AdaptMaxEnt

Adaptive Max-Ent Restricted Evolutions for Local Hamiltonians

This repository contains an implementation of Adaptive Max-Ent Restricted Evolutions for local Hamiltonians.

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Table of Contents

1. Introduction
2. Features
3. Built With
4. Installation
5. Usage
6. Examples
7. Contributing
8. License

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Introduction

Efficiently simulating quantum many-body systems is a fundamental challenge in quantum physics.
This project implements a recipe for adaptive Max-Ent dynamics, which shares conceptual similarities with Krylov techniques and the time-dependent variational principle. This approach closely follows the exact Schrödinger dynamics while avoiding exploration of the full observable space, ${\cal A}$, providing a numerically exact and stable method.

The restricted Max-Ent framework offers a tractable way to simulate dynamics on a smaller subspace via a (weighted-) linear projection $\pi_{B,\sigma}$ from the full observable space ${\cal A}$ onto a smaller, more manageable subspace ${\cal A}_B$, spanned by a reduced basis. Key objects of interest in this framework include:

1. Low-Dimensional Basis of Observables $B_{\ell}$
   Typically, the basis is constructed to include $\ell$ iterated commutators of the Hamiltonian and a relevant seed operator, such as the initial state $\mathbf{K}_0 = \log(\rho(0))$.

2. Scalar Product $(\cdot, \cdot)_{\sigma}^{(s)}$
   The scalar product can take the form of:

   • Kubo-Mori-Bogoliubov (KMB): Accurate but computationally expensive.
   • Covariance Scalar Product: A robust alternative that tightly bounds the KMB product. This is the one which we will use in these examples.

3. Mean-Field Projections Simplify states $\mathbf{K}$ involving costly $n$-body correlations by projecting them into their $m_0$-body components, with $m_0 \ll n$, yielding more tractable states $\mathbf{K}^{\rm (MF)}$.

Despite its advantages, restricted Max-Ent has limitations, including high computational cost and potential errors if the fixed basis $B$ fails to account for all statistically relevant observables.

Adaptive Max-Ent

To overcome these challenges, this project introduces adaptive Max-Ent dynamics, an autonomous method that updates the geometry (basis and scalar product) and applies mean-field projections during discrete simulation steps. This is particularly effective for geometrically local Hamiltonians, as their dynamics are constrained by Lieb-Robinson bounds, which restricted Max-Ent respects by design.

*Figure 1: Illustration of the Lieb-Robinson bounds, an integral idea of Adaptive Max-Ent simulations for geometrically local Hamiltonians.*

The method adapts by monitoring the error measured via a figure of merit, the $p$-th partial sum ${\cal K}_p$ and updating when it exceeds a given tolerance $\epsilon$. This results in a reliable and efficient framework characterized by several parameters $\ell$, $\epsilon$, $m_0$.

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Features

• Restricted Max-Ent Formalism: Restricted evolution using Max-Ent projections and Krylov-like expansions, constrained by Lieb-Robinson bounds.
• Adaptive Max-Ent Formalism:
  • Dynamic Basis and Scalar Product Updates: Automatically update to incorporate the most relevant observables.
  • Mean-Field Projections: Simplify objects for computational efficiency.
  • Customizable Update Criteria: Explore different criteria based on Max-Ent bounds or Lieb-Robinson bounds.
• Examples and Tutorials: Demonstrates the method’s applicability to various quantum many-body dynamics.

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Built With

• Python: Core programming language.
• ALPSQuTip: A library intended to bridge the gap between ALPS condensed matter models and QuTiP calculations.
• ALPS: A collection of libraries for solving models in condensed matter physics, focusing on quantum many-body systems.
• QuTiP: For quantum computing simulations.
• NumPy: Efficient numerical operations.
• Matplotlib: Data visualization.
• SciPy: Advanced scientific computations.

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Installation

To set up the environment for this project:

1. Clone the repository:

git clone https://github.com/your-username/max-ent-evolutions.git
cd max-ent-evolutions

2. Install Dependencies

Use pip to install the required Python libraries:

pip install -r requirements.txt

3. Set up a Virtual Environment (Optional)

To avoid conflicts with your system Python installation, create and activate a virtual environment:

python -m venv venv
source venv/bin/activate  # On Windows: venv\Scripts\activate
pip install -r requirements.txt

Usage

In this documentation, we will be interested in a $L=7$ one-dimensional spin-$1/2$ Heisenberg XYZ system, with Hamiltonian

$$ {\bf H} = -\frac{1}{2}\sum_{j=1}^{L} \bigg( J_x \sigma_{j}^{x} \sigma_{j+1}^{x} +J_y \sigma_{j}^{y} \sigma_{j+1}^{y} +J_z \sigma_{j}^{z} \sigma_{j+1}^{z} \bigg), $$

where $J_x, J_y, J_z$ are the system's parameters, $\sigma_j^{x,y,z}$ are the standard Pauli operators with support on the $j$-th site of the lattice. This Hamiltonian, in the tutorials, can be constructed in a simple way by employing the ALPS library and a dictionary of parameters.

Examples

In Tutorials, a series of tutorials can be found, showcasing different examples and behaviours of interest.