Numerical techniques

This guide provides a comprehensive look at how to access and select elements, subregions, and patterns within arrays and

At a Glance

This guide introduces the core containers provided by the Linear Algebra module, how to inspect them, and the day-to-day

Linear Algebra is the fundamental package for numerical computing. At the core of the Linear Algebra package are the array and matrix objects. These encapsulate n-dimensional arrays of homogeneous data types, with many operations being performed in optimized compiled C++ code for performance. While Aleph is already documented at its core, Linear Algebra serves as the specialized linear algebra and array computation module, much like how NumPy extends Python's capabilities for scientific computing.

Introduction

The eigenspectrum of a Hamiltonian operator encodes fundamental properties of a quantum system. While the ground state and low-lying excitations are often the primary focus – governing low-temperature physics and phase transitions – many problems require access to the interior of the spectrum. For instance, studying thermalization, many-body localization, or excited state quantum phase transitions often necessitates computing eigenstates in the middle of the spectrum.

This section covers numerical methods for simulating quantum systems, including exact diagonalization, time evolution, and tensor network techniques. We provide code examples that can be run in the Workshop to help you understand how to implement these methods and apply them to various quantum models.

Evolving quantum states is a common routine in numerical methods. There are many ways to do this, the most common and accessible being full exact diagonalization and evolving an initial state using knowledge of the full spectrum of a Hamiltonian $H$,

Array vs Matrix: Understanding the Difference