The Asymptotics of Quantum Max-Flow Min-Cut
The Asymptotics of Quantum Max-Flow Min-Cut
复制标题
量子最大流最小割的渐进性
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发表时间：
2016
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通讯作者：
M. Hastings
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作者：
M. Hastings
量子最大流最小割猜想将张量网络的秩与网络中所有张量都相同的情况下的最小割联系起来。（J Am Math Soc 23（1）：107-188，2010）。这一猜想被证明是错误的崔等人。（J Math Phys 57：062206，2016）通过一个明确的反例。在这里，我们证明了这个猜想几乎是正确的，因为当网络边缘的自由度的维数N趋于无穷大时，量子最大流与量子最小割之比收敛于1。证明是基于估计的时刻的奇异值的网络。我们引入了一个推广的“彩虹图”张量网络估计占主导地位的图表。二阶矩和四阶矩的直接比较给出了量子最大流与量子最小割之比的下界。为了显示比趋于1的更紧的界限，我们考虑更高的时刻。此外，我们证明了当N → ∞时的极限矩与网络中张量独立选择的不同系综中的极限矩一致;这用于表明两个不同系综中奇异值的分布弱收敛于相同的极限分布。我们还提出了一个特定的张量网络的数值研究，这表明了一个令人惊讶的依赖N模4的秩赤字，并提出进一步的猜想的限制行为的秩。
The quantum max-flow min-cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical in Calegari et al. (J Am Math Soc 23(1):107–188, 2010). This conjecture was shown to be false in Cui et al. (J Math Phys 57:062206, 2016) by an explicit counter-example. Here, we show that the conjecture is almost true, in that the ratio of the quantum max-flow to the quantum min-cut converges to 1 as the dimension N of the degrees of freedom on the edges of the network tends to infinity. The proof is based on estimating moments of the singular values of the network. We introduce a generalization of “rainbow diagrams” to tensor networks to estimate the dominant diagrams. A direct comparison of second and fourth moments lower bounds the ratio of the quantum max-flow to the quantum min-cut by a constant. To show the tighter bound that the ratio tends to 1, we consider higher moments. In addition, we show that the limiting moments as N → ∞ agree with that in a different ensemble where tensors in the network are chosen independently; this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution. We present also a numerical study of one particular tensor network, which shows a surprising dependence of the rank deficit on N mod 4 and suggests further conjecture on the limiting behavior of the rank.