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Some Results on Matchgates and Holographic Algorithms
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 48 (2006)
, 2006
"... We establish a 11 correspondence between Valiant’s character theory of matchgate/matchcircuit [14] and his signature theory of planarmatchgate/matchgrid [16], thus unifying the two theories in expressibility. In [5], we had established a complete characterization of general matchgates, in terms of ..."
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Cited by 26 (9 self)
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We establish a 11 correspondence between Valiant’s character theory of matchgate/matchcircuit [14] and his signature theory of planarmatchgate/matchgrid [16], thus unifying the two theories in expressibility. In [5], we had established a complete characterization of general matchgates, in terms
Holographic Algorithms Beyond Matchgates
"... Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomialtime and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying famili ..."
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Cited by 2 (1 self)
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Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomialtime and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying
Matchgates and classical simulation of quantum circuits
, 2008
"... Let G(A, B) denote the 2qubit gate which acts as the 1qubit SU(2) gates A and B in the even and odd parity subspaces respectively, of two qubits. Using a Clifford algebra formalism we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n. ..."
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Cited by 14 (1 self)
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.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allowing the same gates to act only
Holographic algorithms without matchgates
, 2009
"... The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a problem to be in the class P. In this article we streamline the i ..."
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Cited by 7 (3 self)
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the implementation of holographic algorithms by eliminating one of the steps in the construction procedure, and generalize their applicability to new signatures. Instead of matchgates, which are weighted graph fragments that replace vertices of a natural bipartite graph ΓP associated to a problem P, our approach
Contraction of matchgate tensor networks on nonplanar graphs
 TENSOR NETWORK CONTRACTIONS FOR #SAT 15
"... ar ..."
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 12 (3 self)
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had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
"... 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 un ..."
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Cited by 17 (7 self)
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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
On Blockwise Symmetric Signatures for Matchgates
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 19 (2007)
, 2007
"... We give a classification of blockwise symmetric signatures in the theory of matchgate computations. The main proof technique is matchgate identities, a.k.a. useful GrassmannPlücker identities.
..."
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Cited by 5 (3 self)
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We give a classification of blockwise symmetric signatures in the theory of matchgate computations. The main proof technique is matchgate identities, a.k.a. useful GrassmannPlücker identities.
Results 1  10
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79