1. Quantum Algorithms for Neural Networks Daniel Shumow

2. Outline Project Motivation Brief overview of Quantum Concepts –Linear superposition –Interference and Entanglement –Quantum Computation Quantum Neural Algorithms –Quantum Associative Memory

3. 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?

4. 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.

5. 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. Where c j is complex and:

6. 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:

7. 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.

8. Quantum-Neural Algorithms Quantum Associative Memory –Ventura and Martinez, 1998. Competitive Learning in a Quantum System –Ventura, 1999.

9. 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.

10. 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.

11. 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.

12. QuAM Results Data SetRecallGeneralization xor100%0% and100%50% Random 3 bit pattern #1100%0% Random 3 bit pattern #2100%66.7% Random 3 bit pattern #3100%33% Parity 4 bits100%0% Random 4 bit pattern #1100%40% Random 4 bit pattern #2100%0% Random 4 bit pattern #3100%20%