• Quantum Algorithms
• Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling
• Measurement-based Quantum Computation
• Topological Quantum Computation
• Quantum error correction
• Classical simulation of quantum systems
• Graph theory in quantum information
• Foundations of quantum mechanics
  

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Quantum Algorithms

Abstract: Quantum information offers the possibility to solve certain problems dramatically faster than is possible with classical computers. In these lectures, we will give an introduction to quantum algorithms. We will begin with an overview of the quantum circuit model and some elementary examples of quantum speedup. Next, we will introduce the quantum Fourier transform and show how it can be used to estimate the eigenvalues of a unitary operator. Using phase estimation, we will describe Shor's algorithm for factoring integers. We will also describe Grover's search algorithm, and will conclude with a discussion of recent quantum algorithms based on quantum analogs of random walks.

Lecturers: Pawel Wocjan, University of Central Florida, Orlando, Florida, USA, and Andrew Childs, University of Waterloo, Ontario, Canada.

Pawel M. Wocjan is Assistant Professor in Computer Science at the University of Central Florida. He received his Ph.D. and his M.S. from the Karlsruhe Institute of Technology in 2003 and 1999, respectively. Prior to joining the University of Central Florida in 2006, he was a postdoctoral scholar in computer science at the Institute for Quantum Information, California Institute of Technology, from 2004 till 2006. He received a National Science Foundation CAREER Award for his research on quantum computing in 2008. His main research interests are quantum algorithms for algebraic problems, quantum-walk-based algorithms, quantum complexity theory, and quantum control theory. His secondary research interests include approximation algorithms, randomized algorithms, online algorithms, algebraic geometry, and number theory.

Andrew Childs received his Ph.D. in physics in 2004 as a Herz Foundation Fellow at the Massachusetts Institute of Technology. He is currently an assistant professor in the Department of Combinatorics and Optimization and the Institute for Quantum Computing at the University of Waterloo. Previously, he was a Lee A. DuBridge Postdoctoral Scholar at the Institute for Quantum Information at the California Institute of Technology. His interests include the theory of quantum information processing, especially algorithms for quantum computers.

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Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling

Lidar
   

Amin
               

Abstract: [Amin] In the first half of the day, adiabatic quantum computation (AQC) will be introduced as a universal scheme of quantum computation and an alternative to the gate model quantum computation. Adiabatic quantum optimization (AQO) will then be introduced as a subset of AQC, and its relation to quantum phase transitions will be discussed. The effect of noise on AQO using realistic environmental noise models will be discussed. We'll end with a presentation of the practical implementation of AQO using superconducting circuitry and some single-qubit and multi-qubit experimental results.

[Lidar] The second part of the day will start by covering several advanced topics in AQC, including a sketch of the proof of the equivalence between AQC and the circuit model, rigorous formulations of the adiabatic theorem, the geometry of AQC, and a time-optimized ("brachistochrone") approach to AQC. We'll then switch gears and provide an introduction to decoherence-free subspaces, noiseless subsystems, dynamical decoupling, and hybrid methods in which they are combined. The emphasis will be on the underlying unifying symmetry principles which enable quantum errors to be avoided by encoding. Time permitting, we'll return to AQC and discuss how it can be made resilient to decoherence.

Lecturers: Mohammad Amin, D-wave Inc., Burnaby, British Columbia, Canada and Daniel Lidar, University of Southern California, Los Angeles, California, USA

Mohammad Amin is a senior scientist at D-Wave Systems Inc. He did a PhD in condensed matter theory under the joint supervision of Profs. Ian Affleck and Philip Stamp at the University of British Colombia (Canada) in 1999, and a short postdoc at Simon Fraser University (Canada), before joining D-Wave in Jan. 2000. His current research interest includes superconducting qubits, adiabatic quantum computation, decoherence, and quantum phase transitions.

Daniel Lidar is a professor of Electrical Engineering and Chemistry at the University of Southern California, and holds a cross-appointment in Physics. He obtained his Ph.D. in Physics from the Hebrew University of Jerusalem in 1997. He was a postdoc at UC Berkeley from 1997 to 2000, then an assistant professor and later associate professor of Chemistry at the University of Toronto from 2000 to 2005, with cross-appointments in Mathematics and Physics.

His research interests lie primarily in the theory and control of open quantum systems, with a special emphasis on quantum information processing. His past interests include scattering theory and disordered systems.

He is the Director and co-founding member of the USC Center for Quantum Information Science & Technology (CQIST). He is a recipient of a Sloan Research Fellowship and is a Fellow of the American Physical Society.

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Measurement-based Quantum Computation

Raussendorf


Browne

Abstract: There are two fundamentally different ways of evolving a quantum state: unitary evolution and projective measurement. Unitary evolution is deterministic and reversible, whereas measurement is probabilistic and irreversible. Despite these differences, it turns out that universal quantum computation can be built on either. In this lecture we are concerned with quantum computation by measurement. We give an introduction to the subject, and discuss various aspects of this field, ranging from physical realization to fault-tolerance and foundations of quantum mechanics.

Lecturers: Dan Browne, University College London, UK, and Robert Raussendorf, University of British Columbia, Vancouver, Canada.

Dan Browne gained his PhD at Imperial College London in 2004 under the supervision of Martin Plenio. The subject of his thesis was implementations of quantum information processing in cavity QED and optics. Before commencing his PhD, he was a DAAD-funded research scholar at LMU university in Munich, working with Hans Briegel and Robert Raussendorf on measurement-based quantum computation. He was a Junior Research Fellow at Merton College Oxford from 2004-2007 and is now a lecturer and Leverhulme research fellow at University College London. His research interests include quantum computation theory, implementations of quantum information processing, and the quantum properties of many-body systems.

Robert Raussendorf is Assistant Professor at the University of British Columbia. He obtained his PhD from the Ludwig Maximilians University in Munich, Germany in 2003. He was postdoc at Caltech and at the Perimeter Institute for Theoretical Physics, and is scholar of the Cifar Quantum Information program and Sloan Research Fellow 2009 - 2011. His main research interest is in models of quantum computation, in particular measurement-based quantum computation, and in fault-tolerance.

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Topological Quantum Computation

     

Abstract: Certain exotic states of matter, so-called non-Abelian states, have the potential to provide a natural medium for the storage and manipulation of quantum information.  In these states, localized particle-like excitations (quasiparticles) possess quantum numbers which are in many ways analogous to ordinary spin quantum numbers. However, unlike ordinary spins, the quantum information associated with these quantum numbers is stored globally, throughout the entire system, and so is intrinsically protected against decoherence. Quantum computation can then be carried out by dragging these quasiparticles around one another so that their trajectories sweep out world-lines in 2+1-dimensional space-time.   The resulting computation depends only on the topology of the braids formed by these world-lines, and thus is robust against error.

In these lectures I will review the theory of non-Abelian states, including the necessary mathematical background for describing the braiding of their quasiparticles.  I will then introduce the basic ideas behind topological quantum computation and demonstrate explicitly that certain non-Abelian quasiparticles can indeed by used for universal quantum computation by showing how any quantum algorithm can be "compiled" into a braiding pattern for them.    I will also discuss the most promising experimental systems for realizing non-Abelian quasiparticles, focusing primarily on fractional quantum Hall states.

Lecturers: Nick Bonesteel, Florida State University, Tallahassee, Florida, USA.

Nick Bonesteel received his Ph.D. from Cornell University in 1991. He is currently Professor of Physics at Florida State University. His research includes generally condensed matter physics, correlated electrons, quantum Hall effect, and quantum computing using topological approaches.

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Quantum error correction

        

Abstract: Errors are likely to be a serious problem for quantum computers, both because they are built of small components and because qubits are inherently more vulnerable to error than classical bits because of processes such as decoherence. Consequently, to build a large quantum computer, we will likely need quantum error-correcting codes, which split up quantum states among a number of qubits in such a way that it is possible to correct for small errors. I will give an overview of the theory of quantum error correction and a discussion of fault-tolerant quantum computation, which applies quantum error-correcting codes to allow more reliable quantum computations. I will cover Shor's 9-qubit code, stabilizer codes, CSS codes, and the threshold theorem, which says that arbitrarily long reliable quantum computations are possible, provided the error rate per gate or time step is below some constant threshold value.

Lecturers: Daniel Gottesman, Perimeter Institute, Waterloo, Ontario, Canada.

Daniel Gottesman is a faculty member at the Perimeter Institute in Waterloo, Ontario. He got his Ph.D. at Caltech in 1997, and did postdocs at Los Alamos National Lab and Microsoft Research, after which he served in the UC Berkeley CS department as a Long-Term CMI Prize Fellow with the Clay Mathematics Institute. His main research interests are quantum error correction, fault-tolerant quantum computation, quantum cryptography, and quantum complexity.

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Classical simulation of quantum systems

Van den Nest


Verstraete
   

Abstract: The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed.

The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’).

The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.

Lecturers: Frank Verstraete, University of Viena, Austria, and Maarten van den Nest, Max-Planck Institute for Quantum Optics, Garching, Germany.

Frank Verstraete received his Ph. D. degree in 2002 at the University of Leuven under supervision of Profs. B. De Moor and H. Verscheld. From 2002 to 2004 he was a research fellow in the theory group of I. Cirac at the Max-Planck Institut für Quantenoptik, Garching, and from 2004 to 2006, a research scholar in the Institute for Quantum Information headed by J. Preskill at Caltech. Since October 2006, he is a Professor at the Fakultät für Physik at the Universität Wien. He is a recipient 2007 HERMANN KÜMMEL Early Achievement Award in Many-body Physics.

Maarten van den Nest received his Ph.D. degree in 2005 at the Katholieke Universiteit Leuven, Belgium. Since 2008, Maarten has been a postdoctoral researcher in the Max Planck Institute for Quantum Optics in Garching, in the group of Ignacio Cirac. In the period 2006-2008 he was a postdoctoral researcher in the group of Hans Briegel in the Institute for Quantum optics and Quantum Information in Innsbruck. He has obtained his PhD in the Katholieke Universiteit Leuven in 2005. Maarten Van den Nest is interested in various theoretical aspects of quantum information and computation. This includes understanding the relationship between quantum and classical computation, the study of measurement-based computing and graph states, as well as the exploration of various connections between quantum information and other fields, such as statistical physics.

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Graph theory in quantum information

               

Abstract: There are a number of significant problems in quantum information where there is an interesting connection with graph theory. Gleason's theorem proves an interesting result about graph coloring.  There are grounds to hope that graph isomorphism can be dealt with more efficiently on a quantum computer.  Discrete quantum walks are defined on graphs.  Graph states underly measurement-based quantum computing and play an important role in quantum codes. Even questions about mub's and sic-povm's, which appear to be entirely geometrical, are related to classical problems in graph theory. I aim to discuss these problems, and to provide an introduction to the related graph theory.

Lecturers: Chris Godsil, University of Waterloo, Ontario, Canada.

Chris Godsil graduated from the University of Melbourne in 1979. He spent two years at the Montanuniversitaet Leoben (Austria) and five years at Simon Fraser University.  In 1987 he moved to the department of Combinatorics and Optimization at the University of Waterloo, where he is still lodged. He has written two books (Algebraic Combinatorics and, with Gordon Royle, Algebraic Graph Theory) and, with Ian Goulden and David Jackson, founded the Journal of Algebraic Combinatorics.  His next book might not have "algebraic" in the title, but nothing is certain.

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Foundations of quantum mechanics

        

Abstract: The field of quantum foundations seeks to answer questions such as: What do the elements of the mathematical formalism of quantum theory represent? From what physical principles can the formalism be derived? What are the precise ways in which a quantum world differs from a classical world and other possible worlds? These lectures will cover some important foundational topics which touch upon these questions, in particular, operational and realist interpretations of the formalism, the quantum measurement problem, nonlocality and contextuality.

Lecturers: Rob Spekkens, Perimeter Institute, Waterloo, Ontario, Canada.

Robert Spekkens received his B.Sc. in physics and philosophy from McGill University and completed his M.Sc. and Ph.D. in Physics at the University of Toronto. He held a postdoctoral fellowship at Perimeter Institute and an International Royal Society Fellowship at the University of Cambridge. He has been a faculty member at Perimeter Institute since November 2008. His research is focused upon identifying the conceptual innovations that distinguish quantum theories from classical theories and investigating their significance for axiomatization, interpretation, and the implementation of various information-theoretic tasks. Topics of current research include: the view that quantum states are states of knowledge rather than states of reality, quantum contextuality, and the information-theoretic approach to reference frames and superselection rules.

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