Approximating permanents and hafnians

Barvinok, Alexander

doi:10.19086/da.1244

Cited by 17 publications

(33 citation statements)

References 36 publications

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“…Computing the Hafnian of a general matrix is in complexity class #P, and formally reduces to the task of computing permanents [22]. If no entry in the matrix is negative, efficient approximation heuristics are known, although their success is only guaranteed under specific circumstances [23,24].…”

Section: B Sampling Photon Counting Eventsmentioning

confidence: 99%

Measuring the similarity of graphs with a Gaussian boson sampler

et al. 2020

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Gaussian Boson Samplers (GBS) have initially been proposed as a near-term demonstration of classically intractable quantum computation. We show here that they have a potential practical application: Samples from these devices can be used to construct a feature vector that embeds a graph in Euclidean space, where similarity measures between graphs -so called 'graph kernels' -can be naturally defined. This is crucial for machine learning with graph-structured data, and we show that the GBS-induced kernel performs remarkably well in classification benchmark tasks. We provide a theoretical motivation for this success, linking the extracted features to the number of r-matchings in subgraphs. Our results contribute to a new way of thinking about kernels as a quantum hardware-efficient feature mapping, and lead to a promising application for near-term quantum computing.

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Section: B Sampling Photon Counting Eventsmentioning

confidence: 99%

Measuring the similarity of graphs with a Gaussian boson sampler

et al. 2020

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“…Barvinok [2,5] found quasipolynomial-time approximation algorithms for computing the permanent of certain matrices, based on absence of zeros. Our method for computing inverse power sums of BIGCPs on bounded degree graphs presented in Section 3 does not seem to apply to permanents of general matrices.…”

Section: Concluding Remarks and Open Questionsmentioning

confidence: 99%

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

Patel

Regts

2017

Electronic Notes in Discrete Mathematics

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In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.More specifically, we define a large class of graph polynomials C and show that if p ∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree graphs G, then for each z in the interior of D there exists a deterministic polynomial-time approximation algorithm for evaluating p(G) at z. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures.Our work builds on a recent line of work initiated by Barvinok [2,3,4,5], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

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“…Whereas the permanent counts the number of perfect matchings in a bipartite graph, the hafnian counts the number of perfect matchings in an undirected graph. For the related problem of approximating the hafnian several methods have been developed for restricted sets of matrices [10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In Gaussian Boson Sampling the probability of the detectors clicking is now proportional to the modulus squared of the hafnian [1] of a complex fullrank submatrix constructed from the unitary matrix representing the circuit and the values of the intensities of the squeezed light going into the device (In appendix C we study in detail the dependence on the rank for low rank matrices using methods developed by Barvinok [21]). Finally, note that the approximate methods developed for counting perfect matchings are aimed at (weighted-)graphs with real or positive entries [10,11,13] making them unsuitable for GBS.…”

Section: Introductionmentioning

confidence: 99%

A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

Björklund

Gupt

Quesada

2019

ACM J. Exp. Algorithmics

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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 × 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 matrices up to size 56 × 56 indicate that one would require the 288000 CPUs of this machine for about a month and a half to compute the hafnian of a 100 × 100 matrix.

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Approximating permanents and hafnians

Cited by 17 publications

References 36 publications

Measuring the similarity of graphs with a Gaussian boson sampler

Measuring the similarity of graphs with a Gaussian boson sampler

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

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