Computing with Quantum Knots
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Quantum Topology and Categorification Seminar, Spring 2017
QUANTUM TOPOLOGY AND CATEGORIFICATION SEMINAR, SPRING 2017 ARUN DEBRAY APRIL 25, 2017 These notes were taken in a learning seminar in Spring 2017. I live-TEXed them using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Michael Ball for finding and fixing a typo. CONTENTS Part 1. Quantum topology: Chern-Simons theory and the Jones polynomial 1 1. The Jones Polynomial: 1/24/171 2. Introduction to quantum field theory: 1/31/174 3. Canonical quantization and Chern-Simons theory: 2/7/177 4. Chern-Simons theory and the Wess-Zumino-Witten model: 2/14/179 5. The Jones polynomial from Chern-Simons theory: 2/21/17 13 6. TQFTs and the Kauffman bracket: 2/28/17 15 7. TQFTs and the Kauffman bracket, II: 3/7/17 17 Part 2. Categorification: Khovanov homology 18 8. Khovanov homology and computations: 3/21/17 18 9. Khovanov homology and low-dimensional topology: 3/28/17 21 10. Khovanov homology for tangles and Frobenius algebras: 4/4/17 23 11. TQFT for tangles: 4/11/17 24 12. The long exact sequence, functoriality, and torsion: 4/18/17 26 13. A transverse link invariant from Khovanov homology: 4/25/17 28 References 30 Part 1. Quantum topology: Chern-Simons theory and the Jones polynomial 1. THE JONES POLYNOMIAL: 1/24/17 Today, Hannah talked about the Jones polynomial, including how she sees it and why she cares about it as a topologist. 1.1. Introduction to knot theory. -
Classical Topology and Quantum States∗
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SU-4240-716 February 2000 Classical Topology and Quantum States∗ A.P. Balachandran Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA Abstract Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its CK− and C∞− structures) can be reconstructed using Gel’fand - Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed. ∗Talk presented at the Winter Institute on Foundations of Quantum Theory and Quantum Op- tics,S.N.Bose National Centre for Basic Sciences,Calcutta,January 1-13,2000. -
MATTERS of GRAVITY, a Newsletter for the Gravity Community, Number 3
MATTERS OF GRAVITY Number 3 Spring 1994 Table of Contents Editorial ................................................... ................... 2 Correspondents ................................................... ............ 2 Gravity news: Open Letter to gravitational physicists, Beverly Berger ........................ 3 A Missouri relativist in King Gustav’s Court, Clifford Will .................... 6 Gary Horowitz wins the Xanthopoulos award, Abhay Ashtekar ................ 9 Research briefs: Gamma-ray bursts and their possible cosmological implications, Peter Meszaros 12 Current activity and results in laboratory gravity, Riley Newman ............. 15 Update on representations of quantum gravity, Donald Marolf ................ 19 Ligo project report: December 1993, Rochus E. Vogt ......................... 23 Dark matter or new gravity?, Richard Hammond ............................. 25 Conference Reports: Gravitational waves from coalescing compact binaries, Curt Cutler ........... 28 Mach’s principle: from Newton’s bucket to quantum gravity, Dieter Brill ..... 31 Cornelius Lanczos international centenary conference, David Brown .......... 33 Third Midwest relativity conference, David Garfinkle ......................... 36 arXiv:gr-qc/9402002v1 1 Feb 1994 Editor: Jorge Pullin Center for Gravitational Physics and Geometry The Pennsylvania State University University Park, PA 16802-6300 Fax: (814)863-9608 Phone (814)863-9597 Internet: [email protected] 1 Editorial Well, this newsletter is growing into its third year and third number with a lot of strength. In fact, maybe too much strength. Twelve articles and 37 (!) pages. In this number, apart from the ”traditional” research briefs and conference reports we also bring some news for the community, therefore starting to fulfill the original promise of bringing the gravity/relativity community closer together. As usual I am open to suggestions, criticisms and proposals for articles for the next issue, due September 1st. Many thanks to the authors and the correspondents who made this issue possible. -
Shaffique Adam a Self-Consistent Theory for Graphene Transport
A self-consistent theory for graphene transport Shaffique Adam Collaborators: Sankar Das Sarma, Piet Brouwer, Euyheon Hwang, Michael Fuhrer, Enrico Rossi, Ellen Williams, Philip Kim, Victor Galitski, Masa Ishigami, Jian-Hao Chen, Sungjae Cho, and Chaun Jang. Schematic 1. Introduction - Graphene transport mysteries - Need for a hirarchy of approximations - Sketch of self-consistent theory: discussion of ansatz and its predictions 2. Characterizing the Dirac Point - What the Dirac point really looks like - Comparison of self-consistent theory and energy functional minimization results 3. Quantum to classical crossover 4. Effective medium theory 5. Comparison with experiments Introduction to graphene transport mysteries High Density Low Density Hole carriers Electron carriers E electrons kx ky holes n Figure from Novoselov et al. (2005) Vg n Fuhrer group (unpublished) 2006 ∝ - Constant (and high) mobility over a wide range of density. Dominant scattering mechanism? - Minimum conductivity plateau ? - Mechanism for conductivity without carriers? What could be going on? Graphene - Honeycomb lattice: Dirac cone with trigonal warping, - Disorder: missing atoms, ripples, edges, impurities (random or correlated) - Interactions: screening, exchange, correlation, velocity/disorder renormalization - Phonons - Localization: quantum interference - Temperature - ... Exact solution is impossible -> reasonable hierarchy of approximations Any small parameters? - For transport, we can use a low energy effective theory i.e. Dirac Hamiltonian. Corrections, -
Topological Quantum Computation Zhenghan Wang
Topological Quantum Computation Zhenghan Wang Microsoft Research Station Q, CNSI Bldg Rm 2237, University of California, Santa Barbara, CA 93106-6105, U.S.A. E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary 57-02, 18-02; Secondary 68-02, 81-02 Key words and phrases. Temperley-Lieb category, Jones polynomial, quantum circuit model, modular tensor category, topological quantum field theory, fractional quantum Hall effect, anyonic system, topological phases of matter To my parents, who gave me life. To my teachers, who changed my life. To my family and Station Q, where I belong. Contents Preface xi Acknowledgments xv Chapter 1. Temperley-Lieb-Jones Theories 1 1.1. Generic Temperley-Lieb-Jones algebroids 1 1.2. Jones algebroids 13 1.3. Yang-Lee theory 16 1.4. Unitarity 17 1.5. Ising and Fibonacci theory 19 1.6. Yamada and chromatic polynomials 22 1.7. Yang-Baxter equation 22 Chapter 2. Quantum Circuit Model 25 2.1. Quantum framework 26 2.2. Qubits 27 2.3. n-qubits and computing problems 29 2.4. Universal gate set 29 2.5. Quantum circuit model 31 2.6. Simulating quantum physics 32 Chapter 3. Approximation of the Jones Polynomial 35 3.1. Jones evaluation as a computing problem 35 3.2. FP#P-completeness of Jones evaluation 36 3.3. Quantum approximation 37 3.4. Distribution of Jones evaluations 39 Chapter 4. Ribbon Fusion Categories 41 4.1. Fusion rules and fusion categories 41 4.2. Graphical calculus of RFCs 44 4.3. Unitary fusion categories 48 4.4. Link and 3-manifold invariants 49 4.5. -
Quantum Topology and Quantum Computing by Louis H. Kauffman
Quantum Topology and Quantum Computing by Louis H. Kauffman Department of Mathematics, Statistics and Computer Science 851 South Morgan Street University of Illinois at Chicago Chicago, Illinois 60607-7045 [email protected] I. Introduction This paper is a quick introduction to key relationships between the theories of knots,links, three-manifold invariants and the structure of quantum mechanics. In section 2 we review the basic ideas and principles of quantum mechanics. Section 3 shows how the idea of a quantum amplitude is applied to the construction of invariants of knots and links. Section 4 explains how the generalisation of the Feynman integral to quantum fields leads to invariants of knots, links and three-manifolds. Section 5 is a discussion of a general categorical approach to these issues. Section 6 is a brief discussion of the relationships of quantum topology to quantum computing. This paper is intended as an introduction that can serve as a springboard for working on the interface between quantum topology and quantum computing. Section 7 summarizes the paper. This paper is a thumbnail sketch of recent developments in low dimensional topology and physics. We recommend that the interested reader consult the references given here for further information, and we apologise to the many authors whose significant work was not mentioned here due to limitations of space and reference. II. A Quick Review of Quantum Mechanics To recall principles of quantum mechanics it is useful to have a quick historical recapitulation. Quantum mechanics really got started when DeBroglie introduced the fantastic notion that matter (such as an electron) is accompanied by a wave that guides its motion and produces interference phenomena just like the waves on the surface of the ocean or the diffraction effects of light going through a small aperture. -
Quantum Gravity: a Primer for Philosophers∗
Quantum Gravity: A Primer for Philosophers∗ Dean Rickles ‘Quantum Gravity’ does not denote any existing theory: the field of quantum gravity is very much a ‘work in progress’. As you will see in this chapter, there are multiple lines of attack each with the same core goal: to find a theory that unifies, in some sense, general relativity (Einstein’s classical field theory of gravitation) and quantum field theory (the theoretical framework through which we understand the behaviour of particles in non-gravitational fields). Quantum field theory and general relativity seem to be like oil and water, they don’t like to mix—it is fair to say that combining them to produce a theory of quantum gravity constitutes the greatest unresolved puzzle in physics. Our goal in this chapter is to give the reader an impression of what the problem of quantum gravity is; why it is an important problem; the ways that have been suggested to resolve it; and what philosophical issues these approaches, and the problem itself, generate. This review is extremely selective, as it has to be to remain a manageable size: generally, rather than going into great detail in some area, we highlight the key features and the options, in the hope that readers may take up the problem for themselves—however, some of the basic formalism will be introduced so that the reader is able to enter the physics and (what little there is of) the philosophy of physics literature prepared.1 I have also supplied references for those cases where I have omitted some important facts. -
A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation
Proceedings of Symposia in Applied Mathematics Volume 68, 2010 A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation Samuel J. Lomonaco, Jr. Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum me- chanics and quantum computation, with enough material to begin reading the research literature in quantum computation, quantum cryptography, and quantum information theory. This paper is a written version of the first of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. Part 1 of the paper is a preamble introducing the reader to the concept of the qubit Part 2 gives an introduction to quantum mechanics covering such topics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schr¨odinger’s equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schr¨odinger picture, quantum entanglement, EPR para- dox, quantum entropy. Part 3 gives a brief introduction to quantum computation, covering such topics as elementary quantum computing devices, wiring diagrams, the no- cloning theorem, quantum teleportation. Many examples are given to illustrate underlying principles. A table of contents as well as an index are provided for readers who wish to “pick and choose.” Since this paper is intended for a diverse audience, it is written in an informal style at varying levels of difficulty and sophistication, from the very elementary to the more advanced. 2000 Mathematics Subject Classification. -
Electronic Structure of Full-Shell Inas/Al Hybrid Semiconductor-Superconductor Nanowires: Spin-Orbit Coupling and Topological Phase Space
Electronic structure of full-shell InAs/Al hybrid semiconductor-superconductor nanowires: Spin-orbit coupling and topological phase space Benjamin D. Woods,1 Sankar Das Sarma,2 and Tudor D. Stanescu1, 2 1Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, USA 2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland, 20742-4111, USA We study the electronic structure of full-shell superconductor-semiconductor nanowires, which have recently been proposed for creating Majorana zero modes, using an eight-band ~k · ~p model within a fully self-consistent Schrodinger-Poisson¨ scheme. We find that the spin-orbit coupling induced by the intrinsic radial electric field is generically weak for sub-bands with their minimum near the Fermi energy. Furthermore, we show that the chemical potential windows consistent with the emergence of a topological phase are small and sparse and can only be reached by fine tunning the diameter of the wire. These findings suggest that the parameter space consistent with the realization of a topological phase in full-shell InAs/Al nanowires is, at best, very narrow. Hybrid semiconductor-superconductor (SM-SC) nanowires butions) and ii) the electrostatic effects (by self-consistently have recently become the subject of intense research in the solving a Schrodinger-Poisson¨ problem). We note that these context of the quest for topological Majorana zero modes are crucial issues for the entire research field of SM-SC hy- (MZMs) [1,2]. Motivated by the promise of fault-tolerant brid nanostructures, but they have only recently started to be topological quantum computation [3,4] and following con- addressed, and only within single-orbital approaches [27–30]. -
Quantum Computing and Quantum Topology
Beyond Anyons: From Fractons to Black Holes Zhenghan Wang Microsoft Station Q & UC Santa Barbara Vanderbilt, May 8, 2019 “Periodic Table” of Topological Phases of Matter Classification of enriched topological order (TQFT) in all dimensions Too hard!!! Special cases: 1): short-range entangled (or symmetry protected--SPT) including topological insulators and topological superconductors: X.-G. Wen et al (group cohomology), …, and A. Ludwig et al (random matrix theory) and A. Kitaev (K-theory)---generalized cohomologies,… 2): Low dimensional: spatial dimensions D=1, 2, 3, n=d=D+1 2a: classify 2D topological orders without symmetry 2b: enrich them with symmetry 2c: 3D much more interesting and harder Model Topological Phases of Quantum Matter Local Hilbert Space Local, Gapped Hamiltonian E Two gapped Hamiltonians 퐻1, 퐻2 realize the same topological phase of matter if there exists a continuous path connecting Egap them without closing the gap/a phase transition. A topological phase, to first approximation, is a class of gapped Hamiltonians that realize the same phase. Topological order in a 2D topological phase is encoded by a TQFT or anyon model. Anyons in Topological Phases of Matter Finite-energy topological quasiparticle excitations=anyons b Quasiparticles a, b, c a c Two quasiparticles have the same topological charge or anyon type if they differ by local operators Anyons in 2+1 dimensions described mathematically by a Unitary Modular Tensor Category Anyonic Objects • Anyons: simple objects in unitary modular categories • Symmetry defects: -
Sankar Das Sarma 3/11/19 1 Curriculum Vitae
Sankar Das Sarma 3/11/19 Curriculum Vitae Sankar Das Sarma Richard E. Prange Chair in Physics and Distinguished University Professor Director, Condensed Matter Theory Center Fellow, Joint Quantum Institute University of Maryland Department of Physics College Park, Maryland 20742-4111 Email: [email protected] Web page: www.physics.umd.edu/cmtc Fax: (301) 314-9465 Telephone: (301) 405-6145 Published articles in APS journals I. Physical Review Letters 1. Theory for the Polarizability Function of an Electron Layer in the Presence of Collisional Broadening Effects and its Experimental Implications (S. Das Sarma) Phys. Rev. Lett. 50, 211 (1983). 2. Theory of Two Dimensional Magneto-Polarons (S. Das Sarma), Phys. Rev. Lett. 52, 859 (1984); erratum: Phys. Rev. Lett. 52, 1570 (1984). 3. Proposed Experimental Realization of Anderson Localization in Random and Incommensurate Artificial Structures (S. Das Sarma, A. Kobayashi, and R.E. Prange) Phys. Rev. Lett. 56, 1280 (1986). 4. Frequency-Shifted Polaron Coupling in GaInAs Heterojunctions (S. Das Sarma), Phys. Rev. Lett. 57, 651 (1986). 5. Many-Body Effects in a Non-Equilibrium Electron-Lattice System: Coupling of Quasiparticle Excitations and LO-Phonons (J.K. Jain, R. Jalabert, and S. Das Sarma), Phys. Rev. Lett. 60, 353 (1988). 6. Extended Electronic States in One Dimensional Fibonacci Superlattice (X.C. Xie and S. Das Sarma), Phys. Rev. Lett. 60, 1585 (1988). 1 Sankar Das Sarma 7. Strong-Field Density of States in Weakly Disordered Two Dimensional Electron Systems (S. Das Sarma and X.C. Xie), Phys. Rev. Lett. 61, 738 (1988). 8. Mobility Edge is a Model One Dimensional Potential (S. -
Table of Contents (Print)
NEWSPAPER 97 Kinetic energy (vertical) of deuterons after fragmentation of deuterium molecules in a pump-probe experiment, for a given time delay (horizontal) between the pump and the probe pulses. Colors denote the number of deuterons, with orange-yellow being the highest. See article 193001. PHYSICAL REVIEW LETTERS PRL 97 (19), 190201– 199901, 10 November 2006 (280 total pages) Contents Articles published 4 November–10 November 2006 VOLUME 97, NUMBER 19 10 November 2006 General Physics: Statistical and Quantum Mechanics, Quantum Information, etc. Quantum Feedback Control for Deterministic Entangled Photon Generation .......................................... 190201 Masahiro Yanagisawa General Approach to Quantum-Classical Hybrid Systems and Geometric Forces ..................................... 190401 Qi Zhang and Biao Wu Condensation of N Interacting Bosons: A Hybrid Approach to Condensate Fluctuations ............................. 190402 Anatoly A. Svidzinsky and Marlan O. Scully Dipole Polarizability of a Trapped Superfluid Fermi Gas . ............................................................ 190403 A. Recati, I. Carusotto, C. Lobo, and S. Stringari Loschmidt Echo in a System of Interacting Electrons ................................................................ 190404 G. Manfredi and P.-A. Hervieux Detection Scheme for Acoustic Quantum Radiation in Bose-Einstein Condensates . ................................. 190405 Ralf Schu¨tzhold Quantum Stripe Ordering in Optical Lattices . ........................................................................