Quantum Algorithms
for
Neural Networks
Daniel Shumow
Outline
• Project Motivation
• Brief overview of Quantum Concepts
– Linear superposition
– Interference and Entanglement
– Quantum Computation
• Quantum Neural Algorithms
– Quantum Associative Memory
Motivation for Project
• By using components that take advantage
of the laws of Quantum Mechanics it has
been shown that there are theoretical
algorithmic performance improvements not
possible from classical computers.
• Can Quantum Computation be used to
improve the performance of Neural
Network Algorithms?
Quantum Mechanics
• Quantum Systems can be in more than one state
at once. This is called a super position of states.
• Quantum systems are described by a wave
function often denoted by the Greek letter 
(psi)
• For state x: (x) evaluates to a complex number
such that (x)·(x)* is the probability that the
quantum system will collapse into state x when it
interacts with the environment.
• Wave functions evolve by unitary
transformations.
Quantum Mechanics:
Linear Superposition
•  can be represented
as a column vector.
•  is a normalized
linear combination of
basis states.
• When  interacts
with the environment
it is projected onto a
basis state.
 c0  Where cj is
 c  complex and:
1 


N
2

|
c
|
1

j
  j 0
c N 
Quantum Mechanics
Interference and Entanglement
• Interference:
States that are in a super position
can interfere with each other causing
probability amplitudes to increase or
decrease. This is like water waves
interfering.
• Entanglement:
A purely quantum phenomenon,
entanglement is when changes to
one part of a quantum system
instantaneously correlate at another
part of the quantum system.
Young’s double slit
experiment:
Quantum Computation
• Quantum algorithms get power from
superposition, interference, and entanglement.
• In a quantum computer the registers will be
quantum systems.
• The two “big” quantum algorithms are:
– Shor’s Algorithm for Factoring:
Factors composite integers in polynomial time.
– Grover’s Searching Algorithm:
Provides a square root speed up over classical
algorithms for searching unordered lists.
Quantum-Neural Algorithms
• Quantum Associative Memory
– Ventura and Martinez, 1998.
• Competitive Learning in a Quantum System
– Ventura, 1999.
Quantum Associative Memory
(QuAM)
A QuAM is analogous to a linear associative memory. It is a neural
network that has an input layer and an output layer. All neurons are
quantum mechanical components.
QuAM
Properties
• The whole network is a quantum system.
• Neurons do not have to be individually
updated because the system will update
itself.
• A QuAM can store an exponential number
of patterns with perfect recall.
• A QuAM can generalize but it is worse than
classical algorithms when generalizing.
QuAM
Training Algorithm
1. Training samples consist of an input pattern and
a desired output pattern.
2. A wave function is designed such that the input
patterns are entangled to the corresponding
output patterns.
3. The wave function is set up so when a pattern is
applied to the network the correct answer
constructively interferes and incorrect answers
destructively interfere with each other.
QuAM
Results
Data Set
xor
and
Random 3 bit pattern #1
Random 3 bit pattern #2
Random 3 bit pattern #3
Parity 4 bits
Random 4 bit pattern #1
Random 4 bit pattern #2
Random 4 bit pattern #3
Recall
100%
100%
100%
100%
100%
100%
100%
100%
100%
Generalization
0%
50%
0%
66.7%
33%
0%
40%
0%
20%