Pymablock: an algorithm and a package for quasi-degenerate perturbation theory

TL;DR

Pymablock introduces an efficient algorithm and Python package for quasi-degenerate perturbation theory, enabling scalable, multi-subspace Hamiltonian reductions with broad applicability in quantum systems.

Contribution

The paper presents a novel algorithm and a Python package that improve the scalability and flexibility of quasi-degenerate perturbation theory calculations.

Findings

Supports calculations of any order and multiple subspaces

Demonstrates effectiveness on quantum models and tight-binding systems

Shows improved performance over existing methods

Abstract

A common technique in the study of complex quantum-mechanical systems is to reduce the number of degrees of freedom in the Hamiltonian by using quasi-degenerate perturbation theory. While the Schrieffer--Wolff transformation achieves this and constructs an effective Hamiltonian, its scaling is suboptimal, it is limited to two subspaces, and implementing it efficiently is both challenging and error-prone. We introduce an algorithm for constructing an equivalent effective Hamiltonian as well as a Python package, Pymablock, that implements it. Our algorithm combines an optimal asymptotic scaling and the ability to handle any number of subspaces with a range of other improvements. The package supports numerical and analytical calculations of any order and it is designed to be interoperable with any other packages for specifying the Hamiltonian. We demonstrate how the package handles constructing a k.p model, analyses a superconducting qubit, and computes the low-energy spectrum of a large tight-binding model. We also compare its performance with reference calculations and demonstrate its efficiency.