Spring, 2012 Lectures
- Retractive Groups
Keye Martin, Naval Research Laborator
When: March 16, 2012, 1:00–1:50pm
Where: Rome Hall (801 22st Street, NW), Room 459
Abstract: One goal of our current research is to define a new area called algebraic information theory. It began with the realization that many important classes of channels, both quantum and classical, possess the structure of a compact affine monoid. The idea is then to use this structure as the basis for new techniques in information theory.
In practice, many classes of channels arise as the convex closure of a certain underlying group, and among these, certain groups distinguish themselves in that they generate very restricted classes, which are remarkable in that they always contain the solution to optimization problems posed over the set of all channels. These groups, which we call retractive, can also be used to derive useful inequalities, devise methods for tomography of quantum channels and appear to always generate channels whose capacities can be expressed in closed form.
Speaker's Bio: Keye Martin is a senior research mathematician at the Naval Research Laboratory. He is the founder of the informatic phenomena group and the director of its quantum optics lab. - An Introduction to Infinite-Dimensional Categorical Quantum Computing
Clarke Smith, GWU
When: March 2, 2012, 2:00–2:50pm
Where: MPA (805 21st Street), Room 303
Abstract: Category theory has proven promising in capturing the logic of quantum information processing at a fairly high level, in similar fashion to Boolean logic and classical computing. In particular, quantum state evolution and quantum teleportation have been able to be depicted by the category of finite-dimensional Hilbert spaces together with linear transformations. Since all categories behave identically by definition, we can then view quantum computation in a highly intuitive, diagrammatic language. By generalizing the category of finite-dimensional Hilbert spaces to the category of infinite-dimensional Hilbert spaces, we can begin to represent categorically the mathematics of quantum mechanics, which involves observables and bases in arbitrary dimension. We show that this generalization might be achieved by expanding our use of unital Frobenius algebras to nonunital Frobenius algebras and H*-algebras.
Fall, 2011 Lectures
- The magic behind quantum computing: Square Root of (-1)
Jerzy Kocik, Southern Illinois University
When: December 4, 2011, 2:00–2:50pm
Where: Rome Hall (801 22nd Street), Room 459
Abstract: The soothingly graspable formalism of Quantum Mechanics (comprising of quite elementary concepts of linear algebra) contrasts strongly with profound interpretational problems of this formalism. Hence, not to discourage a reader, most expositions quickly move to the formalism and technical description of quantum algorithms, leaving a mathematician not trained in physics somewhat perplexed. This gentler introduction to quantum computing honestly presents the strangeness of quantum nature of reality and is aimed to a non-physicist who ponders why quantum computers are possible.
Speaker's Bio: Jerzy Kocik's interests lie in mathematics motivated by physics and range from Lie algebras (including Berezin quantization) to differential geometry to pure geometry. He is currently investigating integral Apollonian disk packings and discovering their unexpectedly rich connections with number theory, geometry, group theory and physics. In 2010 he received the MAA Lester Ford Award for the article "Disentangling a triangle" published in the Monthly. - Adiabatic quantum computing: equivalence with quantum computing
William de la Cruz, Center of Research and Advanced Studies of IPN, Mexico City
When: Thursday, November 3, 2011 1:00–2:00 p.m.
Place: MPA (805 21st Street), Room 303
Abstract: The adiabatic quantum computing (AQC) was originally introduced to solve optimization problems by constructing two Hamiltonian operators where the first one is easy to prepare and the second one codifies the solution of the considered problem. Van Dam et al. (2001) proved that AQC performs universal computing by showing that the adiabatic evolution can be simulated with quantum circuits of polynomial size. In this talk we review van Dam's construction in order to understand the complexity of AQC and its limitations. - Jointly with Math Colloquium and Applied Math Seminar:
Topological Quantum Computation
Zhenghan Wang, Microsoft Research
When: Thursday, October 6, 2011, 3:45–5:00 pm
Where: Funger Hall (2201 G Street), Room 210
Abstract: Quantum computing models have the potential to perform tasks such as factoring integers and simulating quantum physics exponentially faster than any known classical algorithms, thus revolutionizing information science. But the construction of a large-scale quantum computer is still in its infancy due to the decoherence of quantumness. One promising way to defeat decoherence is via topology. I will give an introduction to this approach to building a large-scale quantum computer, as pursued at Microsoft Station Q (http://stationq.ucsb.edu/), and discuss the mathematical and scientific challenges.
Speaker's Bio: Dr. Zhenghan Wang is a senior researcher at Station Q, Microsoft Research located on UC Santa Barbara campus, and an adjunct professor of UC Santa Barbara Math Dept. Before joining Microsoft, he was on the faulty of Indiana University from 1996-2007. His main interests are quantum topology, theoretical models of topological phases of matter, and their application to quantum computing. - Adiabatic quantum computing: application to NP-hard problems
William de la Cruz, Center of Research and Advanced Studies of IPN, Mexico City
When: Thursday, September 22, 2011; 1:00–2:00 p.m.
Place: MPA (805 21st Street), Room 303
Abstract: Adiabatic quantum computing (AQC) have been shown to be a useful tool for approximating optimization problems. We show an experimental study of the AQC applied to the MaxSat problem. - Adiabatic quantum computing: the construction of Hamiltonian operators
William de la Cruz, Center of Research and Advanced Studies of IPN, Mexico City
Place: MPA (805 21st Street), Room 303
When: Tuesday, September 13, 2011, 1:00–2:00 p.m.
Abstract: Adiabatic Quantum Computing (AQC) has been applied to solve optimization problems. It is based on the construction of Hamiltonian operators which codify the optimal solution of the given optimization problem. AQC makes use of the Adiabatic Theorem to approximate solutions of the Schrödinger equation in which a slow evolution occurs. The Hamiltonian operators used in AQC should be local for convenience. Local Hamiltonian operators are expressed as sums of Hamiltonians operating over a reduced number of qubits.
Speaker's Bio: William de la Cruz is visiting GW in Fall 2011. He is a doctoral student from the Center of Research and Advanced Studies of IPN, Mexico City, where his M.Sc. degree in computer science. His visit to GWU is supported by the National Council of Science and Technology at Mexico City. His research interests include computational complexity, quantum computing and quantum information, computer vision and simulation, and second order logic.
Past lectures
- Alexei Kolesnikov, Towson University, Wednesday, May 4, 2011, 3:45–5:00 pm
Special Joint Quantum Computing – Logic Seminar: Generalized Amalgamation and Homology in Model Theory
The first part of this talk will be a non-technical survey of generalized amalgamation properties in model theory, with focus on Shelah's and Zilber's work on excellent classes. The recent research of the speaker with John Goodrick and Byunghan Kim on the construction of homology groups is motivated, in part, by the desire to better understand generalized amalgamation. The second part of the talk will focus on the construction of homology groups for certain families of functors whose properties are motivated by model theory. - Samson Abramsky, University of Oxford, Friday, April 29,
The topology of non-locality and contextuality.
Bell's theorem famously shows that no local theory can account for the predictions of quantum mechanics; while the Kochen-Specker theorem shows the same for non-contextual theories. Non-locality, and increasingly also contextuality, play an important role as computational resources in current work on quantum information. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the inter-relationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. In particular, we give a unified account of contextuality and non-locality in this setting.- A central result is that an empirical model can be extended to all sets of measurements if and only if it can be realized by a factorizable hidden-variable model, where factorizability subsumes both non-contextuality and Bell locality. Thus the existence of incompatible measurements is the essential ingredient in non-local and contextual behavior in quantum mechanics.
- We give hidden-variable-free proofs of Bell style theorems.
- We identify a notion of strong contextuality, with surprising separations between non-local models: Hardy is not strongly contextual, GHZ is.
- We interpret Kochen-Specker as a generic (model-independent) strong contextuality result.
- We give general combinatorial and graph-theoretic conditions, independent of Hilbert space, for such results.
Speaker's Bio: Samson Abramsky is Christopher Strachey Professor of Computing and a Fellow of Wolfson College, Oxford University. Previously he held chairs at the Imperial College of Science, Technology and Medicine, and at theUniversity of Edinburgh. He holds MA degrees from Cambridge and Oxford, and a PhD from the University of London. He is a Fellow of the Royal Society (2004), a Fellow of the Royal Society of Edinburgh (2000), and a Member ofAcademia Europaea (1993). He has played a leading role in the development of game semantics, and its applications to the semantics of programming languages. Other notable contributions include his work on domain theory in logical form, the lazy lambda calculus, strictness analysis, concurrency theory, interaction categories, and geometry of interaction. More recently, he has been working on high-level methods for quantum computation and information, and on quantum foundations as part of the mathematical foundations of information flow. - Special Joint Quantum Computing &ndash Logic Seminar
John Goodrick, University of Andes, Bogotá, Colombia, Wednesday, April 20, 2011. Homology Groups for Types in Model Theory.
We present definitions of homology groups Hn(p) for a complete type p in a stable (or simple, or rosy) theory. We show how these groups relate to certain previously studied amalgamation properties. We can compute H2(p) “explicitly” for strong types in stable theories and show that the groups that can occur as H2(p) are precisely the profinite abelian groups. -
Zbigniew Oziewicz, Universidad Nacional Autonoma de Mexico, Thursday, March 31, 2011. Applied Category Theory: Graph-Operad Logic (Unified Approach to Frobenius Algebras: Associative and Non-Associative)
We are looking for necessary and sufficient conditions on low-dimensional algebras to be Frobenius algebras. We introduce the concept of a solvable Frobenius algebra. We formulate Frobenius algebra within the abelian monoidal category of operad of graphs. -
Peter Selinger, Dalhousie University, Canada, Friday, March 4, 2011. Higher-order quantum computation
In programming languages, a higher-order function is a function for which the input or output is another function. In this talk, I will discuss various approaches for combining higher-order functions and quantum computation. The best-known examples of higher-order quantum functions are functions operating on "black boxes." I will argue that many (perhaps all) of the interesting phenomena of quantum information theory actually take place at higher-order types, although this is not how they are usually presented.Speaker's bio: Peter Selinger received his PhD in mathematics from the University of Pennsylvania in 1997, and held postdoctoral positions at the University of Michigan, Aarhus University in Denmark, and Stanford University, before joining the University of Ottawa. Since 2005, Selinger is an associate professor of mathematics and computer science at Dalhousie University. He is a recipient of NSERC Discovery Grant, of Ontario Premier's Research Excellence Award, as well as a Canada Foundation for Innovation New Opportunities Award. He has served on the editorial board of the journals Mathematical Structures in Computer Science and Logical Methods in Computer Science, and has been the founder and an organizer of the annual workshop on Quantum Physics and Logic.
- Jeffrey Bub, Distinguished University Professor, Philosophy Department and Institute for Physical Science and Technology, University of Maryland, Friday, February 4, 2011. Why the Tsirelson Bound?
Wheeler's question 'why the quantum' has two aspects: why is the world quantum and not classical, and why is it quantum rather than superquantum, i.e., why the Tsirelson bound for quantum correlations? In a recent Nature article (461:1101–1104, October 2009), Pawlowski, et al., provide an information-theoretic derivation of the Tsirelson bound from a principle they call 'information causality.' I review the original derivation and the information-theoretic principle involved, and consider the significance of the result.
Slides are available here. - Wes Campbell, Joint Quantum Institute, NIST and University of Maryland, Friday, November 5, 2010, Speeding Up Trapped Ion Quantum Computing
The basics of quantum information processing with trapped atomic ions will be presented with a focus on the current technical challenges associated with maintaining coherence and scaling up the system to more ions. Our work at the Joint Quantum Institute addresses some of these problems by utilizing ultrashort optical pulses from mode-locked lasers to perform qubit operations. We experimentally demonstrate two distinct regimes of the interaction between hyperfine atomic ion qubits and stimulated Raman transitions driven by picosecond pulses from a far off- resonant mode-locked laser. In the weak pulse regime, the coherent accumulation of successive pulses from an optical frequency comb performs single qubit operations and is used to entangle two trapped atomic ion qubits. In the strong pulse regime, a single pulse is used to implement a fast (less than 50ps) spin flip and we show how a few pulses may be used to address the atom's motion by imparting state-dependent momen tum kicks. To entangle multiple ions, optical frequency combs operated near the strong pulse regime may be used to implement motion-mediated gates that can be performed much faster than a collective motional period. - Ali Eskandarian, GWU, Friday, October 22, 2010, On the Foundations of Quantum Computing: The Case for a Different Logic
After a brief review of the principles of quantum theory, a gedanken experiment (due to Mermin) based on Hardy's two-particle quantum states is investigated. The conclusions are similar to the results of experiments based on Greenberger-Horne-Zeilinger (GHZ) type three-particle quantum states in pointing out the inadequacies of classical rationalism relying on an intuitive notion of "local" reality. The case is made for new connections between physics and logic that explore the empirical roots (or applications) of the latter, and that could have important bearing on quantum computing theory and practice. - Chi-Kwong Li, The College of William & Mary, Friday, October 1, 2010,
Quantum Operations and Completely Positive Linear Maps
Suppose the states of a quantum system are represented as density matrices. A quantum operation on a closed system will be a unitary similarity transform and a quantum operation on an open system will be a (trace preserving) completely positive linear map. [The operations may be time dependent.] We will discuss the conditions on two given families of quantum states for the existence of a quantum operation sending one family to the other one.
Slides available here. - Jennifer Chubb, University of San Francisco, Friday, September 17, 2010
The what and why of quantum computing — for undergraduates (or anyone unacquainted with the subject) - Christopher Monroe, Joint Quantum Institute, University of Maryland & NIST, Friday, February 19, 2010; 4:00–5:00pm
Quantum information hardware - Samuel J. Lomonaco, UMBC, Saturday, December 5, 2009.
Quantum Knots and Lattices, (joint work with Louis K. Kauffman, U. of Illinois at Chicago). - Samuel Lomonaco, UMBC, Friday, October 16, 2009.
A Rosetta Stone for Quantum Computing - Dr. Mark E. Wilde, McGill University, Friday, September 18, 2009
Claude Shannon Meets Quantum Mechanics: An introduction to quantum Shannon theory