The hierarchical (multi-linear) rank of an order-$d$ tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be expected to be efficiently approximable without the curse of dimensionality, i.e., a complexity growing exponentially with $d$. In this work, we show that the ground state projection (GSP) of a class of unbounded Hamiltonians can be approximately represented as an operator of low effective dimensionality that is independent of the (high) dimension $d$ of the GSP. This allows to approximate the GSP without the curse of dimensionality.
翻译:10美元订单的等级(多线)等级(多线级)是确定作为(树)Tensor Network(TN)代表一个阵列的成本的关键。 一般来说,众所周知,如果有固定的准确性,如果没有维度的诅咒(即复杂性以美元指数指数增长,以美元计),一个带有随机条目的阵列不可能被有效地接受。 在这项工作中,我们表明,一组不受约束的汉密尔顿人的地面状态预测(GSP)可以大致作为独立于普惠制(高)维度(美元)的低有效维度操作者,这样可以接近普惠制,而不受维度的诅咒。