Quantum Quantum Computational Computational

IntelligenceIntelligence

Outline

OverviewQuantum search (revisited) Extensions:

Search over arbitrary initial amplitude distributions

Quantum Associative Memory (QuAM)

Beyond Phase Amplification QuAM - Upgraded Competitive Learning

Conclusions

Quantum Computational Intelligence

Computational intelligenceSeeks to produce algorithms that solve

problems which are intractable using traditional methods of computation.

Quantum Computational intelligenceExtension of the field into the domain of

quantum computing.

Current ResearchCurrent 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

Quantum Artificial Neural Networks

Quantum Associative Memory

Competitive Learning

ANN implementation using quantum dots

‘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).

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

Grover’s Search Revisited

Notation

Example

Apply Grover Iteration:

Example (cont)

Iteration 1

Iteration 2

Measure

Arbitrary Initial distribution Example

Pass

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.

Example 3

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

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 lengthm – number of patterns

Quantum Associative Memory (I)

Pattern completionCan be done using the modified search

algorithm presented previously.

Repeat T times (T≈≈√N)

Example 4

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)

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.

Beyond Phase Amplification

Question:

Are Unitary Operators Nessesary ??Are Unitary Operators Nessesary ?? For DB initialization -> For DB initialization -> YesYes

To create a coherent superposition of basis To create a coherent superposition of basis states reversible operators must be used.states reversible operators must be used.

For query Processing -> For query Processing -> NONO The superposition is collapsed anywaysThe superposition is collapsed anyways

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.

Non-Unitary Operators

Given a string q over the alphabet { 0,1,? },{ 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.

Example 5a

Consider a 2 qubit system

And a query string q = ’11’

Example 5b

Consider a 2 qubit system

And a query string q = ’11’

Competitive Learning

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.

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.

References

http://www.cic.unb.br/docentes/weigang/qc/aci.html

http://www.cs.ualberta.ca/~ilya/courses/c605-QC/Refs.pdf

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.