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Optimal spatial searches with long-range tunneling

Emma C. King1, Moritz Linnebacher1, Peter P. Orth1, Matteo Rizzi2, and Giovanna Morigi1

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Phys. Rev. Research 7, 043020 – Published 6 October, 2025

DOI: https://doi.org/10.1103/cv26-dk1q

Abstract

A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time T can exhibit Grover's optimal scaling with the lattice size N, namely, T∼N, on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of d>4 dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance r as 1/rα and 0<α<3d/2. For α<d the dynamics mimics the one of a globally connected graph. For d<α<d+2, the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension ds=2d/(α−d), indicating that the search is optimal for ds>4. Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.

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Phys. Rev. Research 7, 043020– Published 6 October, 2025
  • Received 14 January 2025
  • Revised 17 April 2025
  • Accepted 8 July 2025
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