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.
Quantum Information Processing – Springer Journals
Published: Feb 22, 2018
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