Related papers: A permanent formula for the Jones polynomial
In 2011, Aaronson gave a striking proof, based on quantum linear optics, showing that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and…
Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to "additively" approximate the Jones polynomial V(L,t) at any principal root of unity t.…
A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix…
In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the…
This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order…
We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{\em Adv.\ Appl.\ Math.} {\bf…
We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to…
A simple construction is presented which allows computing the transition amplitude of a quantum circuit to be encoded as computing the permanent of a matrix which is of size proportional to the number of quantum gates in the circuit. This…
This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our…
A celebrated important result due to Freedman, Larsen and Wang states that providing additive approximations of the Jones polynomial at the k'th root of unity, for constant k=5 and k>6, is BQP-hard. Together with the algorithmic results of…
We show that the permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving \alpha-permanents:…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…
Computing the permanent of a non-negative matrix is a computationally challenging, \#P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined…
In this work, we study the computational complexity of quantum determinants, a $q$-deformation of matrix permanents: Given a complex number $q$ on the unit circle in the complex plane and an $n\times n$ matrix $X$, the $q$-permanent of $X$…
The problem of computing the permanent of a matrix has attracted interest since the work of Ryser(1963) and Valiant(1979). On the other hand, trellises were extensively studied in coding theory since the 1960s. In this work, we establish a…
Counting the number of all the matchings on a bipartite graph has been transformed into calculating the permanent of a matrix obtained from the extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach (RM) to…
Counting the number of all the matchings on a bipartite graph has been transformed into calculating the permanent of a matrix obtained from the extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach (RM) to…
We present a method of representing an element of $\mathbb{F}_3^n$ as an element of $\mathbb{F}_n^2 \times \mathbb{F}_n^2$ which in practice will be a pair of unsigned integers. We show how to do addition, subtraction and pointwise…
A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not…
The monomer-dimer model is fundamental in statistical mechanics. However, it is $#P$-complete in computation, even for two dimensional problems. A formulation in matrix permanent for the partition function of the monomer-dimer model is…