output layer corresponds to the number of the output
values of the neural network. The nodes in this layer are active ones.
Fig: illustration of feedforwarded NN’s
TRAINING FEEDFORWARDED NN’s:
• Algorithms for training neural networks can be supervised (i.e. with a ‘teacher’) and unsupervised
(self-organizing).
• Supervised algorithms use a training set — a set of pairs (x, y) of inputs with their corresponding
desired outputs.
• The process of finding a set of weights such that for a given input the network produces the desired
output is called training.
• An outline of a supervised learning algorithm:
1. Initially, set all the weights wij to some random values.
2. Repeat
(a) Feed the network with an input x from one of the examples in the training set.
(b) Compute the network’s output f(x).
(c) Change the weights wij of the nodes.
3. Until the error c(y, f(x)) is small.
BACK PROPAGATION ALGORITHM:
• In a back-propagation neural network, the learning algorithm has two
phases.
• First, a training input pattern is presented to the network input layer.
The network propagates the input pattern from layer to layer until the
output pattern is generated by the output layer.
• If this pattern is different from the desired output, an error is
calculated and then propagated backwards through the network from
the output layer to the input layer. The weights are modified as the
error is propagated.
THREE-LAYER FEED-FORWARD NEURAL NETWORK:
( trained using back-propagation algorithm)
Step 1: Initialisation
Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a
small range.
The weight initialisation is done on a neuron-by-neuron basis.
Step 2: Activation
Activate the back-propagation neural network by applying inputs x1(p), x2(p),…, xn(p) and desired
outputs yd,1(p), yd,2(p),…, yd,n(p).
(a) Calculate the actual outputs of the neurons in the hidden layer:
n

y j ( p)  sigmoid  xi ( p)  wij ( p)   j 
 i 1

where n is the number of inputs of neuron j in the hidden layer, and sigmoid is the sigmoid activation
function.
(b) Calculate the actual outputs of the neurons in the output layer:
m

yk ( p )  sigmoid   x jk ( p )  w jk ( p )   k 
 j 1

where m is the number of inputs of neuron k in the output layer
Step 3: Weight training
Update the weights in the back-propagation network propagating backward the errors associated with
output neurons.
(a) Calculate the error gradient for the neurons in the output layer:
 k ( p)  yk ( p)  1  yk ( p) ek ( p)
where
ek ( p)  yd ,k ( p)  yk ( p)
Calculate the weight corrections:
w jk ( p)    y j ( p)   k ( p)
Update the weights at the output neurons:
w jk ( p  1)  w jk ( p)  w jk ( p)
(b) Calculate the error gradient for the neurons in the hidden layer:
l
 j ( p )  y j ( p )  [1  y j ( p )]    k ( p ) w jk ( p )
k 1
Calculate the weight corrections:
wij ( p)    xi ( p)   j ( p)
Update the weights at the hidden neurons:
wij ( p  1)  wij ( p)  wij ( p)
Step 4: Iteration
Increase iteration p by one, go back to Step 2 and repeat the process until the selected error criterion is
satisfied.
FUNDAMENTALS OF QUANTUM COMPUTING:
Common digital computing
requires data to be encoded
into binary digits (bits), each
of which is always in one of
two definite states (0 or 1),
quantum computation uses
quantum bits or QUBITS,
which can be in superposition
of states.
APPLICATIONS OF ANN’s TO QUANTUM
COMPUTING:
Much recent research has focused on training artificial neural networks to perform
several quantum computing and quantum information processing tasks, which
include:
• Quantum state and process Tomography
• Entanglement characterization
• Quantum gate optimization
• Decoherence mitigation
• Investigate quantum contextuality
QUANTUM STATE TOMOGRAPHY USING ANN’s:
Fig: Flowchart illustrating the FFNN model used to perform QST on two-qubit quantum states generated on an NMR quantum;
on the left, 𝝆𝒊𝒏 represents the state which is to be tomographed; IY denotes a tomographic operation, which is followed by
signal detection, the set of depicted NMR spectra are those obtained after the tomographic measurement. The FFNN with two
hidden layers is represented next, which then uses a reduced data set to reconstruct the final experimental tomographs
represented on the right.
• A multilayer FFNN architecture can be employed to perform QST.
• QST generally refers to the reconstruction of a quantum state and a quantum state is described by
a density matrix. An n-qubit density operator ρ can be expressed as a matrix in the product basis
by:
• The aim of QST is to reconstruct ρ from a set of tomographic measurements.
• The FFNN model is trained on a dataset containing randomly generated pure and mixed states.
• To perform QST the LeakyReLU (α = 0.5) activation function was used for both the input and the
hidden layers of the FFNN:
LeakyReLU(x) =x ; x > 0
=αx ; x < 0
• A linear activation function was used for the output layer.
• Training of this network can be achieved by minimizing a mean-squared-error cost function, with
respect to the network parameters.
• The results demonstrated that FFNN architectures are promising methods for performing QST.
APPLICATIONS OF COMBINING ANN’s WITH
QUANTUM COMPUTING:
Combining quantum computing and artificial neural networks holds the potential for
several advantages, although it's important to note that practical implementation and
realization of these benefits are still areas of active research. Some potential advantages
include:
Improved
Learning
Algorithms
Quantum
Entanglement
Parallelism
and
Superposition
Exponential
speedup of
certain
problems
REFERENCES:
• A. Gaikwad, O. Bihani, Arvind, and K. Dorai, "Neural network assisted quantum state and process
tomography using limited data sets," (2023), arXiv: 2304.04167.
• Laurene Fausett, “Fundamentals of Neural Networks, Architectures, Algorithms and Applications”,
Prentice Hall publications.
• Anderson, J. A., 1995, Introduction to Neural Networks (Cambridge, MA:MIT Press).
• ResearchGate.
• Haykin, S., 1999, Neural Networks: A Comprehensive Foundation, 2nd ed.(Englewood Cliffs, NJ:
Prentice-Hall).