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Quantum computing and the Jones polynomial
 math.QA/0105255, in Quantum Computation and Information
"... 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 evaluati ..."
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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. 1
Teleportation topology
 Optics and Spectroscopy
, 2005
"... The paper discusses teleportation in the context of comparing quantum and topological points of view. 1 ..."
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Cited by 8 (1 self)
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The paper discusses teleportation in the context of comparing quantum and topological points of view. 1
Finding Solutions to NP Problems: Philosophical Differences Between Quantum and Evolutionary Search
 in Proc. 2001 Congress Evolutionary Computation, Seoul, Korea
, 2001
"... This paper uses instances of SAT, 3SAT and TSP to describe how evolutionary search (running on a classical computer) differs from quantum search (running on a quantum computer) for solving NP problems. ..."
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Cited by 7 (0 self)
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This paper uses instances of SAT, 3SAT and TSP to describe how evolutionary search (running on a classical computer) differs from quantum search (running on a quantum computer) for solving NP problems.
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Cited by 6 (2 self)
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Quantum knots and mosaics
 QUANTUM INF PROCESS
"... In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to ..."
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Cited by 5 (2 self)
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In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit nonclassical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.
Krawtchouk polynomials and Krawtchouk matrices,” quantph/0702073
"... Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed SylvesterHadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is then presented. To advertise the breadth and depth o ..."
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Cited by 3 (0 self)
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Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed SylvesterHadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is then presented. To advertise the breadth and depth of the field of Krawtchouk polynomials/matrices through connections with various parts of mathematics, some topics that are being developed into a Krawtchouk Encyclopedia are listed in the concluding section. Interested folks are encouraged to visit the website
Krawtchouk matrices from classical and quantum random walks,” quantph/0702173
"... Krawtchouk’s polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and ..."
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Cited by 2 (1 self)
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Krawtchouk’s polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case. 1
IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM?
, 2006
"... Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go ..."
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Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go on to show that the standard nonabelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard nonabelian QHS algorithm which is equivalent to Grover’s algorithm. Contents
Mathematical Models of Contemporary Elementary Quantum Computing Devices
, 2003
"... Abstract. Computations with a future quantum computer will be implemented through the operations by elementary quantum gates. It is now well known that the collection of 1bit and 2bit quantum gates are universal for quantum computation, i.e., any nbit unitary operation can be carried out by conca ..."
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Abstract. Computations with a future quantum computer will be implemented through the operations by elementary quantum gates. It is now well known that the collection of 1bit and 2bit quantum gates are universal for quantum computation, i.e., any nbit unitary operation can be carried out by concatenations of 1bit and 2bit elementary quantum gates. Three contemporary quantum devices–cavity QED, ion traps and quantum dots–have been widely regarded as perhaps the most promising candidates for the construction of elementary quantum gates. In this paper, we describe the physical properties of these devices, and show the mathematical derivations based on the interaction of the laser field as control with atoms, ions or electron spins, leading to the following: (i) the 1bit unitary rotation gates; and (ii) the 2bit quantum phase gates and the controllednot gate. This paper is aimed at providing a sufficiently selfcontained survey account of analytical nature for mathematicians, physicists and computer scientists to aid interdisciplinary understanding in the research of quantum computation. 1.
Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks
"... Abstract. Within the framework of the cubic honeycomb (cubic tessellation) of Euclidean 3space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3space. This qua ..."
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Abstract. Within the framework of the cubic honeycomb (cubic tessellation) of Euclidean 3space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor’s quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more ”physics friendly ” than the Reidemeister moves because, unlike the Reidemeister moves, they respect the differential geometry of 3space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3space without cutting the rope, and without letting the rope pass through itself. We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions. Contents