QCombo: Automated Commutator Calculation for Quantum Many-Body Operators

QCombo is a Python library for automated computation of commutators for normal-ordered many-body operators based on the generalized Wick theorem. Designed specifically for nuclear physics and condensed matter applications, it provides efficient and accurate tools for handling complex quantum many-body calculations.

Table of Contents

• Overview
• Features
• Installation
• Quick Start
• Command Line Interface
• Python API Usage
• Citation
• License
• Support

Overview

Background and Motivation

In quantum mechanics, operator commutators form the foundation of theoretical calculations. In nuclear physics, ab initio methods (such as NCSM, CC, IMSRG, etc.) are essential tools for studying nuclear many-body systems. These methods start from nuclear forces constructed from chiral effective field theory and provide approximate but rigorous solutions to quantum many-body problems.

As the number of many-body terms increases, the number of terms in commutators grows exponentially, making manual calculations extremely tedious and error-prone. QCombo addresses this challenge by providing an automated solution for computing commutators of normal-ordered many-body operators.

Features

• Based on Generalized Wick Theorem: Implements automated computation of commutators for normal-ordered many-body operators
• Many-Body Coupling Support: Handles commutators of arbitrary-order many-body operators
• Automated Output Generation:
  • Generates LaTeX files for commutator expressions
  • Produces input files for the AMC software package to obtain expressions in J-scheme
• Flexible Configuration: Allows specification of output to specific many-body ranks
• Symbolic Computation: Built on SymPy for precise algebraic manipulations
• Command Line Interface: Easy-to-use terminal commands for quick calculations
• Interactive Mode: Step-by-step guided interface for exploratory calculations

Installation

Install from PyPI (Recommended)

pip install qcombo

Install from Source

git clone https://github.com/chenlh73/qcombo.git
cd qcombo
pip install -e .

System Requirements

• Python >= 3.12
• SymPy >= 1.13.3

Quick Start

Command Line Interface (Recommended)

After installation, you can use QCombo directly from the command line:

# Compute the [2B, 2B] commutator with all possible contractions
qcombo 2 2

# Compute only 0-body and 1-body contractions
qcombo 2 2 -c 0,1

# Compute with range of contractions (0-2 body)
qcombo 1 3 -c 0-2

# Specify custom output filenames
qcombo 2 2 -c 0,1,2 --latex-output my_result.tex --amc-output my_result.txt

# Use specific indices (Python list syntax)
qcombo "[['a','b'], ['c','d']]" "[['e','f','g'], ['h','i','j']]" -c 0,1,2

# Start interactive mode
qcombo
# or explicitly:
qcombo -i

# Show help message
qcombo --help

# Show version information
qcombo --version

Python API

You can also use QCombo within Python scripts:

import qcombo

# Compute the [2B, 2B] commutator coupled to all possible many-body ranks
qcombo.easyCombo(2, 2)

# Output only terms coupled to 0-body and 1-body ranks
qcombo.easyCombo(2, 2, [0, 1])

Command Line Interface

Basic Usage

QCombo provides a powerful command-line interface that allows you to perform calculations directly from your terminal:

qcombo LEFT RIGHT [OPTIONS]

Where:

• LEFT: Number of particles in left operator (integer) or index list (Python list string)
• RIGHT: Number of particles in right operator (integer) or index list (Python list string)

Command Line Options

Option	Short	Description	Example
--contraction	-c	Contraction body numbers: single integer(0), range(0-2), list(0,1,2), or 'all' (calculate all)	qcombo 2 2 -c 0,1,2
--latex-output	-lo	LaTeX output filename	qcombo 2 2 -lo result.tex
--amc-output	-ao	AMC program input file output name	qcombo 2 2 -ao result.txt
--version	-v	Show version information	qcombo --version
--quiet	-q	Quiet mode, reduce output	qcombo 2 2 -q
--interactive	-i	Start interactive mode	qcombo -i

Interactive Mode

For exploratory calculations, QCombo offers an interactive mode that guides you through the process step-by-step:

qcombo

Or explicitly:

qcombo -i

In interactive mode, you will be prompted for:

1. Left operator (integer or index list)
2. Right operator (integer or index list)
3. Contraction body numbers
4. Output filenames (optional)

Examples

Basic calculations:

# 2-body with 2-body operator commutator
qcombo 2 2

# 1-body with 3-body operator commutator
qcombo 1 3

# 2-body with 2-body, only 0-body and 1-body terms
qcombo 2 2 -c 0,1

With custom indices:

# Using specific index lists
qcombo "[['a','b'], ['c','d']]" "[['e','f'], ['g','h']]"

# Using range contractions
qcombo "[['i','j'], ['k','l']]" "[['m','n','o'], ['p','q','r']]" -c 0-2

With output files:

# Save results to specific files
qcombo 2 2 --latex-output commutator.tex --amc-output commutator.txt

# Using short options
qcombo 2 2 -c 0,1,2 -lo result.tex -ao result.txt

Python API Usage

Function Parameters

qcombo.easyCombo(left, right, contraction=None, latexOutput=None, amcOutput=None)

• left: Many-body rank of the first operator (integer) or index list
• right: Many-body rank of the second operator (integer) or index list
• contraction: Optional parameter specifying output ranks
  • Example: [0, 1] outputs only zero-body and one-body terms
  • None (default) outputs all possible many-body terms
• latexOutput: Custom LaTeX output filename
• amcOutput: Custom AMC input filename

Basic Examples

import qcombo

# Compute [2B, 2B] commutator
result = qcombo.easyCombo(2, 2)

# Compute with specific contractions
result = qcombo.easyCombo(2, 2, contraction=[0, 1])

# Compute with custom output files
result = qcombo.easyCombo(2, 2, 
                          contraction=[0, 1, 2],
                          latexOutput="my_commutator.tex",
                          amcOutput="my_commutator.txt")

# Compute with specific indices
left_indices = [['a', 'b'], ['c', 'd']]
right_indices = [['e', 'f', 'g'], ['h', 'i', 'j']]
result = qcombo.easyCombo(left_indices, right_indices, contraction=[0, 1])

Output Description

After execution, QCombo generates:

1. Console Output: Displays the computed commutator expressions in formatted text
2. LaTeX Files: Generated expressions in LaTeX format (ready for inclusion in papers)
3. AMC Input Files: Input files for the AMC software package to obtain expressions in J-scheme

The output filenames follow the pattern:

• LaTeX: commutator_[left]B[right]B_to_[contractions]B.tex
• AMC: commutator_[left]B[right]B_to_[contractions]B_amcInput.txt

For example: commutator_2B2B_to_0_1_2B.tex

Output Format

The returned result is a dictionary where keys are strings in the format '{filter_body}B_lambda{lambda_body}B' and values are SymPy expressions:

{
    '0B_lambda1B': sympy_expression_1,
    '0B_lambda2B': sympy_expression_2,
    # ...
}

Applications

Nuclear Physics Applications

In nuclear ab initio methods, QCombo can be used for:

1. IMSRG Flow Equations: Computing commutators between generators and Hamiltonians
2. Similarity Renormalization Group (SRG): Evaluating flow equations
3. Configuration Interaction: Deriving effective interactions

Education and Research

QCombo is also valuable for quantum mechanics education and research:

• Verifying theoretical derivations
• Exploring contributions from higher-order many-body terms
• Automating tedious symbolic calculations
• Teaching quantum many-body theory concepts

Citation

If you use QCombo in your research, please cite:

@software{qcombo2024,
  title = {QCombo: Automated Commutator Calculation for Quantum Many-Body Operators},
  author = {Chen, L.H. and Li, Y. and Hergert, Heiko and Yao, J.M.},
  year = {2024},
  url = {https://github.com/chenlh73/qcombo},
  note = {Based on generalized Wick theorem for automatic commutator calculation of many-body operators}
}

License

This project is licensed under the MIT License - see the LICENSE file for details.

Support

• Issues: Report bugs or request features on GitHub Issues: https://github.com/chenlh73/qcombo/issues
• Documentation: Visit the GitHub repository for more information: https://github.com/chenlh73/qcombo
• Questions: Contact the development team via email

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*QCombo - Quantum Commutator of Many-Body operator *