Valiant's theory of holographic algorithms is a novel methodology to achieve exponential speed-ups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2-dimensional bases are universal for holographic ... more >>>

We give a classification of block-wise symmetric signatures in the theory of matchgate computations. The main proof technique is matchgate identities, a.k.a. useful Grassmann-Pl\"{u}cker identities. more >>>

Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This ... more >>>

We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, ... more >>>

The most intriguing aspect of the new theory of matchgate computations and holographic algorithms by Valiant~\cite{Valiant:Quantum} \cite{Valiant:Holographic} is that its reach and ultimate capability are wide open. The methodology produces unexpected polynomial time algorithms solving problems which seem to require exponential time. To sustain our belief in P $\not =$ ... more >>>