The Asymptotics of Quantum Max-Flow Min-Cut

TL;DR

This paper proves that the ratio of quantum max-flow to min-cut converges to 1 as the edge dimension increases, using moments of singular values and diagram techniques, with numerical evidence for specific networks.

Contribution

It introduces a generalized rainbow diagram method for tensor networks and establishes the asymptotic convergence of quantum max-flow to min-cut ratio.

Findings

The ratio converges to 1 as N approaches infinity.

Higher moments provide tighter bounds on the ratio.

Numerical results show dependence on N mod 4.

Abstract

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\cite{mfmc1}. This conjecture was shown to be false in Ref. \onlinecite{mfmc2} 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"\cite{rainbow} 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 $N \rightarrow \infty$ 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.