Positive bias makes tensor-network contraction tractable

TL;DR

The work rigorously analyzes how the sign structure of tensor-network entries influences contraction complexity. By leveraging Barvinok's approximate counting framework and a root-free strip approach, it shows that a small positive bias mu>=1/d in intermediate-d bond-dimension 2D random TNs yields a quasi-polynomial-time algorithm for multiplicative approximation, while exact contraction remains #P-hard. It further characterizes the complexity landscape for fully positive tensor networks, proving BPP-completeness for additive 1-norm approximation and StoqMA-hardness for multiplicative accuracy, and connects randomness to a 2D Ising model that underpins the analysis. Collectively, the results reveal a sign-driven phase-like transition in TN contraction complexity and open avenues for both theoretical and practical exploration of tensor-network methods in quantum physics and chemistry.

Abstract

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.

Figures (7)

• Figure 1: Tensor and operations on tensors. (a) A rank-$k$ tensor. (b) The product of a rank-$k$ and a rank-$l$ tensor. (c) Contracting two tensors by identifying edges $i_1$ and $j_1$. (d) Contracting two free edges in the same tensor. (e) A special tensor which can be factorized into a product of rank-$1$ tensors. (f) If all tensors have a factorized structure, then the contraction value of tensor network can be computed by contracting the rank-$1$ tensors.
• Figure 2: (a) The expectation of the product of $A^{[v]}$ and its conjugate is a product of delta tensors. (b) The expectation of the product $M^{[v]} \otimes \overline{ M^{[v]}}$ decomposes into a linear combination of delta tensors and a product of rank-$1$ tensors.
• Figure 3: (a) Pair the tensors $M^{[v]}$ and its conjugate in $E_A |g_A(z)|^2.$ (b) The tensors with respect to edge $(v,w)$ in $T(s)$. From top to button, the value of $(s_v,s_w)$ are $(1,-1),(1,1),(-1,-1)$ respectively.
• Figure 4: The radius of the small disk and the big disk is $\lambda$ and $1-\lambda$ respectively. We divide the big disk $\mathcal{B}(1-\lambda)$ into $M$ circular sectors. In each sector we choose a strip of width $w$. The first strip $\mathcal{T}_0$ starts from $-w$ and ends at $1-2\lambda+w$. Other strips are rotations of $\mathcal{T}_0$. All the strips are disjoint outside the small disk $\mathcal{B}(\lambda)$.
• Figure 5: $F_i$ is the edges connecting $\{v_1,...,v_{i-1}\}$ and $\{v_i,...,v_n\}$. The edges attached to $v_i$ are partitioned into the in-edges $K_i$ and out-edges $L_i$. When contracting the tensor $M^{[v_i]}$ we map the in-edges to out-edges. The edges in $F_i$ but not in $K_i$ are called $J_i$.

Theorems & Definitions (56)

• Theorem 1: Informal version of Theorem \ref{['thm:exactH']}
• Theorem 2: Informal version of Theorem \ref{['thm:random']}
• Theorem 3: Informal version of Theorem \ref{['thm:StoqMA']}
• Theorem 4: Informal version of Theorem \ref{['thm:BPP']}
• Example 5
• Lemma 6: Approximation using a root-free disk, see the proof of Lemma 1.2 in barvinok2016computing
• Lemma 7: barvinok2016approximating
• Lemma 8: Embedding a disk into a strip, Lemma 8.1 in barvinok2016approximating
• Corollary 9: Approximation using a root-free strip
• proof