🤖 AI Summary
A long-standing gap exists between time complexity and query complexity for the $k$-distinctness problem.
Method: We introduce a novel multidimensional quantum walk framework grounded in extended electrical network theory, integrating graph-structural encoding, amplitude amplification, and tailored quantum state evolution.
Contribution/Results: Our approach achieves the first simultaneous optimality in both time and query complexity for $k$-distinctness, attaining time complexity $widetilde{O}(n^{3/4 - 1/(4(2^k-1))})$, which matches the known optimal query complexity exactly. Moreover, it yields exponential speedup for the welded tree problem, solving it with $O(n)$ queries and $O(n^2)$ time—improving over prior superpolynomial quantum time bounds. The core innovation lies in transcending the limitations of one-dimensional quantum walks, establishing a scalable, multidimensional quantum search paradigm. This provides a general-purpose tool for designing quantum algorithms on complex combinatorial structures.
📝 Abstract
While the quantum query complexity of $k$-distinctness is known to be $Oleft(n^{3/4-1/4(2^k-1)}
ight)$ for any constant $k geq 4$, the best previous upper bound on the time complexity was $widetilde{O}left(n^{1-1/k}
ight)$. We give a new upper bound of $widetilde{O}left(n^{3/4-1/4(2^k-1)}
ight)$ on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in $O(n)$ queries and $O(n^2)$ time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.