Quantum Computational Intelligence

1. Quantum Computational Intelligence

2. Outline • Overview • Quantum search (revisited) • Extensions: • Search over arbitrary initial amplitude distributions • Quantum Associative Memory (QuAM) • Beyond Phase Amplification • QuAM - Upgraded • Competitive Learning • Conclusions

3. Quantum Computational Intelligence Computational intelligence • Seeks to produce algorithms that solve problems which are intractable using traditional methods of computation. Quantum Computational intelligence • Extension of the field into the domain of quantum computing.

4. Current Research • Quantum Artificial Neural Networks • Structured Search • Quantum inspired Genetic Algorithm • Decision Tree evaluation using quantum computation. • Quantum algorithm for learning DNF formulas • Quantum Bayesian Networks

5. Quantum Artificial Neural Networks • Quantum Associative Memory • Competitive Learning • ANN implementation using quantum dots

6. ‘Classical’ problems with ANNs • Repetition of training examples • humans do not undergo the prolonged and repeated exposure to information that ANNs require to learn training patterns. • Memoryless learning / Catastrophic forgetting • humans remember and recall information that is provided to them as a part of learning. • For ANNs, training examples are used for the changing of the weights and are immediately forgotten. • new information forces a neural network to forget older information. (ICNN´97, Menneer and Narayanan, 1998).

7. Quantum Based ANN’s • Pros: • Exponential Memory Capacity • Fast (Linear/Polynomial) training times • Closer resemblance to human cognitive processing (???) • Cons • Input collapses the quantum wave function • I.e.. The superposition of stored patterns is collapsed to a single pattern. • So the QuANN needs to be retrained constantly

8. Grover’s Search Revisited

9. Notation

10. Example Apply Grover Iteration:

11. Example (cont) • Iteration 1 • Iteration 2 • Measure

12. Arbitrary Initial distribution Example • Pass

13. Arbitrary Initial distribution (cont) • Need to create a new operator which synchronizes the phase of non-solution states. • Define an operator which performs a phase shift on set P containing all states with non-zero amplitudes.

14. Example 3

15. Associative Memory (AM) • Performs pattern completion • Learning a set (P) of complete prototypical patterns • Output a complete pattern upon presentation of an incomplete or noisy input pattern

16. Quantum Associative Memory • Initial pattern memorization / Learning (not covered) • Can be performed in O(nm) number of steps using 2n+1 qubits where • n – pattern length • m – number of patterns

17. Quantum Associative Memory (I) • Pattern completion • Can be done using the modified search algorithm presented previously. • Repeat T times (T≈≈√N)

18. Example 4

19. QuAM summary • Need to create a pattern DB represented as a superposition of states • At the same time need to create the operator • To query the system need to create the operator , where is(are) the target pattern(s)

20. Classical vs Quantum Associative Memory • Classical Case (Hopfield Network) • Can store approx. 0.15n patterns, where n is the number of nodes • QuAM • Stores 2^n patterns, where n is the number of qubits. • Quantum search is quite slow for pattern recall.

21. Beyond Phase Amplification • Question: Are Unitary Operators Nessesary ?? • For DB initialization -> Yes • To create a coherent superposition of basis states reversible operators must be used. • For query Processing -> NO • The superposition is collapsed anyways

22. Non-Unitary Operators • Ex: Observation of a system is not unitary nor evolutionary • Since pattern recall requires decoherance and collapse of the quantum system into one basis state, pattern recall is a non-evolutionnary and non-unitary operation.

23. Non-Unitary Operators • Given a string q over the alphabet { 0,1,? }, define a new (non-unitary) operator • Where h(a,b) is the hamming distance • The ‘1’s entries allow states with non-zero hamming disnace the possibility of being chosen. • The ‘-1’ perform destructictive interference to allow the state with maximal hamming distance to be chosen.

24. Example 5a • Consider a 2 qubit system • And a query string q = ’11’

25. Example 5b • Consider a 2 qubit system • And a query string q = ’11’

26. Competitive Learning

27. Conclusions • QuAM offers exponential increase instorage capasity • Using phase ampliphication offers polynomial(quadratic) speedup in pattern retreval over 1-nearest neighbour alg. • Using non-unitary operators offers an exponential speedup for pattern retreval.

28. Conclusions • QuAM is the most practical application of Quantum Computing • A 30 qubit system can store over a billion patterns • Shor’s factoring technique will only be practical if we can factor very large numbers (> 512 bits) requiring a large number of qubits.

29. References • http://www.cic.unb.br/docentes/weigang/qc/aci.html • http://www.cs.ualberta.ca/~ilya/courses/c605-QC/Refs.pdf

30. Final Thoughts • Prof. Sham: Well, we factored number 15 into 3 times 5. It was a thorny problem, but, by Jove, we did it. We started building our NMR computer in 1997 with a 5-million-dollar grant from DARPA. So it took 5 years, a few million dollars, and much hard work to build our room sized computer. But just think how powerful it is! And according to theory, NMR computers have tremendous growth potential. They can have as many as 10 qubits! • Reporter: Could you please tell us something about your future plans? • Prof. Sham: Well. I plan to continue publishing in Nature Magazine. Another exciting development is that I plan to add one more qubit to NMR computers in the next few years. We've won another 3-million-dollar grant from DARPA to continue our work. Isn't our government just great! We can't wait to start factoring number 18.