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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 c 1 c N Where cj is complex and: N 2 | c | j 1 j 0 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%