Quantum Simulations Become Far More Efficient with New Polfed.jl Package

A new Julia package, Polfed.jl, implements the Polynomially Filtered Exact Diagonalization algorithm and enables the efficient calculation of mid-spectrum eigenpairs of quantum many-body Hamiltonians. Rok Pintar and colleagues from University of Ljubljana, Jagiellonian University, Jožef Stefan Institute, Institute of Theoretical Physics, University of Science and Technology, and Barcelona Supercomputing Centre developed Polfed.jl to address computational limitations in understanding complex quantum systems. The package circumvents the exponential growth of computational demands by preserving Hamiltonian sparsity and reducing memory costs, providing a key method for investigating non-equilibrium physics. Benchmarking against existing methods reveals Polfed.jl’s ability to access larger system sizes, alongside sharp speedups achieved through GPU acceleration, offering a flexible set of tools for exploring many-body phenomena and potentially applicable to large sparse matrices beyond quantum physics.

Expanded computational access to quantum many-body systems via efficient mid-spectrum eigenpair

A six-fold increase in the size of quantum systems accessible via diagonalization techniques is now achievable, routinely simulating spin chains and fermionic models containing up to 40 sites, a significant improvement over the previous limit of approximately 30. Previously, modelling these systems required prohibitive computational resources. This breakthrough circumvents the exponential growth of computational demands that restricted the study of larger, more complex quantum phenomena. The new Polfed.jl package, an open-source Julia implementation of the POLFED algorithm, efficiently calculates mid-spectrum eigenpairs, the energy levels and corresponding states of a quantum system, essential for understanding non-equilibrium physics and ergodicity-breaking transitions. Benchmarks reveal that this new approach outperforms alternatives when applied to disordered systems, allowing calculations on systems 50 percent larger than previously possible with comparable accuracy.

The computational challenge arises from the exponential scaling of the Hilbert space, the mathematical space encompassing all possible states of a quantum system, with the number of constituent particles or lattice sites. For a system of N spins-1/2 particles, the Hilbert space dimension is 2^N. Direct diagonalization of the Hamiltonian matrix, which describes the system’s energy, becomes intractable for even moderately sized systems. POLFED mitigates this by employing a polynomial spectral transformation, effectively reshaping the Hamiltonian’s eigenvalue spectrum without altering its essential physics. This transformation concentrates the spectral density in a region where efficient filtering can be applied, reducing the computational cost of finding the desired mid-spectrum eigenpairs. The package incorporates GPU acceleration, reducing the computational burden of repeated sparse matrix-vector multiplications, and CPU-GPU comparisons confirm sharp speedups in processing these complex calculations. Despite these advances, current system sizes still fall short of fully capturing the behaviour of macroscopic quantum materials, and scaling to hundreds of sites remains a considerable challenge. Polfed.jl also provides pre-built code for modelling the quantum sun model Hamiltonian, a valuable tool for investigating ergodicity-breaking transitions, a phenomenon where systems fail to explore all accessible states, allowing scientists to focus on the physics rather than the implementation details.

The POLFED algorithm itself builds upon the well-established Lanczos iteration, a method for finding the largest eigenvalues of a symmetric matrix. However, standard Lanczos suffers from spectral pollution, where unwanted eigenstates contaminate the calculated eigenpairs. POLFED addresses this by applying the polynomial filter, effectively projecting out these spurious states and isolating the desired mid-spectrum region. The choice of the polynomial filter is crucial for performance and accuracy, and Polfed.jl offers options for customising this parameter. The implementation leverages Julia’s strengths in high-performance computing, including its just-in-time compilation and support for parallel processing. The package is designed to be user-friendly, with a clear API and comprehensive documentation, facilitating its adoption by researchers in diverse fields. Furthermore, the open-source nature of Polfed.jl encourages community contributions and further development of the algorithm and its applications.

Eigenpair calculations enable simulation of complex quantum material properties

Systems where the bizarre rules of quantum mechanics govern behaviour are now being investigated with new computational tools. Polfed.jl expands the scale of these simulations, though its current validation focuses on specific models like disordered spin-chains and fermionic systems. Its potential extends far beyond these initial tests, offering a pathway to investigate more realistic materials and phenomena. Accessing mid-spectrum eigenpairs is important for understanding complex quantum behaviours, and this software tackles a major computational hurdle in that field.

The package efficiently calculates these energy levels and corresponding states by employing a technique called polynomial spectral filtering within a Lanczos iteration, which are important for understanding a system’s properties. This advancement enables simulations of larger, more complex models than previously possible, overcoming limitations imposed by the exponential growth of computational demands. The ability to accurately determine these eigenpairs is crucial for studying dynamical properties, such as response functions and transport coefficients, which govern how quantum materials interact with external stimuli. For instance, understanding the mid-spectrum behaviour is vital for characterising many-body localisation, a phenomenon where quantum systems become trapped in insulating states due to strong disorder. This has implications for designing novel quantum materials with tailored electronic and magnetic properties.

Its broader utility is suggested by the ability to apply it to any large, sparse matrix, potentially aiding research in diverse areas beyond quantum physics itself. Sparse matrices, those containing predominantly zero elements, arise frequently in fields like structural mechanics, data analysis, and image processing. The efficient sparse matrix-vector multiplication routines within Polfed.jl, coupled with the polynomial filtering technique, could be adapted to accelerate computations in these domains. The package’s modular design and well-defined interface facilitate such extensions. Future work may focus on further optimising the algorithm for specific hardware architectures, such as exascale computers, and extending its applicability to more complex quantum models, including those with long-range interactions and topological order. The availability of Polfed.jl represents a significant step towards bridging the gap between theoretical models and experimental observations in the realm of quantum many-body physics.

The researchers developed Polfed.jl, an open-source Julia package that efficiently calculates mid-spectrum eigenvalues and eigenvectors of quantum many-body Hamiltonians. This is important because determining these energy levels is computationally demanding, limiting the size and complexity of models that can be simulated. Polfed.jl overcomes this limitation through polynomial spectral filtering and GPU acceleration, enabling access to larger system sizes than alternative approaches. The authors intend to further optimise the algorithm for advanced hardware and more complex quantum models.