# A faster hafnian formula for complex matrices and its benchmarking on the Titan supercomputer

@article{Bjrklund2018AFH, title={A faster hafnian formula for complex matrices and its benchmarking on the Titan supercomputer}, author={Andreas Bj{\"o}rklund and Brajesh Gupt and Nicol{\'a}s Quesada}, journal={ArXiv}, year={2018}, volume={abs/1805.12498} }

We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of $n \times n $ complex matrices run in $O(n^3 2^{n/2})$ time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest exact algorithms to compute these quantities. Despite our highly optimized algorithm, numerical benchmarks on the Titan supercomputer with…

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## References

SHOWING 1-10 OF 32 REFERENCES

Counting perfect matchings as fast as Ryser

- Computer ScienceSODA
- 2012

This work shows that there is a polynomial space algorithm that counts the number of perfect matchings in an n-vertex graph in O*(2n/2) ⊂ O(1.415n) time, and presents it in the more general setting of computing the hafnian over an arbitrary ring analogously to Ryser's algorithm for permanent computation.

Polynomial Time Algorithms to Approximate Permanents and Mixed Discriminants Within a Simply Exponential Factor

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 1999

We present real, complex, and quaternionic versions of a simple ran-domized polynomial time algorithm to approximate the permanent of a non-negative matrix and, more generally, the mixed discriminant…

Fast Polynomial-Space Algorithms Using Möbius Inversion: Improving on Steiner Tree and Related Problems

- Computer Science, MathematicsICALP
- 2009

The concept of branching walks is introduced and the Inclusion-Exclusion algorithm of Karp is extended for counting Hamiltonian paths and polynomial-space $\mathcal{O}^*(2^n)$ algorithms for several spanning tree and partition problems are obtained.

Hafnians, perfect matchings and Gaussian matrices

- Mathematics, Computer Science
- 2014

It is concluded that Barvinok's estimator gives a polynomial-time algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.

Faster exponential-time algorithms in graphs of bounded average degree

- Mathematics, Computer ScienceInf. Comput.
- 2015

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

- Computer Science, MathematicsAlgorithmica
- 2007

This work shows that the Exact Satisfiability problem of size l with m clauses can be solved in time 2mlO(1) and polynomial space, and shows how to count the number of perfect matchings in time O(1.732n) and exponential space.

Approximating permanents and hafnians

- Mathematics, Computer Science
- 2016

We prove that the logarithm of the permanent of an nxn real matrix A and the logarithm of the hafnian of a 2nx2n real symmetric matrix A can be approximated within an additive error 1 > epsilon > 0…

Computing Permanents for Boson Sampling on Tianhe-2 Supercomputer

- Computer ScienceNational Science Review
- 2018

The permanent of the largest matrix is computed using up to 312,000 CPU cores of Tianhe-2, and it is inferred from the current most efficient permanent-computing algorithms that an upper bound on the performance of Tiane-2 is one 50-photon sample per ~100 min.

Gaussian boson sampling for perfect matchings of arbitrary graphs

- Computer SciencePhysical Review A
- 2018

With the proposed method, a Gaussian Boson Sampling device can be used to estimate the number of perfect matchings significantly faster and with lower energy consumption compared to a classical computer.

Alternative Algorithms for Counting All Matchings in Graphs

- Computer Science, MathematicsSTACS
- 2003

Two new methods for counting all matchings in a graph are presented, one of which is a generalization of a Godman-Godsil estimator and the other uses importance sampling.