Fermionic tensor network contraction for arbitrary geometries

TL;DR

We address the challenge of simulating fermionic quantum many-body states on arbitrary geometries, where fermionic signs complicate contraction order. We implement two fermionic TN formalisms—globally ordered and locally ordered—in the quimb library and use hyperoptimized contraction strategies for exact and approximate contractions. Benchmarks on the Fermi-Hubbard model with nearest-neighbor hopping $t$ and on-site interaction $U$ are performed on a 3D diamond lattice and on random regular graphs, showing rapid convergence of energies with bond dimension $D$ and the use of cluster-based approximations. The framework enables geometry-agnostic, scalable fermionic TN simulations with broad potential applications in condensed matter, quantum information, and quantum chemistry.

Abstract

We describe our implementation of fermionic tensor network contraction on arbitrary lattices within both a globally ordered and locally ordered formalism. We provide a pedagogical description of these two conventions as implemented for the quimb library. Using hyperoptimized approximate contraction strategies, we present benchmark fermionic projected entangled pair states simulations of finite Hubbard models defined on the three-dimensional diamond lattice and random regular graphs.

Figures (10)

• Figure 1: Examples of contracting a tensor network (a) exactly and (b) approximately. Computation flows from bottom to top. In each case, pairwise contractions form a tree (green lines), with intermediate tensors shown as smaller green circles. In the approximate case, compressions between intermediates also occur, shown here as horizontal orange lines connecting crosses.
• Figure 2: Swap rule for two fermionic tensors $A$ and $B$ for the fermionic contraction, $\mathcal{F}(AB)=\mathcal{F}(\tilde{B}A)=\mathcal{F}(BgA)$ (where we have identified $\tilde{A}=A$). A diagonal bond parity tensor $g$ is inserted between the tensors $A$ and $B$ after the swap.
• Figure 3: (a) Given the ordered fermionic tensor network, $\mathcal{T}=\mathcal{F}(A^{[0]}B^{[1]}C^{[2]}D^{[3]})$ on the graph, the relative ordering in the indexes determines the arrow directions in the bonds. The graph with arrows forms a directed acyclic graph (DAG). When contracting $A^{[0]}$ and $C^{[2]}$, the relative ordering between $B^{[1]}$ and $C^{[2]}$ needs to be swapped to be $\Tilde{C}^{[1]}$ and $B^{[2]}$. During the swap, the direction of the arrow between $B^{[1]}$ and $C^{[2]}$ is converted. (b) When the graph has a cycle, the global ordering of the tensors cannot be assigned. Nonetheless, any directed graph with cycles can be made into a DAG by converting some arrows to assign the order of the tensors.
• Figure 4: (a) Norm tensor network for $| \Psi \rangle = \mathcal{F}(A^{[0]}B^{[1]}C^{[2]})$ as $\langle \Psi | \Psi\rangle = \mathcal{F}(C^{\dag[0]} B^{\dag[1]} A^{\dag[2]} A^{[3]} B^{[4]} C^{[5]})$. (b) Norm for the partial 'cluster' formed by taking only the central site $\mathcal{F}(B^{\dag[0]} B^{[1]})$, requiring some change of arrow directions.
• Figure 5: Geometry of a $3 \times 3 \times 3$ diamond lattice. The central sites in blue correspond to the primitive cell.