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## Route Finding: A Quantum [Non] Algorithm

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### Route Finding: A Quantum [Non] Algorithm Jason Clemons EECS 598 November 7, 2001 – PowerPoint PPT presentation

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Title: Route Finding: A Quantum [Non] Algorithm

1
Route Finding A Quantum Non Algorithm
• Jason Clemons
• EECS 598
• November 7, 2001

2
The Paper
• Narayanan, A., Wallace, J. A Quantum Algorithm
for Route Finding. Proceedings of the 15th
European Meeting on Cybernetics and Systems
Research (EMCSR 2000), Vienna, Austria, 25-28
April 2000, Vol. 1, pp 140-143.

3
Outline for the day
• Introduction to Graphs
• Review of basic graphs
• Review of representation
• The QAND
• The Basic Algorithm
• Interesting Findings
• Final Thoughts and Questions

4
Graphs
• Common representation in CS

A
B
C
D
5
Graphs
• Terminology
• Edges connect two nodes
• Node Basic state or representation there of
• Weight Cost associated with a node

6
Graphs
• A few characteristics
• Weighted
• Directional
• Problems from graphs
• Graph Coloring
• Hamiltonian Circuit Finding
• Traveling Salesman (Shortest Route)

7
Our Graph
• Shortest Route Finding for
• Weighted finite edges
• Finite number of nodes
• Undirected
• No self connection edges

8
Our Graph (Example)
A
2
5
C
S
1
4
2
B
9
Representation As Matrix
• N x N Matrix where N is the number of nodes
• Mij wi to j where w is the weight of the edge
from node I to node J and zero or zero if there
is no edge

10
Representation as Matrix
S A B C
S 0 2 4 0
A 2 0 1 5
B 4 1 0 2
C 0 5 2 0
11
Our Graph as a Tree
S
4
2
B
A
8
6
4
5
7
1
C
B
S
S
A
C
12
9
6
8
11
8
10
12
7
4
5
7
6
10
C
A
B
A
B
A
B
A
A
C
S
B
C
S
12
The Quantum And
• a QSUMAND b a b if (a gt 0 and b gt0)
• 0 if a0 or b 0
• where a, b ? R ? 0
• Example
• 2 QSUMAND 2 4
• 2 QSUMAND 0 0

13
The Matrix creation
• Each Row is superposition of states representing
the edges
• Example
• From our Matrix before the row Sgt is the vector
a000gt b010gt c100gt d000gt
• An element can be identified using the row and
column labels
• Example ltA1b1gt 1 (note not an IP)

14
Quantum Registers
• Similar to what is used by Shor
• 3 Registers
• Reg1 Holds the original adjacency matrix
• Reg2 Holds Matrix created when choose start node
• Reg3 Interacts with Reg1 to tell us whether a
route exists.
• During the Algorithm we alternate between Reg2
and Reg3 to hold info

15
Algorithm Step 1
• Select the Starting node and perform the QSUMAND
manipulation process (Perform QSUMAND between
start node row and all other rows) to produce a
new matrix M2

16
Algorithm Step 1
• S1gtQSUMANDS1gt
• 0 2 4 0gt QSUMAND 0 2 4 0gt 0 4 8 0gt
• S1gtQSUMANDA1gt
• 0 2 4 0gt QSUMAND 2 0 1 5gt 0 0 5 0gt
• S1gtQSUMANDB1gt 0 3 0 0gt
• S1gtQSUMANDC1gt 0 7 6 0gt

17
M2
s2gt a2gt b2gt c2gt
S2gt 0 4 8 0
A2gt 0 0 5 0
B2gt 0 3 0 0
C2gt 0 7 6 0
18
What M2 Says
• M2 shows that from S to the row node is connected
through column node with a total length of the
entry ltrowcolumngt
• Example
• S to B to A is 5
• We have a new level of the search tree!

19
What M2 Says
• The column shows from the start point to that
node and from that node to each of the rows.

20
Algorithm Step 2
• Manipulate each non-zero column of the matrices
constructed in previous step by performing
QSUMAND with each non zero column and the entire
row in matrix M1 that is associated with each
element from the non zero column

21
Step 2 example
• ltS2a2gt QSUMAND S1gt
• 4gt QSUMAND 0 2 4 0gt 0 6 8 0gt
• ltA2a2gt QSUMAND A1gt 0 0 0 0gt
• ltB2a2gt QSUMAND B1gt 7 4 0 5gt
• ltC2a2gt QSUMAND C1gt 0 12 9 0gt

22
M3
s3gt a3gt b3gt c3gt
S3gt 0 6 8 0
A3gt 0 0 0 0
B3gt 7 4 0 5
C3gt 0 12 9 0
23
M4
s4gt a4gt b4gt c4gt
S4gt 0 10 12 0
A4gt 7 0 6 10
B4gt 0 0 0 0
C4gt 0 11 8 0
24
M3 and M4
• They are each a separate side of the search tree
• M3 is path from S down Node A
• M4 is path from S down Nobe B

25
M3 and M4
• Entry ltRowColumngt is the distance from start to
Row by path computed so far and then from row to
column
• Example
• ltB3s3gt in M3 has the cumulative weight for
S-gtA-gtB-gtS which is 7
• ltB4s4gt in M4 has the cumulative weight for
S-gtB-gtB-gtS which is 0

26
M3 and M4
S
M4
M3
4
2
B
A
8
6
4
5
7
C
B
S
S
A
C
12
9
6
8
11
8
10
12
7
4
5
7
6
10
C
A
B
A
B
A
B
A
A
C
S
B
C
S
27
Step 3
• Repeat step 2 adequate amount of times to reach
bottom of tree and for each new matrix at each
level

28
Step 4
• Route exists if there is non zero value in a
column
• Thus measure columns and non zero values are path

29
• Each row and column are a super position of
quantum state vectors with N states where N is
number of nodes
• The N states either show a weight if there is an
edge or show that there is no weight ie the n
states represent a weight

30
Issues
• Even if have weights, information on which node
is connected to which is lost
• ?sgt ?sgt ?agt ?bgt ?cgt looks okay but
• S1gt a000gt b010gt c100gt d000gt

31
Issues
• Author calls for n distinct values but opens up
problem of dealing with states that have the same
weight
• Author agrees saying
• There are problems describing the adjacency
matrix as quantum state vectors

32
Issues
• Algorithm calls for acting on single state in a
superposition of states ie acting solely on state
?agt for the vector ?sgt ?sgt ?agt ?bgt
?cgt

33
Alternate Representation
• Add in qubits for the node in which this weight
is for.
• One issue is make sure you have enough qubits to
represent the weight
• Furthermore the fact that the Qubits are no
longer in a single state will complicate the
QSUMAND

34
Review
• Graph Basics
• Route Finding in general
• QAND
• The basic Algorithm
• Issues with Algorithm
• Fixes to be explored

35
Questions?
36
References
• Narayanan, A., Wallace, J., A Quantum Algorithm
for Route Finding. Proceedings of the 15th
European Meeting on Cybernetics and Systems
Research (EMCSR 2000), Vienna, Austria, 25-28
April 2000, Vol. 1, pp 140-143.
• Penrose, R. Shadows of the Mind A search for
the missing science of consciousness. Oxford
University Press 1994.
• Shor, P., Algorithms for quantum computation
Discrete logarithms and factoring 1994.

37