  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.
