Pfaffians and Valiant's holographic algorithms

If one computed the determinant of an n by n matirx using the naive formula, it would involve performing a number of arithmetic operations that is exponential in $n$. Gaussian elimination enables us to compute determinants in less than $n^4$ operations. There are many classes of problems and sequences of polynomials which naively require performing a number of arithmetic operations that is exponential in the input data to solve. Some, like the determinant, also admit fast algorithms. Valiant has obtained a class of such algorithms, which he calls ``holographic'' algorithms, which, from his perspective, allow one to simulate a quantum computer on a classical computer. I'll explain Valiant's algorithms, their potential applicability, how they relate to Pfaffians (as well as some identites regarding Pfaffians of interest in their own right), and what light they shed on the P vs NP problem.