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## Quantum Algorithms for Neural Networks Daniel Shumow

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**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.** 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. Quantum Mechanics:Linear Superposition Where cj is complex and:**Quantum MechanicsInterference 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.**QuAMProperties**• 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.**QuAMTraining Algorithm**• Training samples consist of an input pattern and a desired output pattern. • A wave function is designed such that the input patterns are entangled to the corresponding output patterns. • 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.