Numerical study has proven to be one of the core tools for exploring the emergent properties of many-body quantum systems. A wide array of numerical techniques have been developed for the analysis of quantum many-body physics, some of which rely on the specific physics in question (for example, matrix product state methods whose accuracy relies on the relevant physics having low entanglement) and some of which can treat more general cases.

Dynamite

Dynamite is a numerical package that uses massively parallel Krylov subspace algorithms for quantum dynamics and eigensolving. It has a straightforward Python interface for user-specified Hamiltonians and states. The backend is highly optimized for modern supercomputing environments, including distributed memory parallelism with MPI and hand-tuned computational kernels for GPUs. Other features include symmetry subspaces such as charge conservation, and “matrix-free” methods to drastically reduce memory usage. For the documentation, see https://dynamite.readthedocs.io/. 

Density Matrix Truncation

One method [https://github.com/Jack-Kemp/dmt] the group uses for calculating long-time quantum dynamics is density matrix truncation or DMT, originally proposed by [1]. Unlike the exact Krylov methods discussed above, this algorithm is approximate, but the truncation method is chosen to preserve local observables, making it particularly well-suited for calculating hyrdrodynamic properties such as transport coefficients. The group has used DMT in various setting from Floquet heating [2] to universal Kardar-Parisi-Zhang dynamics in integrable spin chains [3].

Neural Quantum States

In the last few years AI has brought significant advances to multiple branches of science and technology. Some of those AI methods (such as neural networks) have been recently used to variationally study classical simulation of quantum many body systems. Concrete tasks include solving for the system’s ground state or simulating its time dynamics. In our group we are trying to utilize recent advances from the ML community to perform accurate, interpretable and large-scale classical simulations of challenging quantum many-body physics problems.

LOBPCG

For systems that exhibit interesting physical behavior in excited states, such as the phenomenon of many-body localization, interrogating novel physics requires solving for eigenvalues and eigenvectors in the middle of a Hamiltonian’s spectrum. This is generally challenging for large-scale iterative numerical methods, because the transformations required use large amounts of computer memory, ultimately limiting the system sizes that can be studied. We have approached the eigenproblem with a different spectral transform for which the memory usage is dramatically decreased, while still achieving good convergence by using a cutting-edge eigensolver called LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient) [4, 5].

References

1. White, C.D., Zaletel, M., Mong R.S.K., et al. Quantum dynamics of thermalizing systems. Phys. Rev. B 97, 035127 (2018). https://doi.org/10.1103/PhysRevB.97.035127
2. Ye, B., Machado, F., White, C.D., et al. Emergent Hydrodynamics in Nonequilibrium Quantum Systems. Phys. Rev. Lett. 125, 030601 (2020). https://doi.org/10.1103/PhysRevLett.125.030601
3. Ye, B., Machado, F., Kemp, J., et al. Universal Kardar-Parisi-Zhang Dynamics in Integrable Quantum Systems. Phys. Rev. Lett. 129, 230602 (2022). https://doi.org/10.1103/PhysRevLett.129.230602
4. Van Beeumen, R., Kahanamoku-Meyer, G.D., Yao, N.Y., and Yang, C. A Scalable Matrix-Free Iterative Eigensolver for Studying Many-Body Localization. In Proceedings of the International Conference on High Performance Computing in Asia-Pacific Region (HPCAsia2020). Association for Computing Machinery, New York, NY, USA, 179–187 (2020). https://doi.org/10.1145/3368474.3368497
5. Van Beeumen, R., Ibrahim, K.Z., Kahanamoku–Meyer, G.D., Yao, N.Y., Yang, C. Enhancing scalability of a matrix-free eigensolver for studying many-body localization. The International Journal of High Performance Computing Applications 36(3):307-319 (2022). doi:10.1177/10943420211060365