Vertices cannot be hidden from quantum spatial search for almost all random graphs

Vertices cannot be hidden from quantum spatial search for almost all random graphs In this paper, we show that all nodes can be found optimally for almost all random Erdős–Rényi $$\mathcal G(n,p)$$ G ( n , p ) graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $$p=\omega (\log ^8(n)/n)$$ p = ω ( log 8 ( n ) / n ) , while the second requires $$p\ge (1+\varepsilon )\log (n)/n$$ p ≥ ( 1 + ε ) log ( n ) / n , where $$\varepsilon >0$$ ε > 0 . The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ norm. At the same time for $$p<(1-\varepsilon )\log (n)/n$$ p < ( 1 - ε ) log ( n ) / n , the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Vertices cannot be hidden from quantum spatial search for almost all random graphs

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Publisher
Springer US
Copyright
Copyright © 2018 by The Author(s)
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-018-1844-7
Publisher site
See Article on Publisher Site

Abstract

In this paper, we show that all nodes can be found optimally for almost all random Erdős–Rényi $$\mathcal G(n,p)$$ G ( n , p ) graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $$p=\omega (\log ^8(n)/n)$$ p = ω ( log 8 ( n ) / n ) , while the second requires $$p\ge (1+\varepsilon )\log (n)/n$$ p ≥ ( 1 + ε ) log ( n ) / n , where $$\varepsilon >0$$ ε > 0 . The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ norm. At the same time for $$p<(1-\varepsilon )\log (n)/n$$ p < ( 1 - ε ) log ( n ) / n , the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.

Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 22, 2018

References

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