• Corpus ID: 44093237

# 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}
}
• Published 31 May 2018
• Computer Science
• ArXiv
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…
9 Citations

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## References

SHOWING 1-10 OF 32 REFERENCES
Counting perfect matchings as fast as Ryser
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
• A. Barvinok
• Mathematics, Computer Science
Random 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
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.
Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings
• Computer Science, Mathematics
Algorithmica
• 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
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 Science
National 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 Science
Physical 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
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.