{
  "id": "2106.02005",
  "version": "v1",
  "published": "2021-06-03T17:22:08.000Z",
  "updated": "2021-06-03T17:22:08.000Z",
  "title": "Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks",
  "authors": [
    "Harry Buhrman",
    "Bruno Loff",
    "Subhasree Patro",
    "Florian Speelman"
  ],
  "categories": [
    "quant-ph",
    "cs.CC"
  ],
  "abstract": "Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n^3)-time or O(n^2)-time classic algorithm can be solved by a known O(n^1.5)-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro, and Speelman (arXiv:1911.05686), and by Aaronson, Chia, Lin, Wang, and Zhang (arXiv:1911.01973). In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-a}), for any a>0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-b})-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-03T17:22:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum speed-up",
      "fine-grained complexity",
      "computational geometry",
      "quantum walks",
      "quantum algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}