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Database Searching in Quantum and Natural Computing

Michael Heather & Nick Rossiter, Northumbria University, England

nick.rossiter@unn.ac.uk

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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

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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.

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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)

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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

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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

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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

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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 n2 steps rather than (n*2)n steps

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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

)N/MO(or )( NO

)/( MNO

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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

)(|| || : xfqxqxO

2/)1| 0(|

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Steps (continued)

2) Walsh-Hadamard Transformation

is entanglement of qubits where f(x)=0 and f(x) =1

xN xfxf

|1

|1)(0)(

|

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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

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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

Solutions |β> Entanglement |Ψ> θ Non-solutions ||α>

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Number of Iterations (R)

where R is the number of calls to the Oracle.

Note R includes the number of solutions M

M

NR

2

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Categorical View

|0> πl α

Σ |Ψ> Δ α|0>+β|1>

Π πr β

|1> r

•All solutions of |Ψ> map through β

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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 106 in 5 disk reads)

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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)?