Grover Algorithm

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Grover Algorithm
Marek Perkowski

• I used slides of Anuj Dawar, Jake Biamonte,
  Julian Miller and Orlin Grabbe but I am to be
  blamed for extensions and (possible) mistakes

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From Grover to general search
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Graph Coloring

• Building oracle for graph coloring is a better
  example.
• This is not an optimal way to do graph coloring
  but explains well the principle of building
  oracles.

The Graph Coloring Problem
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Color every node with a color. Every two nodes
that share an edge should have different colors.
Number of colors should be minimum
1
3
2
3
2
5
4
5
4
6
7
6
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This graph is 3-colorable
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Simpler Graph Coloring Problem
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3
2
We need to give all possible colors here
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Two wires for color of node 1
Two wires for color of node 2
Two wires for color of node 3
Two wires for color of node 4
?
?
?
?
?
Gives 1 when nodes 1 and 2 have different colors
1?3
2?3
2?4
3?4
1?2
Value 1 for good coloring
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Simpler Graph Coloring Problem
We need to give all possible colors here
Give Hadamard for each wire to get superposition
of all state, which means the set of all colorings
H
0gt
0gt
H
0gt
H
H
H
?
?
?
?
?
1?3
2?3
2?4
3?4
1?2
Value 1 for good coloring
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Block Diagram
All good colorings are encoded by negative phase
Think about this as a very big Kmap with -1 for
every good coloring
Vector Of Hadamards
Vector Of Basic States 0gt
Oracle with Comparators, Global AND gate
Output of AND
Work bits
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What Grover algorithm does?

• Grover algorithm looks to a very big Kmap and
  tells where is the -1 in it.

Here is -1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 -1 1 1 1 1 1
1 1 1 1 1 1 1 1
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What Grover for multiple solutions algorithm
does?

• Grover algorithm looks to a very big Kmap and
  tells where is the -1 in it.
• Grover for many solutions will tell all
  solutions.

Here is -1, and here is -1, and here
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 -1 1 1 1 1 1
1 1 1 -1 -1 1 1 1
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Variants of Grover

• With this oracle the Grover algorithm for many
  solutions will find all good colorings of the
  graph.
• If we want to find the coloring, that is good and
  in addition has less than K colors, we need to
  add the cost comparison circuit to the oracle.
• Then the oracles answers will be one only if
  the coloring is good and has less colors than K.
• The oracle thus becomes more complicated but the
  Grover algorithm can be still used.

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Another example
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This is a better example of problems for Grover
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Advantage of Grover, although not tractable
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Example of oracle
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Another example of Oracle, more realistic
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tour
Answer bit or oracle bit
Working bits
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Possible Applications

• Formulate an oracle (reversible circuit) for the
  following problems
• 1. Graph coloring with of a planar map.
• 2. Graph coloring with the minimum number of
  colors of an arbitrary graph.
• 3. Satisfiability.
• 4. Set Covering.
• 5. Euler Path in a graph.
• 6. Hamiltonian Path in a graph.
• 7. Cryptographic Puzzle like SENDMOREMONEY.