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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
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
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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of a set of useful GrassmannPlücker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planarmatchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we
Toward an instance theory of automatization
 Psychological Review
, 1988
"... This article presents a theory in which automatization is construed as the acquisition of a domainspecific knowledge base, formed of separate representations, instances, of each exposure to the task. Processing is considered automatic if it relies on retrieval of stored instances, which will occur ..."
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Cited by 613 (37 self)
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This article presents a theory in which automatization is construed as the acquisition of a domainspecific knowledge base, formed of separate representations, instances, of each exposure to the task. Processing is considered automatic if it relies on retrieval of stored instances, which will occur only after practice in a consistent environment. Practice is important because it increases the amount retrieved and the speed of retrieval; consistency is important because it ensures that the retrieved instances will be useful. The theory accounts quantitatively for the powerfunction speedup and predicts a powerfunction reduction in the standard deviation that is constrained to have the same exponent as the power function for the speedup. The theory accounts for qualitative properties as well, explaining how some may disappear and others appear with practice. More generally, it provides an alternative to the modal view of automaticity, arguing that novice performance is limited by a lack of knowledge rather than a scarcity of resources. The focus on learning avoids many problems with the modal view that stem from its focus on resource limitations. Automaticity is an important phenomenon in everyday mental life. Most of us recognize that we perform routine activities quickly and effortlessly, with little thought and conscious awarenessin short, automatically (James, 1890). As a result, we often perform those activities on "automatic pilot " and turn our minds to other things. For example, we can drive to dinner while conversing in depth with a visiting scholar, or we can make coffee while planning dessert. However, these benefits may be offset by costs. The automatic pilot can lead us astray, causing errors and sometimes catastrophes (Reason & Myceilska, 1982). If the conversation is deep enough, we may find ourselves and the scholar arriving at the office rather than the restaurant, or we may discover that we aren't sure whether we put two or three scoops of coffee into the pot. Automaticity is also an important phenomenon in skill acquisition (e.g., Bryan & Harter, 1899). Skills are thought to consist largely of collections of automatic processes and procedures
A New Kind of Science
, 2002
"... “Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplit ..."
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Cited by 850 (0 self)
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“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplitudes you told me about, they’re so complicated and absurd, what makes you think those are right? Maybe they aren’t right. ’ Such remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out.” —Richard Feynman [1, p.161]
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.
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
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