Approximating the Permanent via Nonabelian Determinants

Since the celebrated work of Jerrum, Sinclair, and Vigoda, we have known that the permanent of a 0-1 matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain to sample perfect matchings of a bipartite graph. A separate strand of the literature has pursued the possibility of an alternate, algebraic polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. In the case where A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first---and therefore the number of trials we need to obtain a good estimate of the permanent---is (1 + O(1/d))^n when A is the algebra of d-by-d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, if d is constant---the only case in which an efficient algorithm is known---we show that the critical ratio exceeds 2^n / n^O(d). Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.