Holographic algorithms: from art to science
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Holographic algorithms with unsymmetric signatures
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Holographic algorithms: guest column
ACM SIGACT News
Basis Collapse in Holographic Algorithms
Computational Complexity
Holographic algorithms: The power of dimensionality resolved
Theoretical Computer Science
On blockwise symmetric signatures for matchgates
Theoretical Computer Science
Holographic algorithms: From art to science
Journal of Computer and System Sciences
On block-wise symmetric signatures for matchgates
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
Holographic algorithms: the power of dimensionality resolved
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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Valiant has proposed a new theory of algorithmic computation based on perfect matchings and Pfaffians. We study the properties of matchgates--the basic building blocks in this new theory. We give a set of algebraic identities which completely characterizes these objects for arbitrary numbers of inputs and outputs. These identities are derived from Grassmann-Plücker identities. The 4 by 4 matchgate character matrices are of particular interest. These were used in Valiant's classical simulation of a fragment of quantum computations. For these 4 by 4 matchgates, we use Jacobi's theorem on compound matrices to prove that the invertible matchgate matrices form a multiplicative group. Our results can also be expressed in the theory of Holographic Algorithms in terms of realizable standard signatures. These results are useful in establishing limitations on the ultimate capabilities of Valiant's theory of matchgate computations and Holographic Algorithms.