Optimal spatial searches with long-range tunneling
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917005808893952 |
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| author | King, Emma C. Linnebacher, Moritz Orth, Peter P. Rizzi, Matteo Morigi, Giovanna |
| author_facet | King, Emma C. Linnebacher, Moritz Orth, Peter P. Rizzi, Matteo Morigi, Giovanna |
| contents | 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 $1/r^α$ 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 $d_s=2d/(α-d)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal spatial searches with long-range tunneling King, Emma C. Linnebacher, Moritz Orth, Peter P. Rizzi, Matteo Morigi, Giovanna Quantum Physics Other Condensed Matter Statistical Mechanics 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 $1/r^α$ 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 $d_s=2d/(α-d)$, 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. |
| title | Optimal spatial searches with long-range tunneling |
| topic | Quantum Physics Other Condensed Matter Statistical Mechanics |
| url | https://arxiv.org/abs/2501.08148 |