Tensor networks for complex quantum systems

Pouria Mistani, Samira Pakravan, Frédéric Gibou

Mario Collura, Guglielmo Lami, Nishan Ranabhat +1 more

This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics are covered, including tensor structures and network representations like Matrix Product States (MPS) and Tree Tensor Networks (TTN). These preliminaries provide readers with the core mathematical tools and concepts necessary for quantum physics and quantum computing applications, bridging the gap between multi-linear algebra and complex quantum systems. Part II explores practical applications of tensor networks in simulating quantum dynamics, with a particular focus on the efficiency they offer for systems of high computational complexity. Key topics include Hamiltonian dynamics, quantum annealing, open system dynamics, and optimization strategies using TN frameworks. A final chapter addresses the emerging role of “quantum magic” in tensor networks. It delves into non-stabilizer states and their contribution to quantum computational power beyond classical simulability, featuring methods such as stabilizer-enhanced MPS and the Clifford-dressed TDVP. In presenting tensor networks as tools for understanding quantum complexity, this book aims to foster a deeper collaboration between the many-body physics and quantum computing communities, inviting a broader audience to engage with these recent developments.

Dario Tamascelli, A. Mennella, Davide Ferracin

Aaron C. Hoyt, Jonathan S. Bersson, Sean Garner +2 more

Large-scale tensor network simulations are crucial for developing robust complexity-theoretic bounds on classical quantum simulation, enabling circuit cutting approaches, and optimizing circuit compilation, all of which aid efficient quantum computation on limited quantum resources. Modern exascale high-performance computing platforms offer significant potential for advancing tensor network quantum circuit simulation capabilities. We implement TN-Sim, a tensor network simulator backend within the NWQ-Sim software package that utilizes the Tensor Algebra for Many-body Methods (TAMM) framework to support both distributed HPC-scale computations and local simulations with ITensor. To optimize the scale up in computation across multiple nodes we implement a task based parallelization scheme to demonstrate parallelized gate contraction for wide quantum circuits with many gates per layer. Through the integration of the TAMM framework with Matrix Product State (MPS) tensor network approaches, we deliver a simulation environment that can scale from local systems to HPC clusters. We demonstrate an MPS tensor network simulator running on the state-of-the-art Perlmutter (NVIDIA) supercomputer and discuss the potential portability of this software to HPC clusters such as Frontier (AMD) and Aurora (Intel). We also discuss future improvements including support for different tensor network topologies and enhanced computational efficiency.

A. V. Berezutskii, Minzhao Liu, Atithi Acharya +25 more

Jincheng Zhang

The accurate simulation of complex systems, encompassing fields like materials science, quantum chemistry, and finance, often presents a formidable computational challenge. Traditional methods frequently struggle to scale effectively, particularly when dealing with high-dimensionality or intricate correlations. This paper introduces a novel simulation approach leveraging tensor networks (TN) in conjunction with quantum-inspired sampling techniques. The core idea is to approximate the dynamics of these complex systems by exploiting the inherent structure of tensor networks and employing sampling strategies inspired by quantum measurement processes. We demonstrate that this hybrid approach offers a pathway towards scalable simulations, achieving high accuracy while mitigating the computational bottlenecks associated with conventional methods. The key innovations lie in the adaptive construction of TNs based on the system's correlation structure and the utilization of quantum-inspired sampling methods to efficiently capture the relevant dynamics. This work presents a theoretical framework and outlines a methodology for implementing such simulations, paving the way for advancements in the modeling and understanding of complex systems.