Problem
Research questions and friction points this paper is trying to address.
Subset Sum
k-SUM
quantum algorithms
worst-case complexity
Improved Quantum Algorithms for Subset Sum and $k$-SUM
🤖 AI Summary
This work addresses the worst-case quantum complexity bottlenecks of Subset Sum and its parameterized variant, the k-SUM problem. The authors propose a novel quantum search framework that integrates quantum walks, block reduction, and a refined modular-7 classification scheme. By leveraging structured subproblem decomposition and an improved search strategy, they achieve the first faster worst-case quantum algorithm for instances where \(k \equiv 3\) or \(6 \pmod{7}\), attaining a time complexity of \(\tilde{O}(n^{\Psi_k})\). Extending this approach to the general Subset Sum problem yields a quantum algorithm with complexity \(O^*(2^{2n/7})\), surpassing the previous best-known bound of \(O^*(2^{n/3})\).
📝 Abstract
The Subset Sum problem asks whether, given $n$ integers and a target, some subset of the integers sums to the target. Its best known worst-case running time is $O^*(2^{n/2})$ (Horowitz and Sahni, 1974), whereas the best quantum upper bound is $O^*(2^{n/3})$ (Bernstein, Jeffery, Lange, and Meurer, 2013). The $k$-SUM problem is a parameterized version of Subset Sum asking whether there are $k$ integers that sum to the target. The best classical upper bound for it is $\widetilde O(n^{\lceil k/2\rceil})$, whereas the best quantum running time is $\widetilde O(n^{k/3})$ (Tani, 2009). For random instances, a quantum algorithm with running time $\widetilde O(n^{Φ_k})$ is known, where
$$ Φ_k=\frac{2k-\lfloor k/7\rfloor-\lfloor (k+3)/7\rfloor}{6} $$
(Schrottenloher, 2021).
We present a new quantum algorithm solving worst-case $k$-SUM in time $\widetilde O(n^{Ψ_k})$, where
$$ Ψ_k=Φ_k-\frac{[k\equiv 3\bmod 7]}{9}-\frac{[k\equiv 6\bmod 7]}{18}. $$
The algorithm is not only faster for all $k$ congruent to $3$ or $6$ modulo $7$, but also gives a worst-case guarantee rather than a guarantee restricted to single-solution random instances. Combining our algorithm for $7$-SUM with the standard block reduction technique yields an $O^*(2^{2n/7})$ quantum algorithm for Subset Sum, improving the previously known $O^*(2^{n/3})$ algorithm.
Innovation
Methods, ideas, or system contributions that make the work stand out.
quantum algorithm
k-SUM
Subset Sum
worst-case complexity
block reduction
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