1 Database Searching in Quantum and Natural Computing Michael Heather & Nick Rossiter, Northumbria University, England

2 Traditional Databases Databases store, organise and search collections of real-world data Run on traditional computers which are effectively examples of the universal Turing Machine Rely on some theoretical schema in the form of separate metadata which is not 1:1 with the internal structure of the data

3 Natural Computing Data can be input neat without any reductionist pre-processing New era possible in databases Very appropriate for applications of current interest like –biological and medical data, –environmental and geophysical data, –image and moving picture data, etc.

4 Construction of Natural Computers Molecular computers have been constructed [Adleman, 1996] But still tendency to resort to models like the sticker-based model [Roweis et al, 1998] Execution in vivo in DNA is a reality in nature (e.g. linked list addition)

5 Concept of Quantum Computers Most progress to date in natural systems seems to be with quantum information systems Concept of quantum computer realised during 1980s and 1990s Draws heavily on standard quantum theory and computational theory of the time to postulate an analogous Church-Turing hypothesis

6 Realising a Quantum Computer Realising concept of a quantum computer is not the same as realising a quantum computer Literature on quantum computer is mainly bottom-up (as with Turing) –qubit corresponds to bit –quantum logic to propositional logic –quantum algorithms to NP methods

7 Quantum Machines should be Faster Quantum parallelism could at least double the speed or be up to ten times faster with a single program [Maurer 2001] Chuang estimated that a quantum computer on average required one evaluation for a function compared to 2.25 for a classical computer. –He employed nuclear magnetic resonance to carbon-13 in chloroform molecules dissolved in acetone

8 Quantum Algorithms Deutsch and Jozsa found that a quantum algorithm was fast –for determining whether an unknown mathematical function is constant or balanced (for instance as many 1s as 0s) Shor and Deutsch-Jozsa algorithms are a quantum version of the fast Fourier transform – requiring only n 2 steps rather than (n*2) n steps

9 Quantum Database Algorithms Grover algorithm Time for searching for solutions is: –where N is number of entries, M is number of solutions and O is order Conventional timing is: So Grover looks faster

10 What does Grover Algorithm do? Steps: 1) –|x> register of existing qubits –|q> simple qubit –O is Oracle –f(x) =1 if solution –f(x) = 0 if no solution –Initial state of |q> is –The state remains unchanged if f(x) is not a solution in subsequent iterations

11 Steps (continued) 2) Walsh-Hadamard Transformation is entanglement of qubits where f(x)=0 and f(x) =1

12 Steps (continued) 3) Phase shift – every state except |0> receives a phase shift of –1 4) Then further Walsh-Hadamard transform Steps 1)-4) are repeated until solutions are maximally identified

13 Visual View of Grover Algorithm Steps involve multiple reflections. Product of two reflections is a rotation. –Then move |Ψ> towards |β> in each rotation –When get sufficiently close, Oracle chooses |Ψ> as answer

14 Number of Iterations (R) where R is the number of calls to the Oracle. Note R includes the number of solutions M

15 Categorical View All solutions of |Ψ> map through β

16 Significance of Grover Question of structure inherent in information. –Database scheme utilises this inherent structure in the construction and storage of the data. Tree constructions with lexicographical ordering may typically give the order of log N comparisons. –Elementary structuring (B-trees) can give faster conventional systems than by the use of Grover's algorithm (1 record in 10 6 in 5 disk reads)

17 Discussion Can quantum algorithms be realised on physical machines? –Re-examine the various interpretations of the physics to be found in quantum theory to check that they can be converted into constructible systems. Use of non-maximally entangled states has been claimed as promising. –Do the published algorithms really exhibit non-local operability of true quantum processing? Is the language of quantum theory an adequate basis for computation (as a programming language)?