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Connectivity is a poor indicator of fast quantum search
David A Meyer1, Thomas G Wong2
1Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112, USA.
Physical Review Letters
|April 4, 2015
Summary
Quantum search on graphs is not always faster on highly connected structures. This study demonstrates graphs with low connectivity that allow fast quantum search, challenging previous assumptions.
Area of Science:
- Quantum physics
- Graph theory
- Computational complexity
Background:
- Quantum walks offer a potential speedup for search algorithms compared to classical random walks.
- Previous research suggested a correlation between high graph connectivity and fast quantum search performance.
- The efficiency of quantum search on d-dimensional cubic lattices is known to be O(√N) for d≥5.
Purpose of the Study:
- To investigate the relationship between graph connectivity and quantum search efficiency.
- To challenge the prevailing intuition that high connectivity guarantees fast quantum search.
- To present counterexamples demonstrating this relationship is not always true.
Main Methods:
- Analysis of quantum particle search on various graph structures using Schrödinger's equation.
- Development of a novel two-stage quantum walk algorithm.
- Adjustment of walking rates in quantum walk algorithms to optimize search probability.
Main Results:
- Demonstrated that quantum search is not solely dependent on graph connectivity.
- Provided examples of graphs with low connectivity exhibiting fast quantum search.
- Presented a graph with high connectivity that results in slow quantum search.
- Introduced a two-stage quantum walk requiring adjusted walking rates for efficient search.
Conclusions:
- Graph connectivity is not a definitive indicator of quantum search speed.
- The efficiency of quantum search can be influenced by factors beyond simple connectivity metrics.
- Novel quantum walk strategies can achieve efficient search even on less connected graphs.
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