Remove the label-symmetry assumption from quantum collision-finding lower bounds

Establish a space-preserving symmetrisation argument for arbitrary quantum algorithms solving collision finding in uniformly random functions, or prove the corresponding time-space tradeoff without imposing label symmetry, thereby extending the lower bound from label-symmetric algorithms to all quantum algorithms.

Background

The paper proves tight quantum time-space lower bounds for finding a collision in a uniformly random function, and for the search version of Element Distinctness, only within the class of label-symmetric algorithms. Label symmetry requires that the algorithm’s reduced input state be invariant under arbitrary permutations of the function’s range labels.

Because both the uniform input distribution and the condition that an output is a collision are invariant under range relabelling, the authors argue that asymmetric treatment of particular labels does not appear intrinsically useful. However, the standard procedure for symmetrising an arbitrary algorithm stores a random permutation of the range labels, which can require Θ(N log N) additional qubits. This additional memory destroys the space parameter relevant to the paper’s lower bound, leaving open the problem of extending the tradeoff to unrestricted quantum algorithms.

References

The main open problem is to remove the label-symmetry assumption. Because both the uniform input distribution and the collision success condition are invariant under range relabelling, it is difficult to imagine how treating particular labels asymmetrically could help. Nevertheless, the usual method of symmetrising an arbitrary algorithm stores a random permutation and can require $\Theta(N\log N)$ additional space, so it does not preserve the parameter that our lower bound tracks. A space-preserving symmetrisation argument, or a proof that avoids symmetry altogether, would extend the tradeoff to all algorithms.

— Tight Time-Space Lower Bounds for Collision Finding and Element Distinctness under Label Symmetry  (2609.10808 - Magniez et al., 9 Sep 2026) in Section 1, subsection “Open problems”