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We describe our implementation of fermionic tensor network contraction on arbitrary lattices within both a globally ordered and a 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 state simulations of finite Hubbard models defined on the three-dimensional diamond lattice and random regular graphs. Published by the American Physical Society2025 

Abstract Recent experimental advances have stimulated interest in the use of large, two-dimensional arrays of Rydberg atoms as a platform for quantum information processing and to study exotic many-body quantum states. However, the native long-range interactions between the atoms complicate experimental analysis and precise theoretical understanding of these systems. Here we use new tensor network algorithms capable of including all long-range interactions to study the ground state phase diagram of Rydberg atoms in a geometrically unfrustrated square lattice array. We find a greatly altered phase diagram from earlier numerical and experimental studies, revealed by studying the phases on the bulk lattice and their analogs in experiment-sized finite arrays. We further describe a previously unknown region with a nematic phase stabilized by short-range entanglement and an order from disorder mechanism. Broadly our results yield a conceptual guide for future experiments, while our techniques provide a blueprint for converging numerical studies in other lattices. 

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications. 

Tensor contractions are ubiquitous in computational chemistry andphysics, where tensors generally represent states or operators andcontractions express the algebra of these quantities. In this context,the states and operators often preserve physical conservation laws,which are manifested as group symmetries in the tensors. These groupsymmetries imply that each tensor has block sparsity and can be storedin a reduced form. For nontrivial contractions, the memory footprint andcost are lowered, respectively, by a linear and a quadratic factor inthe number of symmetry sectors. State-of-the-art tensor contractionsoftware libraries exploit this opportunity by iterating over blocks orusing general block-sparse tensor representations. Both approachesentail overhead in performance and code complexity. With intuition aidedby tensor diagrams, we present a technique, irreducible representationalignment, which enables efficient handling of Abelian group symmetriesvia only dense tensors, by using contraction-specific reduced forms.This technique yields a general algorithm for arbitrary group symmetriccontractions, which we implement in Python and apply to a variety ofrepresentative contractions from quantum chemistry and tensor networkmethods. As a consequence of relying on only dense tensor contractions,we can easily make use of efficient batched matrix multiplication viaIntel’s MKL and distributed tensor contraction via the Cyclops library,achieving good efficiency and parallel scalability on up to 4096 KnightsLanding cores of a supercomputer. 

The quantitative description of correlated electron materials remains a modern computational challenge. We demonstrate a numerical strategy to simulate correlated materials at the fully ab initio level beyond the solution of effective low-energy models and apply it to gain a detailed microscopic understanding across a family of cuprate superconducting materials in their parent undoped states. We uncover microscopic trends in the electron correlations and reveal the link between the material composition and magnetic energy scales through a many-body picture of excitation processes involving the buffer layers. Our work illustrates a path toward a quantitative and reliable understanding of more complex states of correlated materials at the ab initio many-body level.