Quantum computation of zeta functions of curves

Kedlaya, Kiran S.

doi:10.1007/s00037-006-0204-7

Cited by 21 publications

(20 citation statements)

References 22 publications

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“…A quantum algorithm for finding the Zeta function of a curve is given in [24]. One could replace the role that the Gauss sum estimation [25] plays in our scheme with this quantum algorithm for the Zeta function.…”

Section: Previous Quantum Complexity Resultsmentioning

confidence: 99%

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On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

Geraci

Lidar

2008

Commun. Math. Phys.

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We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCCǫ) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date.

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Section: Previous Quantum Complexity Resultsmentioning

confidence: 99%

“…Note that there is a quantum algorithm for finding zeta functions of curves which is exponentially faster in q than the classical algorithm in [23] (as is ours). This is given in [24]. The use of this algorithm instead of the Gauss sum approximation algorithm is left for a future publication.…”

Section: Classical and Quantum Complexity Of The Schemementioning

confidence: 99%

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

Geraci

Lidar

2008

Commun. Math. Phys.

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“…On the other hand, Kedlaya (2006) explained how the quantum algorithm for determining the structure of an unknown finite Abelian group (Section IV.G) can be used to count the number of points on a planar curve of genus g over F q in time poly(g, log q). It is probably fair to say that this constitutes not so much a new quantum algorithm, but rather a novel application of known quantum algorithms to algebraic geometry.…”

Section: H Counting Points On Curvesmentioning

confidence: 99%

Quantum algorithms for algebraic problems

2010

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Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic

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“…Subsequent results by Pila [17], Adleman and Huang [1] generalized this result to hyper-elliptic curves, giving an algorithm with running time (log q) O(g for counting points with time complexity poly(p, r, g). All these classical algorithms are bested by the quantum algorithm that Kedlaya [13] developed, which solves the same counting problem with time complexity poly(g, log q).…”

Section: Counting Points Of Finite Field Equationsmentioning

confidence: 99%

Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory

Dam¹,

Sasaki²

2012

Diversities in Quantum Computation and Quantum Information

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Abstract. Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.

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Quantum computation of zeta functions of curves

Cited by 21 publications

References 22 publications

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

Quantum algorithms for algebraic problems

Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory

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