Quantum Hidden Subgroup Algorithms: The Devil Is in the Details
We conjecture that one of the main obstacles to creating new non-abelian quantum hidden subgroup algorithms is the correct choice of a transversal.
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arXiv subjects
Louis H. Kauffman
Publications and source records attributed to Louis H. Kauffman.
On the classification of rational tangles
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rational tangles). This paper forms the basis for a sequel on the classification of rational knots and links.
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Review of "Knots" by Alexei Sossinsky, Harvard University Press, 2002, ISBN 0-674-00944-4
This paper is a review of the book "Knots" by Alexei Sossinsky. The review includes a short personal history of knot theory at the end of the twentieth century.
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On the Classification of Rational Knots
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.
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From tangle fractions to DNA
This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangle fraction from the point of view of the bracket polynomial, accounts of the classification theorems for rational tangles and rational knots (and of our new proofs of these results) and a description of the use of rational tangles in studying DNA recombination.
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Reformulating the Map Color Theorem
This paper discusses reformulations of the problem of coloring plane maps with four colors. The context is the edge-coloring with three colors of cubic graphs such that three distinct colors occur at each vertex. We include discussion of the Eliahou-Kryuchkov conjecture, the Penrose formula, the vector cross product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown. The latter includes a proof of the Spencer-Brown parity lemma and discussion of the parity-pass algorithm.
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Entanglement Criteria - Quantum and Topological
This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The entanglement criterion is used to explore this connection. The paper discusses non-locality in the light of this criterion.
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Countinuous Quantum Hidden Subgroup Algorithms
In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L^2 paths x:[0,1]-->R^n in real n-space R^n. Also included is a curious discrete QHS algorithm which is dual to Shor's algorithm.
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Non-commutative Calculus and Discrete Physics
Sequences of actions do not commute.. For example, the tick of a clock and the measurement of a position do not commute with one another, since the position will have moved to the next position after the tick. We adopt non-commutative calculus, with derivatives represented by commutators. In the beginning distinct derivatives do not commute with one another, providing curvature formalism so that the form of the curvature of a gauge field appears almost as soon as the calculus is defined. This provides context for the Feynman-Dyson derivation of electromagnetic formalism from commutators, and generalizations including the early appearance of the form of the Levi-Civita connection dervived from the Jacobi identity. In this version of non-commutative physics bare quantum mechanics (its commutation relations) appears as the flat background for all other constructions. Ascent to classical physics is obtained by replacing commutators with Poisson brackets that satisfy the Leibniz rule. An appendix on matrix algebra from a discrete point of view (Iterants) is provided. This paper will appear in the proceedings of the ANPA conference held in Cambridge, England in the summer of 2002.
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On E.D.Jones' Microcosmology
By taking seriously the limits on observability which come from combining relativistic quantum mechanics with general relativity, Ed Jones has shown that the current measurements of the cosmological constant density $Ω_Λ\sim 0.7$ imply that the temperature scale at which it becomes possible to discuss cosmological models is $\sim 5 Tev$ ($5.8\times 10^{16} \ ^oK$). This is self-consistent with the assumption that the number of Planck masses which make some sort of ``phase transition'' to this state is $N_{Pk}\sim 4\times 10^{61}$. We review Jones' argument and the {\it bit-string physics} calculation which gives the baryon-photon ratio at nucleosynthesis as $\sim 2/256^4$, the dark matter-baryon ratio as $\sim 12.7$, and hence $Ω_m \sim 0.3$, in agreement with current observations. Accepting these values for the two energy densities $Ω_Λ+Ω_m \sim 1$ in accord with recent analyses of fluctuations in the CMB showing that space is flat to about 6%. We conclude that experiments with particle accelerators in the 5-10 Tev range must either show that current theory can adequately describe the currently observed structure of our universe or force us to revise our ideas about physics at a very fundamental level.
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On Filamentations and Virtual Knots
In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. We introduce a class of flat diagrams for which no filamentations exist. This class gives the first example of an infinite set of virtual knots for which the Jones polynomial and the fundamental group both evaluate trivially. The related generalized Alexander polynomials for the virtual knots in this class turn out to be distinct, proving that the class is indeed infinite.
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Quantum Hidden Subgroup Problems: A Mathematical Perspective
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).
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Comparing Quantum Entanglement and Topological Entanglement
This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison it is fundamental to view topological entanglements such as braids as "entanglement operators" and to associate to them unitary operators that are capable of creating quantum entanglement.
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Biologic
In this paper we explore the boundary between biology and the study of formal systems (logic). In the end, we arrive at a summary formalism, a chapter in "boundary mathematics" where there are not only containers <> but also extainers ><, entities open to interaction and distinguishing the space that they are not. The boundary algebra of containers and extainers is to biologic what boolean algebra is to classical logic. We show how this formalism encompasses significant parts of the logic of DNA replication, the Dirac formalism for quantum mechanics, formalisms for protein folding and the basic structure of the Temperley Lieb algebra at the foundations of topological invariants of knots and links.
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Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Joyce (which, in turn, is a generalization of the fundamental group of the knot or link). We then introduce the concept of a bi-oriented quantum algebra which provides an algebraic context for the associated quantum invariant.
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Quantum Computing and the Jones Polynomial
This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open.
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Oriented Quantum Algebras, Categories and Invariants of Knots and Links
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
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Oriented Quantum Algebras and Invariants of Knots and Links
In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a more algebraic perspective, and develop a more detailed theory for them and their associated invariants.
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