Tensor network contraction is a powerful computational tool in quantum many-body physics, quantum information and quantum chemistry. The complexity of contracting a tensor network is thought to mainly depend on its entanglement properties, as reflected by the Schmidt rank across bipartite cuts. Here, we study how the complexity of tensor-network contraction depends on a different notion of quantumness, namely, the sign structure of its entries. We tackle this question rigorously by investigating the complexity of contracting tensor networks whose entries have a positive bias. We show that for intermediate bond dimension d>~n, a small positive mean value >~1/d of the tensor entries already dramatically decreases the computational complexity of approximately contracting random tensor networks, enabling a quasi-polynomial time algorithm for arbitrary 1/poly(n) multiplicative approximation. At the same time exactly contracting such tensor networks remains #P-hard, like for the zero-mean case [HHEG20]. The mean value 1/d matches the phase transition point observed in [CJHS24]. Our proof makes use of Barvinok's method for approximate counting and the technique of mapping random instances to statistical mechanical models. We further consider the worst-case complexity of approximate contraction of positive tensor networks, where all entries are non-negative. We first give a simple proof showing that a multiplicative approximation with error exponentially close to one is at least StoqMA-hard. We then show that when considering additive error in the matrix 1-norm, the contraction of positive tensor network is BPP-Complete. This result compares to Arad and Landau's [AL10] result, which shows that for general tensor networks, approximate contraction up to matrix 2-norm additive error is BQP-Complete.

翻译：张量网络缩并是量子多体物理、量子信息和量子化学中一种强大的计算工具。张量网络缩并的复杂度通常被认为主要取决于其纠缠特性，这反映在二分切割上的施密特秩。本文研究张量网络缩并的复杂度如何依赖于另一种量子性概念，即其分量的符号结构。我们通过严格研究具有正偏置分量的张量网络的缩并复杂度来探讨这个问题。我们证明，对于中等键维数 d>~n，当张量分量具有小正均值 >~1/d 时，随机张量网络的近似缩并计算复杂度会急剧下降，使得任意 1/poly(n) 乘法近似可在拟多项式时间内实现。同时，精确缩并此类张量网络仍然保持 #P-困难性，与零均值情况相同 [HHEG20]。均值 1/d 与 [CJHS24] 中观察到的相变点一致。我们的证明利用了 Barvinok 的近似计数方法以及将随机实例映射到统计力学模型的技术。我们进一步考虑了正张量网络（所有分量非负）近似缩并的最坏情况复杂度。首先给出一个简单证明，表明误差指数趋近于 1 的乘法近似至少是 StoqMA-困难的。随后证明，当考虑矩阵 1-范数下的加法误差时，正张量网络的缩并是 BPP-完全的。该结果可与 Arad 和 Landau [AL10] 的结果相比较，后者表明对于一般张量网络，达到矩阵 2-范数加法误差的近似缩并是 BQP-完全的。