route finding: a quantum [non] algorithm jason clemons eecs 598 november 7, 2001
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TRANSCRIPT
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 GraphsReview of basic graphsReview of representation
The QAND
The Basic Algorithm
Interesting Findings
Final Thoughts and Questions
Graphs
TerminologyEdges – connect two nodesNode – Basic state or representation there
ofWeight – Cost associated with a node
Graphs
A few characteristics:WeightedDirectional
Problems from graphsGraph Coloring Hamiltonian Circuit FindingTraveling Salesman (Shortest Route)
Our Graph
Shortest Route Finding for:Weighted finite edgesFinite number of nodesUndirectedNo self connection edges
Representation As Matrix
Adjacency Matrix:N x N Matrix where N is the number of
nodesMij = 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
Our Graph as a TreeS
BA
CBS S A C
CA
A CS
BA BA
B CS
BA
2 4
4 7
12 97 4 5
6 8 10 12
7 6 10
11 8
8 5 61
The Quantum And
a QSUMAND b = a + b if (a > 0 and b >0)
0 if a=0 or b =0 where a, b R+ {0}
Example:2 QSUMAND 2 = 42 QSUMAND 0 = 0
The Matrix creation
Each Row is superposition of states representing the edgesExample:
From our Matrix before the row |S> is the vector a|000> + b|010> +c|100> + d|000>
An element can be identified using the row and column labelsExample: <A1|b1> = 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
|S1>QSUMAND|S1> |0 2 4 0> QSUMAND |0 2 4 0> = |0 4 8 0>
|S1>QSUMAND|A1> |0 2 4 0> QSUMAND |2 0 1 5> = |0 0 5 0>
|S1>QSUMAND|B1> = |0 3 0 0>
|S1>QSUMAND|C1> = |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 <row|column>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
<S2|a2> QSUMAND |S1> |4> QSUMAND | 0 2 4 0> = |0 6 8 0>
<A2|a2> QSUMAND |A1> = |0 0 0 0>
<B2|a2> QSUMAND |B1> = |7 4 0 5>
<C2|a2> QSUMAND |C1> = |0 12 9 0>
M3 and M4
They are each a separate side of the search treeM3 is path from S down Node A
M4 is path from S down Nobe B
M3 and M4
Entry <Row|Column> is the distance from start to Row by path computed so far and then from row to columnExample:
<B3|s3> in M3 has the cumulative weight for S->A->B->S which is 7
<B4|s4> in M4 has the cumulative weight for S->B->B->S which is 0
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 |s> = |s> + |a> + |b> + |c> looks okay but
leads to: |S1> = a|000> + b|010> + c|100> + d|000>
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 |a> for the vector |s> = |s> + |a> + |b> + |c>
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 weightFurthermore 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
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.