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Quantum spatial search using continuous-time quantum walks is optimal for most random graphs. This finding extends to high-fidelity quantum communication tasks in disordered qubit networks.

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Area of Science:

  • Quantum Information Science
  • Graph Theory
  • Computer Science

Background:

  • Spatial search algorithms based on continuous-time quantum walks (CTQW) offer efficient solutions for locating marked nodes in graphs.
  • The optimal performance of CTQW search is typically limited to specific graph structures.

Purpose of the Study:

  • To determine the conditions under which CTQW spatial search is optimal on Erdös-Renyi random graphs.
  • To investigate the broader implications of this optimality for quantum information tasks in disordered systems.

Main Methods:

  • Analysis of CTQW spatial search on Erdös-Renyi random graphs with edge probability p.
  • Proving optimality by examining the ratio of graph eigenvalues.
  • Extending search results to quantum communication protocols like state transfer and entanglement generation.

Main Results:

  • CTQW search is almost surely optimal on Erdös-Renyi random graphs when p≥log^{3/2}(n)/n.
  • Quantum spatial search is optimal for a vanishingly small fraction of graphs as n approaches infinity.
  • Optimality is demonstrated for graphs with a bounded ratio between their second largest and largest eigenvalues.

Conclusions:

  • Quantum spatial search retains its optimal performance across a vast range of disordered graph structures.
  • The findings enable high-fidelity quantum communication, including state transfer and entanglement generation, in random qubit networks.
  • Quantum information tasks can be effectively implemented even in highly disordered systems.