This is a set of Mathematica packages built to compute the cohomology of the chain complexes introduced in the paper 'Homological Tools for the Quantum Mechanic' (arXiv:1901.02011).
Don't have Mathematica? See Wolfram Engine as a Mathematica Alternative.
For software that can compute ranks of cohomology components in Octave/Matlab, see: https://github.com/tmainiero/homological-tools-4QM-octave.
Unlike the Octave software, these packages are written in a functional style, and also allow one to output explicit representatives of generators of cohomology components.
An interactive quick start guide/minimal documentation is included in quick_start.nb.
Most functions are documented. (If we are interested in a function functionOfInterest, a description can be found by evaluating ??functionOfInterest.)
The package BasicStable.wl provides useful quantum mechanical operations and can be used independently of the remainder of the packages.
The repository (https://github.com/tmainiero/quantum-info) contains only this package.
git clone https://github.com/tmainiero/homological-tools-4QM-mathematica.git
Click that fancy green "<> Code" button on the top right! Then "Download ZIP".
- Go to the file you want to download and click it to view the contents
- Locate the "Raw" button (On the top right at the time of writing) and right click.
- Save as...
Make sure that all files are located in Mathematica's $Path (just type $Path in Mathematica to see what this is).
Unfortunately, this software was written before the GNS and commutant cochain complexes were fully understood. The techniques used to compute cohomology, present in CechOpsStable, are meant to compute Cech homology of a co-presheaves, rather than Cech cohomology of presheaves---the latter being the technique used in the paper. One can, however, recover the appropriate cohomology after some degree shifting (and sign corrections) in the form of some wrapper functions present in StateHomologyStable. A solution to this confusion would be a rewrite of the functions in the intermediate package CechOpsStable; this might be planned for future versions of this software.
If you do not have Mathematica, there is "free" (as in free beer) alternative: the Wolfram Engine, which can be run as a Jupyter kernel. See Wolfram research's official installation instructions here. This YouTube video (circa Feb 2023) demonstrates some capabilities of this kind of setup, and this accompanying blog post for a Window user's setup.
As far as I know, the Wolfram Engine has the same full computational capabilities as in Mathematica, the only drawback of running it in a Jupyter notebook seems to be the loss of some graphical capabilities of Mathematica notebook front end. For the purposes of interacting with the code here, nothing serious is lost.
Mathematica has a severe drawback: it is not free or open-source software. Among other things, this makes it practically inaccessible to many. (See the section above for an excellent workaround.) Yet, the choice was made to work with it for this codebase was for two reasons:
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Its tight integration of symbolic and numerical capabilities in a streamlined interface: a somewhat key feature that allows one to explore the kind of ideas in (arXiv:1901.02011) with ease. Alternatives such as SageMath just do not have this tight integration, or have implementation drawbacks when constructing software that simultaneously requires both numerical and symbolic capabilities.
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Mathematica is used widely by high-energy physicists, I'm a high-energy physicist by training and pitched the ideas of (arXiv:1901.02011) to high energy physicists first.
In the future this software may be ported to alternatives for pragmatic reasons.