Quantum many-body systems

Maximally mixed state

This section explores specific condensed matter systems and their properties. We cover free fermions, the Su-Schrieffer-Heeger (SSH) model, single-electron transistors, and operator dynamics in quantum systems. Each tutorial includes code examples that can be run in the Workshop to reproduce results and gain hands-on experience with these models.

- The Fbit class represents a fermionic product state in the occupation number representation $|f\rangle = |n0\rangle \otimes \dots \otimes |n\rangle$ of $L$ sites and $n_{i} \in \{ 0,1\}$.

- Operator transformations manipulate quantum expressions symbolically.

This section introduces the basics of quantum many-body physics, including lattice models, spin models, and ground state calculations.

Area Laws

At a Glance

This sub-section introduces key concepts relating to quantum many-body physics. These topics will introduce foundational concepts used to implement high-level algorithms in aleph.

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.

In this tutorial we use the quantum toolkit, and specifically the operator framework to efficiently validate dual unitary chaos. For further reading you can read the general analytical setup and proof here. We are going to use the numerical setup here. This problem is interesting for a few reasons. First it showcases the power of the operator framework. We will use features that can build quantum circuits as well as construct a custom kernel for a global, diagonal operator that is not directly supported in the quantum toolkit. Second it is physically interesting because it features a space-time swap. We will be computing things on the time lattice and computing thermodynamic limit properties of our system in real space.

Overview

The tensors will eventually be replaced by the linear algebra module's array.

This section focuses on the time evolution of quantum systems, including both unitary and non-unitary dynamics. We explore how to simulate the time evolution of quantum states and operators, and how to analyze the resulting dynamics. Each tutorial includes code examples that can be run in the Workshop to reproduce results and gain hands-on experience with these concepts.