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, modulo certain Mersenne type integers. These counting problems are #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the Grassmann-Plücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant’s classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi’s theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant’s theory of matchgate computations and his closely related theory of Holographic Algorithms. 1

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 problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2. 1

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.

Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but tractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of k-bit matchgates form a group, extending the recent work of Cai and Choudhary [1] of the same result for the case of k = 2, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant [4].