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_transform your quantum weight enumerators_
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[](https://github.com/Mc-Zen/qsalto/blob/main/LICENSE)
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---
**qsalto** provides transformations between several (normalized) quantum weight enumerators, including
- Shor-Laflamme enumerators[^1] $a$, $b$,
- Rains unitary enumerators[^2] $a'$, $b'$,
- and Rains shadow enumerators[^3] $a''$.
We provide both
- a python package, available on PyPI: https://pypi.org/project/qsalto,
- a web viewer for visualizing the transformation matrices at https://mc-zen.github.io/qsalto.
<img src="https://github.com/Mc-Zen/qsalto/raw/main/docs/media/transformation-diagram.svg" width="420">
## Python package
The python package `qsalto` can be installed via `pip install qsalto` and features functions to generate nine classes of transformation matrices.
|Matrix | Function | Transforms from ... | ... to |is self-inverse|
|-----------|----------|-------------------------------|-------------------------------|---------------|
|$M$ |`M(n)` |$\mathbf{a}$ |$\mathbf{b}$ |✅ |
|$M'$ |`M1(n)` |$\mathbf{a'}$ |$\mathbf{b'}$ |✅ |
|$M''$ |`M2(n)` |$\mathbf{a''}$ |$\mathbf{b''}$ |✅ |
|$T'$ |`T1(n)` |$\mathbf{a}$, $\mathbf{b}$ |$\mathbf{a'}$, $\mathbf{b'}$ |❌ |
|$T'^{-1}$ |`iT1(n)` |$\mathbf{a'}$, $\mathbf{b'}$ |$\mathbf{a}$, $\mathbf{b}$ |❌ |
|$T''$ |`T2(n)` |$\mathbf{a}$, $\mathbf{b}$ |$\mathbf{a''}$, $\mathbf{b''}$ |❌ |
|$T''^{-1}$ |`iT2(n)` |$\mathbf{a''}$, $\mathbf{b''}$ |$\mathbf{a}$, $\mathbf{b}$ |❌ |
|$T'''$ |`T3(n)` |$\mathbf{a'}$, $\mathbf{b'}$ |$\mathbf{a''}$, $\mathbf{b''}$ |❌ |
|$T'''^{-1}$|`iT3(n)` |$\mathbf{a''}$, $\mathbf{b''}$ |$\mathbf{a'}$, $\mathbf{b'}$ |❌ |
To compute the full matrices, an optimized algorithm making use of recursive patterns is employed. Each matrix generator also features the computation of single elements through, e.g., `M(100, entry=[3,4])` where `entry` specifies the row and column of the entry (in that order).
### Single-shot estimators
Furthermore, the function `single_shot_estimators(n)` generates single-shot estimators for $a$, $b$, $a'$, $b'$, $a''$, and $b''$ for all possible numbers $m=0,...,n$ of singlets as an outcome of a two-copy Bell measurement. This function returns six 2D-arrays (one for each quantum weight enumerator in the order as given above) with the estimator for $m$ singlets in the $m$-th column.
### High-precision transformation matrices
For some applications, a higher precision than 64 bit floating point is needed for the transformation matrices. For this purpose, each transformation features a `precise` argument (which defaults to `false`). If set to `true`, an `mpmath.matrix` is returned instead of an `np.array`. This requires [`mpmath`](https://mpmath.org/) to be installed. The precision can for example be set via `mpmath.mp.dps = 120` (more on precision with mpmath, see [here](https://mpmath.org/doc/current/basics.html#setting-the-precision)) before calling the transformation generator.
## License
This library is distributed under the MIT License.
If you want to support work like this, please cite our paper: tba
[^1]: [P. Shor and R. Laflamme, Phys. Rev. Lett. **78**, 1600 (1997)](http://dx.doi.org/10.1103/PhysRevLett.78.1600)
[^2]: [E. M. Rains, IEEE Trans. Inf. Th., **44**, 1388 (1998)](http://dx.doi.org/10.1109/18.681316)
[^3]: [E. M. Rains, IEEE Trans Inf. Th. **45**, 2361 (1999)](http://dx.doi.org/10.1109/18.796376)
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