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Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... 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 ..."
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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 #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
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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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 characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have 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 and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ̸ = 2. 1
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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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 #Phard 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 realvalued 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 #Phard on general graphs but tractable on planar graphs, or (3) those which are #Phard even on planar graphs. The classification criteria
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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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.) 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 on n.n. and nextn.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the JordanWigner representation of a Clifford algebra, we show how one may generalise the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits. 1
Some Observations on Holographic Algorithms
"... Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities ..."
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Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities for degree three planar undirected graphs: the number of 3colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case the parity can be computed for any slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes. These holographic algorithms use bases of three components, rather than two. 1
Signature Theory in Holographic Algorithms
"... In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable ..."
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In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable signatures we prove a Birkhofftype theorem which determines this class. It gives a complete structural understanding of the relationship between 2realizability and 2admissibility. This is followed by characterization theorems for 1realizability and 1admissibility. Finally, using this theory of general (i.e., unsymmetric) It is generally conjectured that many combinatorial problems in the class NP or #P are not computable in polynomial time. The prevailing opinion is that these problems seem to require the accounting or processing of exponentially many potential solution fragments to the problem. However it is rather natural, and it should not cause any surprise, that the answer to such a problem can in general be
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups 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.
On Blockwise Symmetric Signatures for Matchgates
, 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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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.
The Complexity of Planar Boolean #CSP with Complex Weights
"... We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generali ..."
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We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting the number of Eulerian orientations over planar 4regular graphs to show the latter is #Phard. This strengthens a theorem by Huang and Lu to the planar setting.
A THEORY FOR VALIANT’S MATCHCIRCUITS (EXTENDED ABSTRACT)
, 2008
"... 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 kbit matchgates form a group, extending the recen ..."
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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 kbit 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 twobit matchgates are universal for matchcircuits, answering a question of Valiant [4].