Matrix permanent and determinant from a spin system

Generalized matrix functions, determinant and permanent

In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the de...

Permanent Uncertainty: on the Quantum Evaluation of the Determinant and the Permanent of a Matrix

We investigate the possibility of evaluating permanents and determinants of matrices by quantum computation. All current algorithms for the evaluation of the permanent of a real matrix have exponential time complexity and are known to be in the class P #P. Any method to evaluate or approximate the permanent is thus of fundamental interest to complexity theory. Permanents and determinants of mat...

Determinant versus permanent

Determinant Versus Permanent

We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two re...

The Number of Terms in the Permanent and the Determinant of a Generic Circulant Matrix

Let A = (aij) be the generic n×n circulant matrix given by aij = xi+j , with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n...