Optimal spatial searches with long-range tunneling

• URL: http://arxiv.org/abs/2501.08148v1
• Date: Tue, 14 Jan 2025 14:25:02 GMT
• Title: Optimal spatial searches with long-range tunneling
• Authors: Emma C. King, Moritz Linnebacher, Peter P. Orth, Matteo Rizzi, Giovanna Morigi,
• Abstract summary: We show that Grover's optimal scaling can be reached in lower dimensions on lattices of long-range interacting particles. Our work identifies an exact relation between criticality of long-range and short-range systems.
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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 $\alpha<d$ the dynamics mimics the one of a globally connected graph. For $d<\alpha<d+2$, the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension $d_s=2d/(\alpha-d)$, indicating that the search is optimal for $d_s>4$. 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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