1. 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

2. From Grover to general search

6. The Graph Coloring Problem
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 1
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 7
This graph is 3-colorable

7. Simpler Graph Coloring Problem1 3 2
We need to give all possible colors here 4
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

8. 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
|0>
|0> H
|0> H H H     
13
23
24
34
12
Value 1 for good coloring

9. Oracle with Comparators,
All good colorings are encoded by negative phase
Block Diagram
Think about this as a very big Kmap with -1 for every good coloring
Vector Of
Basic
States
|0>
Vector Of
Hadamards
Oracle with Comparators,
Global AND gate
Output of AND
Work bits

10. 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

11. 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

12. 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 oracle’s 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.

13. Another example

15. This is a better example of problems for Grover

16. Advantage of Grover, although not “tractable”

17. Example of oracle

18. Another example of Oracle, more realistic

19. Answer bit or oracle bit
tour
Answer bit or oracle bit
Working bits

21. 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 SEND+MORE=MONEY.