Add support for calculating the permanent using the BBFG algorithm

[mlxd]

Issue description

Currently, The Walrus uses the Ryser formula to calculate permanents of arbitrary matrices (https://en.wikipedia.org/wiki/Computing_the_permanent#Ryser_formula). Another useful algorithm to calculate permanents is the Balasubramanian–Bax–Franklin–Glynn (BBFG) formula (https://en.wikipedia.org/wiki/Computing_the_permanent#Balasubramanian%E2%80%93Bax%E2%80%93Franklin%E2%80%93Glynn_formula). Having both methods implemented will allow for performance and accuracy comparisons and trade-offs.

Required tasks:

• Implement a templated version of the BBFG formula in C++, located in the ./include directory (see existing implementation)
• Add tests for the new algorithm to the /tests directory
• The implemented algorithm should also be callable from python (Note: this can be achieved by adding the correct headers and wrapper functions in setup.py, /include/libwalrus.hpp and /include/libwalrus.pyx)

[maliasadi]

Sorta done with adding BBFG support, I think. However, as I've changed the list of requirements just a bit, I'll review what's been done this evening:

• Implemented a templated version of the BBFG formula in C++. It's a sequential algorithm (nthreads=1) from "The permanent of a square matrix" by David G. Glynn with using Gray code ordering. This is added in the ./include/permanent.hpp.
• Added two wrappers for Python integration (perm_BBFG_qp and perm_BBFG_cmplx) and one for the future parallel vs. serial integration (perm_BBFG).
• Added tests to ./tests/libwalrus_unittest.cpp
• Integrated three algorithms in ./thewalrus/libwalrus.pyx:

  T perm_BBFG[T](vector[T] &mat) 
  double perm_BBFG_qp(vector[double] &mat)
  double complex perm_BBFG_cmplx(vector[double complex] &mat)

• Added tests to ./thewalrus/tests/test_permanent.py for integrated python methods naming perm_BBFG_real and perm_BBFG_complex.

My script to test changes before commit:

make clean && \
pip install -e . && \
make test-cpp && \
make test 

Please, let me know if the list of changes cover all concerns within this issue.

[mlxd]

Hi @maliasadi sounds good. Feel free to open a PR with your changes. The team can review and discuss in there.

[maliasadi]

Feel free to open a PR with your changes. The team can review and discuss in there.

Sorry for the late reply @mlxd. I was looking for some time to optimize my code and implement the parallel version (that's not done yet!) before making a PR. I collected some promising benchmarks comparing my two versions of BBFG formula with the Ryser's one. I'll open a PR and continue the conversation there.

[nquesada]

Solved via https://github.com/XanaduAI/thewalrus/pull/267