Adversary Lower Bound for the k-sum Problem

We prove a tight quantum query lower bound $\Omega(n^{k/(k+1)})$ for the problem of deciding whether there exist $k$ numbers among $n$ that sum up to a prescribed number, provided that the alphabet size is sufficiently large. This is an extended and simplified version of an earlier preprint of one of the authors arXiv:1204.5074.

💡 Research Summary

The paper addresses the quantum query complexity of the k‑sum problem, where one is given an input string of length n over a sufficiently large alphabet and a target value t, and must decide whether there exist k distinct indices whose values sum to t. Classical algorithms require O(n^k) time, while quantum algorithms can achieve sub‑linear query complexity. Prior work had established an upper bound of O(n^{k/(k+1)}) using quantum walk techniques, but a matching lower bound was missing. The authors fill this gap by proving a tight lower bound of Ω(n^{k/(k+1)}) under the condition that the alphabet size grows faster than n^k, ensuring that different k‑tuples of positions produce distinct sums with high probability.
The core of the proof relies on the adversary method, a powerful technique for deriving quantum query lower bounds. The authors construct an adversary matrix Γ that captures the difficulty of distinguishing “YES’’ instances (where a suitable k‑tuple exists) from “NO’’ instances (where no such tuple exists). They partition the input space into two families: one consisting of inputs that deliberately avoid any k‑tuple summing to t, and another consisting of inputs that contain exactly one such k‑tuple. By carefully assigning weights to pairs of inputs that differ on a minimal set of positions, they ensure that each row and column of Γ has uniform weight, satisfying the normalization constraints required by the adversary bound.
A key technical ingredient is the use of a large alphabet. Because the alphabet size is assumed to be at least on the order of n^k, the probability that two different k‑tuples produce the same sum is negligible. This eliminates potential collisions that could otherwise reduce the spectral norm of Γ. Consequently, the spectral norm ‖Γ‖ scales as Θ(n^{k/(k+1)}), while the maximum column‑wise change Δ_i remains bounded by a constant. Applying the general adversary bound formula, which states that the quantum query complexity is at least √(‖Γ‖ / max_i‖Δ_i‖), yields the desired Ω(n^{k/(k+1)}) lower bound.
The authors also discuss the tightness of their result. Since an O(n^{k/(k+1)}) quantum algorithm for k‑sum is already known (based on quantum walks on Johnson graphs), the lower bound matches the upper bound up to constant factors, establishing that the quantum query complexity of the k‑sum problem is Θ(n^{k/(k+1)}). They further analyze the dependence on the alphabet size, showing that if the alphabet is too small, the lower bound may degrade because different k‑tuples could collide, reducing the effectiveness of the adversary construction.
In the concluding section, the paper highlights the broader implications of the technique. The adversary matrix design used here can be adapted to other combinatorial search problems, such as the k‑distinctness problem and multi‑collision detection, suggesting a pathway to tight quantum lower bounds for a wider class of problems. The work thus not only resolves an open question about the optimal quantum query complexity of k‑sum but also enriches the toolbox for quantum complexity theorists.