Efficient studies of large-scale topological and unconventional superconductors
Abstract
Topological superconductors is a class of materials with features uniquely advantageous for quantum computing due to being able to host Majorana fermions at surfaces, vortices, and other defects. Approximately one can say that a Majorana fermion is half an electron, or more accurately, in a system with Majorana fermions the wave function of an electron has delocalized into two separate parts. This non-local property can be used for exceptionally fault-tolerant quantum computing. Topological superconductors avoid the extreme sensitivity of standard quantum computing platforms by using the non-local nature of the Majorana fermions. The long-term goal of this project is to theoretically investigate the currently most promising topological superconductors as well as discover new topological superconductors.
We have many years of experience studying topological superconductors using a microscopic lattice tight-binding Bogoliubov-de Gennes (BdG) formalism. Thanks to medium SNIC/NAISS grants, we have successfully investigated a wide range of systems (see webpage for a full publication list) using GPU resources. The method traditionally involves diagonalizing large matrices and since we often need a self-consistent solution for superconductivity, this requires finding all eigenvalues. To avoid costly and poorly scalable calculations we have developed our own code, called TBTK (SoftwareX 9, 205 (2019)) that efficiently treat these types of systems using a Chebychev polynomial expansion of the Green’s functions. The method takes advantage of recursion relationships and can work with just matrix-vector multiplications giving a (A+BN)*N scaling, with A and B being constants and N the number of degrees of freedom (lattice sites, spin, orbitals, etc), instead of the brute force N^6 scaling of exact diagonalization. Due to the calculations to a large part consisting of repeated matrix-vector multiplications it is optimally suited for GPUs. For problems where exact diagonalization is still needed, we have developed Python scripts that use CuPy and JAX for efficient array computations on the GPU. The scripts use CuPy to diagonalize large NxN matrices because of its efficient memory management and employs JAX to compute physical observables by batch processing all the eigenvectors and eigenvalues through vectorized operations. Furthermore, the iterative steps in solving the self-consistent equations for superconductivity are optimized by JAX’s Just-In-Time (JIT) compilation.
We have the last few years also used quasiclassical theory to be able to study even larger, mesoscopic, topological superconducting systems. We are here using SuperConga (Appl. Phys. Rev. 10, 011317 (2023)), an open-source framework for simulating physics of mesoscopic superconductors using the quasiclassical theory of superconductivity. This results in a complex ray-tracing problem, which due to it being a quantum field theory, consists of multiple source points, very high degrees of freedoms, and non-linear differential equations. We also here require a self-consistent solution, which results in an optimization problem solved using an involved adaptive scheme.
By running our software on high-end GPUs, we are continuously pushing the boundaries of scientific research in our field.
