稠密展开图上二部量子最大割的经典算法
Classical Algorithms for Bipartite Quantum Max-Cut on Dense Expanders
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中文总结 AI 辅助
针对稠密平衡二部展开图上的量子最大割问题，提出多项式时间经典随机算法，通过马尔可夫链估计基态能量与关联，达到加性误差ε。
中文摘要 AI 辅助
量子最大割是一个被广泛研究的局域哈密顿量问题，其中二部情形的经典复杂性仍然开放。我们给出了自旋-1/2反铁磁海森堡模型（等价于量子最大割）在稠密平衡二部展开图上的多项式时间经典随机算法。该类别以高概率包含G(n,n,p)对于每个固定p>0。该算法在n和1/ε的多项式时间内估计基态能量和任意基态边关联至加性误差ε。我们在完全二部图的完美匹配上构造了一个马尔可夫链，每个匹配表示单重态的乘积，该链收敛到基态，并利用图的展开性质证明该链的多项式步数和样本数就足够了。
英文摘要
Quantum Max-Cut is a well-studied local Hamiltonian problem for which the classical complexity of the bipartite case remains open. We give a polynomial-time classical randomized algorithm for the spin-$\tfrac12$ antiferromagnetic Heisenberg model, equivalently Quantum Max-Cut, on dense balanced bipartite expanders. This class includes $G(n,n,p)$ with high probability for every fixed $p>0$. The algorithm estimates the ground energy and any ground state edge correlation to additive error $\varepsilon$ in time polynomial in $n$ and $1/\varepsilon$. We construct a Markov chain on perfect matchings of the complete bipartite graph, each matching representing a product of singlets, that converges to the ground state, and use the expansion of the graph to show that polynomially many steps and samples of this chain suffice.
