Numerical Methods
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.
When translating a numerical method into Aleph code, use Kai to reason through the algorithm, validate your implementation, or debug unexpected results.
Matrix times vector: dense vs. sparse vs. symbolic
Introduction
Solving the 1D XY Model with DMRG
Overview
Mid-spectrum calculations with Chebyshev filters
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.
Calculating the trace with Haar random pure states
Maximally mixed state