Computing with Quantum Knots

Collins, Graham P.

doi:10.1038/scientificamerican0406-56

Cited by 63 publications

(61 citation statements)

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“…The reason for the tremendous amount of attention they have received is two-fold. One, they have been predicted to exhibit a number of exotic phenomena such as the magnetoelectric effect 1 , magnetic monopole-like behavior 2 and the existence of topologically protected Majorana modes 3 with potential applications for topological quantum computing 4 . Two, a number of materials have already been theoretically predicted [5][6][7][8][9] and experimentally found [10][11][12][13][14][15][16] to be in this fascinating phase.…”

Section: Introductionmentioning

confidence: 99%

Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response

2011

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We study theoretically the optical response of the surface states of a topological insulator, especially the generation of helicity-dependent direct current by circularly polarized light. Interestingly, the dominant current, due to an interband transition, is controlled by the Berry curvature of the surface bands. This extends the connection between photocurrents and Berry curvature beyond the quasiclassical approximation where it has been shown to hold. Explicit expressions are derived for the (111) surface of the topological insulator Bi2Se3 where we find significant helicity dependent photocurrents when the rotational symmetry of the surface is broken by an in-plane magnetic field or a strain. Moreover, the dominant current grows linearly with time until a scattering occurs, which provides a means for determining the scattering time. The dc spin generated on the surface is also dominated by a linear-in-time, Berry curvature dependent contribution.

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Section: Introductionmentioning

confidence: 99%

Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response

2011

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“…We let K (n) denote the subset of M (n) of all knot n-mosaics. 3 The previous two 4-mosaics shown above are examples respectively of a non-knot 4-mosaic and a knot 4-mosaic. Other examples of knot (or link) mosaics are the Hopf link 4-mosaic, the figure eight knot 5-mosaic, and the Borromean rings 6-mosaic, respectively illustrated below:…”

Section: Fig 1 Tile Connection Pointsmentioning

confidence: 99%

“…Within the context of the mosaic construction, a quantum knot in [7] corresponds to an element of the orbit Hilbert space K (n) /A(n). In [3] and in [26] the phrase "quantum knot" refers not to knots, but to the use of representations of the braid group to model the dynamic behavior of certain quantum systems. In this contexts, braids are used as a tool to model topological obstructions to quantum decoherence that are conjectured to exist within certain quantum systems.…”

Section: Propositionmentioning

confidence: 99%

Quantum knots and mosaics

Lomonaco

Kauffman

2008

Quantum Inf Process

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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 non-classical 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.

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“…The biggest problem in the field of quantum computation is decoherence or the loss of quantum properties [8]. The topological quantum computer overcomes this problem by using anyons (quasi particles) that possess higher stability [9]. Recent efforts have shown the possible practical development of the topological quantum computer [9].…”

Section: Introductionmentioning

confidence: 99%

“…The topological quantum computer overcomes this problem by using anyons (quasi particles) that possess higher stability [9]. Recent efforts have shown the possible practical development of the topological quantum computer [9].Biocomputing, as the name suggests, is computation performed using biomolecules such as DNA and proteins. A biological process such as glycolysis [10] or bioluminescence [11] can be viewed as a genetic regulatory circuit.…”

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confidence: 99%

Circuit development using biological components: Principles, models and experimental feasibility

Krishnan

Purdy

2007

Analog Integr Circ Sig Process

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Today most VLSI circuits are built in silicon using CMOS transistors. Developments in design automation and process fabrication have resulted in the progressive increase of the number of transistors per chip and decrease in the size of the transistors. But transistor designers are fast approaching fundamental physical barriers to further size reduction. Thus engineers are looking at alternate technologies such as nanodevices and biocircuits for next-generation circuits. In our research, we concentrate on the development of biocircuits and their applications. Our eventual goal is the design and simulation of complete systems integrating biocircuits and VLSI technology appropriately. Biocircuits are circuits developed in vivo or in vitro, using DNA and proteins. A biological process such as glycolysis or bioluminescence can be viewed as a genetic regulatory circuit, a complex set of biochemical reactions regulating the behavior of genes, operons, DNA, RNA and proteins. Similar to voltage in an electrical circuit, a genetic regulatory circuit produces an output protein in response to an input stimulus. We can engineer biocircuits to meet design specifications, using genetic engineering. Our aim is to build a library of in vitro biocircuits representing the Boolean functions. The biocircuits from this library can be further cascaded to form larger circuits. In this paper, we review the feasibility of building biocircuits. We discuss the construction of Boolean logic gates such as NOT, AND and OR and their verification by simulation. We also address important aspects such as cascading of the biocircuits and practical implementation. In addition, we describe an algorithm Box that can help to control biocircuit characteristics such as gain and switching behavior. This approach is similar to design space exploration in traditional VLSI, but takes into account biological knowledge obtained through experiments. We also provide insight into the robustness of biocircuits in the presence of noise. This paper is intended to pave the way for electrical engineers to start exploring the field of biocircuits.Keywords -biocircuits, bio-inverter, noise model, sensitivity analysis, optimization. IntroductionConventionally, VLSI circuits are built using CMOS transistors and the circuits are typically fabricated on silicon. Developments in the areas of design automation and process fabrication have resulted in the progressive increase of the number of transistors per chip and the progressive decrease in the size of the transistors. Transistors are fast approaching the size of atoms and this presents a fundamental barrier to further reduction in transistor sizes [1]. Scientists are already looking at alternate technologies that can be used to develop the next-generation circuits. The three main research areas in this direction are nanotechnology, quantum computing and biocomputing.Nanotechnology deals with the design of electronic circuits at the atomic range of 0.1-100nm [2]. At the time this paper is being written, Intel has ...

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Computing with Quantum Knots

Cited by 63 publications

References 0 publications

Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response

Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response

Quantum knots and mosaics

Circuit development using biological components: Principles, models and experimental feasibility

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