Database Searching in Quantum and Natural Computing

1
Database Searching in Quantum and Natural
Computing

• Michael Heather Nick Rossiter, Northumbria
  University, England
• nick.rossiter_at_unn.ac.uk

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 11 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 n2 steps rather than (n2)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)
• xgt register of existing qubits
• qgt simple qubit
• O is Oracle
• f(x) 1 if solution
• f(x) 0 if no solution
• Initial state of qgt 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 0gt 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 ?gt towards ßgt in each rotation
• When get sufficiently close, Oracle chooses ?gt
  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 ?gt map through ß

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 106 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)?