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PPT – Route Finding: A Quantum [Non] Algorithm PowerPoint presentation | free to download - id: 68526a-OTA1O

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Route Finding A Quantum Non Algorithm

- Jason Clemons
- EECS 598
- November 7, 2001

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.

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

Graphs

- Common representation in CS

A

B

C

D

Graphs

- Terminology
- Edges connect two nodes
- Node Basic state or representation there of
- Weight Cost associated with a node

Graphs

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

Our Graph

- Shortest Route Finding for
- Weighted finite edges
- Finite number of nodes
- Undirected
- No self connection edges

Our Graph (Example)

A

2

5

C

S

1

4

2

B

Representation As Matrix

- Adjacency 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

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

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

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

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)

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

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

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

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

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!

What M2 Says

- The column shows from the start point to that

node and from that node to each of the rows.

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

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

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

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

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

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

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

Step 3

- Repeat step 2 adequate amount of times to reach

bottom of tree and for each new matrix at each

level

Step 4

- Route exists if there is non zero value in a

column - Thus measure columns and non zero values are path

Expanding the Adjacency Matrix Creation

- 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

Issues

- Even if have weights, information on which node

is connected to which is lost - ?sgt ?sgt ?agt ?bgt ?cgt looks okay but

leads to - S1gt a000gt b010gt c100gt d000gt

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

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

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

Review

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

Questions?

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.

Follow Up

- I need a drink after all this.