Problem
Research questions and friction points this paper is trying to address.
Maximum-Weight Perfect Matching
Quantum Algorithm
General Graphs
Quantum Speed-up
A Quantum Scaling Algorithm for Maximum-Weight Perfect Matching in General Graphs
🤖 AI Summary
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📝 Abstract
Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in \(
\widetilde{O}(n m^{2/3}\log W) \) time, where $W$ is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of \(
\widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum algorithm
Maximum-weight perfect matching
Quantum speed-up
QRAM model
General graphs
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