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An HPC Approach to Accelerate Tensor Computations in Quantum Physics

Markus Hellgren

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Quantum systems grow in complexity so rapidly that even modest models become difficult to simulate, creating a strong need for methods that can handle high-dimensional data. In this work, we investigate a Jacobi-type tensor algorithm in this context and develop a CUDA implementation that supports tensors of arbitrary order on a single GPU. We test the implementation on NVIDIA H100 hardware and show that the algorithm converges correctly for diagonalizable tensors up to nine dimensions, with runtime increasing in a predictable way as tensor order grows.We also apply the method to Heisenberg spin chains and show that it can operate on tensors corresponding to systems of up to N=15 spins, that is, tensors with up to 30 dimensions, which correspond to Hamiltonian matrices of dimension 2N by 2N (for N=15, this is 32768 by 32768). In addition, we demonstrate that the implementation can be used to approximate eigenvalues of the corresponding Hamiltonian matrices. These results confirm that the method scales to high-dimensional problems and is applicable to spectral analysis in quantum physics settings.

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Quantum systems grow in complexity so rapidly that even modest models become difficult to simulate, creating a strong need for methods that can handle high-dimensional data. In this work, we investigate a Jacobi-type tensor algorithm in this context and develop a CUDA implementation that supports tensors of arbitrary order on a single GPU. We test the implementation on NVIDIA H100 hardware and show that the algorithm converges correctly for diagonalizable tensors up to nine dimensions, with runtime increasing in a predictable way as tensor order grows.We also apply the method to Heisenberg spin chains and show that it can operate on tensors corresponding to systems of up to N=15 spins, that is, tensors with up to 30 dimensions, which correspond to Hamiltonian matrices of dimension 2N by 2N (for N=15, this is 32768 by 32768). In addition, we demonstrate that the implementation can be used to approximate eigenvalues of the corresponding Hamiltonian matrices. These results confirm that the method scales to high-dimensional problems and is applicable to spectral analysis in quantum physics settings.
Quantum systems grow in complexity so rapidly that even modest models become difficult to simulate, creating a strong need for methods that can handle high-dimensional data, also known as tensors. In this work, we investigate a novel Jacobi-type tensor algorithm for tensor decomposition and develop a CUDA-based algorithm that supports tensors of arbitrary order on a single GPU. We test the implementation on NVIDIA H100 GPUs and show that the algorithm converges correctly for diagonalizable tensors up to nine dimensions, with runtime scaling in a predictable way as tensor order grows. Finally, our general algorithm outperforms the original MATLAB reference by more than two orders of magnitude.
High Resolution Image Download MS PowerPoint Slide The interplay of quantum and classical simulation and the delicate divide between them is in the focus of massively parallelized tensor network state (TNS) algorithms designed for high performance computing (HPC). In this contribution, we present novel algorithmic solutions together with implementation details to extend current limits of TNS algorithms on HPC infrastructure building on state-of-the-art hardware and software technologies. Benchmark results obtained via large-scale density matrix renormalization group (DMRG) simulations on single node multiGPU NVIDIA A100 system are presented for selected strongly correlated molecular systems addressing problems on Hilbert space dimensions up to 4.17 × 10 35 .
We present the NVIDIA cuQuantum SDK [1], a state-of-the-art library of composable primitives for GPU-accelerated quantum circuit simulations. As the size of quantum devices continues to increase, making their classical simulation progressively more difficult, the availability of fast and scalable quantum circuit simulators becomes vital for quantum algorithm developers, as well as quantum hardware engineers focused on the validation and optimization of quantum devices. The cuQuantum SDK was created to accelerate and scale up quantum circuit simulators developed by the quantum information science community by enabling them to utilize efficient scalable software building blocks optimized for NVIDIA GPU-based platforms. The functional building blocks provided cover the needs of both state vector- and tensor network- based simulators, including approximate tensor network simulation methods based on matrix product state, projected entangled pair state, and other factorized tensor representations. By leveraging the enormous computing power of the latest NVIDIA GPU architectures, quantum circuit simulators that have adopted the cuQuantum SDK demonstrate significant acceleration, compared to CPU-only execution, for both the state vector and tensor network simulation methods. Furthermore, by utilizing the parallel primitives available in the cuQuantum SDK, one can easily transition to distributed GPU-accelerated platforms, including those furnished by cloud service providers and high-performance computing systems deployed by supercomputing centers, extending the scale of possible quantum circuit simulations. The rich capabilities provided by the cuQuantum SDK are conveniently made available via both Python and C application programming interfaces, where the former is directly targeting a broad Python quantum community and the latter allows tight integration with simulators written in any programming language.
Abstract We present an efficient tensor-network-based approach for simulating large-scale quantum circuits exemplified by quantum support vector machines (QSVMs). Experimentally, leveraging the cuTensorNet library on multiple GPUs, our method effectively reduces the exponential runtime growth to near-quadratic scaling with respect to the number of qubits in practical scenarios. Traditional state-vector simulations become computationally infeasible beyond approximately 50 qubits; in contrast, our simulator successfully handles QSVMs with up to 784 qubits, executing simulations within seconds on a single high-performance GPU. Furthermore, utilizing the message passing interface for multi-GPU environments, our method demonstrates strong linear scalability, effectively decreasing computation time as dataset sizes increase. We validate our framework using the MNIST and Fashion MNIST datasets, achieving successful multiclass classification and highlighting the potential of QSVMs for high-dimensional data analysis. By integrating tensor-network techniques with advanced high-performance computing resources, this work demonstrates both the feasibility and scalability of simulating large-qubit quantum machine learning models, providing a valuable validation tool within the emerging Quantum-HPC ecosystem.
Abstract Large-scale classical simulation of quantum computers is crucial for benchmarking quantum algorithms, establishing boundaries of quantum advantage and exploring heuristic quantum algorithms. We present a full-state vector simulation algorithm and software implementation designed to perform HPC simulation of layers of rotations around a string of Pauli operators. We demonstrate robust scalability of the simulation method on large distributed CPU and GPU systems. Our distributed computation harnessed up to 16384 CPU cores and 512 NVIDIA H100 GPUs, using 32 TB of memory. The simulator significantly outperforms other high-performance libraries, showing a typical speedup of 10-100 for large-scale multi-GPU workloads. As a first application of our approach, we report a numerical experiment aimed at simulating exactly Hamiltonian dynamics of up to 40 qubits to investigate the Trotter error for a quantum chemistry problem. Bounding the Trotter error is important for evaluating the cost of quantum algorithms for chemistry, and rigorous bounds are often conservative, as our simulations confirm. Our software, specifically designed for quantum time evolution applications, is also well equipped to manage circuits that utilize standard gate sets.
Due to the increasing computing power of the graphics processing units (GPU), they are becoming more and more popular when solving general purpose algorithms. As the simulation of quantum computers results on a problem with exponential complexity, it is advisable to perform a parallel computation, such as the one provided by the SIMD multiprocessors present in recent GPUs. In this paper, we focus on an important quantum algorithm, the quantum Fourier transform (QTF), in order to evaluate different parallelization strategies on a novel GPU architecture. Our implementation makes use of the new CUDA software/hardware architecture developed recently by NVIDIA.