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Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices [article]

Nima Anari, Leonid Gurvits, Shayan Oveis Gharan, Amin Saberi
2017 arXiv   pre-print
We design a deterministic polynomial time c^n approximation algorithm for the permanent of positive semidefinite matrices where c=e^γ+1≃ 4.84.  ...  We write a natural convex relaxation and show that its optimum solution gives a c^n approximation of the permanent.  ...  Prior to our paper, no efficient algorithm (deterministic, randomized, or quantum) was known for simply exponential approximation of the permanent of general positive semidefinite matrices.  ... 
arXiv:1704.03486v1 fatcat:bspbnw7jlrdoznuqyiqnoasi7u

Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

L. Chakhmakhchyan, N. J. Cerf, R. Garcia-Patron
2017 Physical Review A  
Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a  ...  We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson  ...  to approximate the permanent of Hermitian positive semidefinite matrices.  ... 
doi:10.1103/physreva.96.022329 fatcat:dcwwa7ybtvap5da4hej4lvfj2e

A simple polynomial time algorithm to approximate the permanent within a simply exponential factor [article]

Alexander Barvinok
1997 arXiv   pre-print
We present a simple randomized polynomial time algorithm to approximate the mixed discriminant of n positive semidefinite n × n matrices within a factor 2^O(n).  ...  When applied to approximating the permanent, the algorithm turns out to be a simple modification of the well-known Godsil-Gutman estimator.  ...  Hence we get a randomized polynomial time algorithm approximating the mixed discriminant of positive semidefinite matrices (and hence the permanent of a non-negative matrix) within a simply exponential  ... 
arXiv:math/9704218v1 fatcat:4az67x6idffenlueils2fvrkkq

Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices [article]

Chenyang Yuan, Pablo A. Parrilo
2021 arXiv   pre-print
We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices.  ...  By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates  ...  [AGGS17] gave the first polynomial-time algorithm for approximating the permanent of HPSD matrices with a simply exponential multiplicative approximation factor of n!  ... 
arXiv:2002.04149v2 fatcat:ra37myimgrfn3e5ro5iqqqonrq

Relative entropy optimization and its applications

Venkat Chandrasekaran, Parikshit Shah
2016 Mathematical programming  
We provide solutions based on REPs to a range of problems such as permanent maximization, robust optimization formulations of GPs, and hitting-time estimation in dynamical systems.  ...  We conclude with a discussion of quantum analogs of the relative entropy function, including a review of the similarities and distinctions with respect to the classical case.  ...  Acknowledgements The authors would like to thank Pablo Parrilo and Yong-Sheng Soh for helpful conversations, and Leonard Schulman for pointers to the literature on Von-Neumann entropy.  ... 
doi:10.1007/s10107-016-0998-2 fatcat:5xjfffz5yjcxrpsh5fslvq24ce

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

Gurvits, Samorodnitsky
2002 Discrete & Computational Geometry  
We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within an exponential multiplicative factor.  ...  To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n-tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling.  ...  The realistic goal, then, is to try and the permanent efficiently approximate as well as possible, for large classes of matrices. How well can the permanent be approximated in polynomial time?  ... 
doi:10.1007/s00454-001-0083-2 fatcat:kkaajcxvabfv7fjwjscpegsqjy

When is the Stability and Complexity of a mixed Discriminant is described [article]

Chandra Sekhar Giri
2019 Zenodo  
We show that the mixed discriminant of n positive semidefinite n×n real symmetric matrices can be approximated within a relative error > 0 in quasipolynomial time, provided the distance of each matrix  ...  As is shown by Gurvits, for m = 2 the problem is #P-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2  ...  for many inspiring conversations about mixed discriminants during the "Geometry of Polynomials" program at the Simons Institute for the Theory of Computing.  ... 
doi:10.5281/zenodo.5835395 fatcat:xatvk6u3cjh5zlf75wybsiy3da

Approximation of the joint spectral radius using sum of squares

Pablo A. Parrilo, Ali Jadbabaie
2008 Linear Algebra and its Applications  
We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices.  ...  We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming.  ...  Acknowledgements We thank the referees for their careful reading of the manuscript, and their many useful suggestions.  ... 
doi:10.1016/j.laa.2007.12.027 fatcat:p2rz22sl2jg3pbxiepwn6gbcjq

Stability and complexity of mixed discriminants [article]

Alexander Barvinok
2019 arXiv   pre-print
We show that the mixed discriminant of n positive semidefinite n × n real symmetric matrices can be approximated within a relative error ϵ >0 in quasi-polynomial n^O( n -ϵ) time, provided the distance  ...  As is shown by Gurvits, for m=2 the problem is #P-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.  ...  for many inspiring conversations about mixed discriminants during the "Geometry of Polynomials" program at the Simons Institute for the Theory of Computing.  ... 
arXiv:1806.05105v2 fatcat:gzlsceryc5ewtoim47gmmg5vge

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography [article]

Alex Meiburg
2021 arXiv   pre-print
In the process, we find that it reduces to the problem of approximately computing the permanent of a Hermitian positive semidefinite (HPSD) matrix.  ...  This implies that HPSD permanents are also NP-Hard to approximate, resolving a standing question with applications in quantum information and BosonSampling.  ...  approximate the permanent of positive semidefinite matrices within a factor of C.  ... 
arXiv:2111.03142v1 fatcat:bltbloylj5ai5ikrdjynqofaxy

Page 643 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
of positive semidefinite matrices.  ...  I.] (1-MI; Ann Arbor, MI) Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor.  ... 

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Leonid Gurvits
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. (Here operator refers to maps from matrices to matrices.)  ...  First, we reformulate the Edmonds Problem in terms of of completely positive operators, or equivalently, in terms of bipartite density matrices .  ...  time algorithm to approximate within a simply exponential factor quantum permanents of separable unnormalized bipartite density matrices (more details on this matter can be found in [7] A Proof of  ... 
doi:10.1145/780542.780545 dblp:conf/stoc/Gurvits03 fatcat:4idkxs332nhwzjlqjqqyr237qm

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Leonid Gurvits
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. (Here operator refers to maps from matrices to matrices.)  ...  First, we reformulate the Edmonds Problem in terms of of completely positive operators, or equivalently, in terms of bipartite density matrices .  ...  time algorithm to approximate within a simply exponential factor quantum permanents of separable unnormalized bipartite density matrices (more details on this matter can be found in [7] A Proof of  ... 
doi:10.1145/780543.780545 fatcat:adtssg67jzep5gixqnf5fwj3qm

Classical deterministic complexity of Edmonds' problem and Quantum Entanglement [article]

Leonid Gurvits
2003 arXiv   pre-print
The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix .  ...  This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .  ...  time algorithm to approximate within a simply exponential factor quantum permanents of separable unnormalized bipartite density matrices (more details on this matter can be found in [7] A Proof of  ... 
arXiv:quant-ph/0303055v1 fatcat:sf237x3kdbg5fdnekcobmrxyai

A polynomial time algorithm to approximate the mixed volume within a simply exponential factor [article]

Leonid Gurvits
2009 arXiv   pre-print
for the approximation of the volume of a convex set.  ...  We prove the mixed volume analogues of the Van der Waerden and Schrijver-Valiant conjectures on the permanent.  ...  And the mixed volume of ellipsoids is approximated by (D(A 1 , ..., A n )) 1 2 of the corresponding positive semidefinite matrices A i 0.  ... 
arXiv:cs/0702013v4 fatcat:waymgofs3bcq3cdg2ubwjtv3f4
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