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Optimal Quantum Spatial Search with One-Dimensional Long-Range Interactions.

Dylan Lewis1, Asmae Benhemou1, Natasha Feinstein1

  • 1Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom.

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Optimal spatial search using continuous-time quantum walks is now possible in 1D spin chains with long-range interactions. This quantum computing approach achieves a quadratic speedup, offering high fidelity and robustness to noise.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing Algorithms
  • Condensed Matter Physics

Background:

  • Continuous-time quantum walks offer quadratic speedups for spatial search problems in quantum algorithms.
  • Previous studies focused on nearest-neighbor interactions in 1D, which do not support optimal spatial search.
  • The potential of natural models, particularly those with long-range interactions, remains largely unexplored.

Purpose of the Study:

  • To investigate the feasibility of optimal spatial search in 1D spin chains with long-range interactions.
  • To determine the conditions (interaction decay exponent α) required for achieving O(sqrt[n]) search time and high fidelity.
  • To analyze the robustness of this quantum search method against dephasing noise.

Main Methods:

  • Theoretical analysis of continuous-time quantum walks on 1D spin chains with power-law decaying interactions (1/r^α).
  • Derivation of conditions for achieving optimal spatial search runtime and fidelity.
  • Numerical simulations to assess performance and robustness to dephasing noise.

Main Results:

  • Optimal spatial search with O(sqrt[n]) runtime and high fidelity is proven possible for 1D spin chains with long-range interactions.
  • Near-unit fidelity is achieved for interaction decay exponents α ≈ 1.
  • A continuous phase transition is identified at α = 1.5, separating regions where optimal search is possible (α < 1.5) and impossible (α > 1.5).

Conclusions:

  • 1D spin chains with specific long-range interactions can support efficient quantum spatial search.
  • The interaction decay exponent α plays a critical role, with α < 1.5 being necessary for optimal performance.
  • The method demonstrates robustness to dephasing noise, suggesting experimental feasibility with α ≲ 1.2 for near-unit fidelity.