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arXiv · 2501.08148

Optimal spatial searches with long-range tunneling

Emma C. King·Moritz Linnebacher·Peter P. Orth·Matteo Rizzi·Giovanna Morigi
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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, $T\sim \sqrt{N}$, 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 $1/r^{\alpha}$ and $0<\alpha<3d/2$. For $\alpha 4$. 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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Emma C. King, Moritz Linnebacher, Peter P. Orth, Matteo Rizzi, Giovanna Morigi. 2025-01-14. Optimal spatial searches with long-range tunneling. https://doi.org/10.1103/cv26-dk1q

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