May 28, 2018

August 17-21, 2009
Canadian Quantum Information Summer School

Lecture Abstracts


NMR Quantum Information Processing
Jonathan Baugh

Nuclear magnetic resonance (NMR) was the first testbed system in which many of the fundamental ideas of QIP were physically implemented, such as error correction, teleportation, and benchmarking of quantum control with non-trivial numbers of qubits (recently up to 12). The "language" of NMR QIP, which translates quantum algorithms into RF pulses and spin evolutions, can be applied in some analogous form to nearly every potential implementation, and therefore provides a useful conceptual basis for understanding QIP experiments in a wide variety of systems. I will provide an introduction to both NMR and NMR QIP, from the rotating frame and concept of resonance, to implementation of simple algorithms such as Deutsch-Jozsa.

Advances and prospects in QIP implementations
Jonathan Baugh

The last few years have witnessed an explosion of compelling experimental advances driven by the ultimate goal of realizing quantum computers. I will give a partial survey of recent progress across several fields, including quantum dots, ion traps and superconducting qubits. Major advances and current challenges will be highlighted.

Quantum Error Correction
Daniel Gottesman

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. I will cover Shor's 9-qubit code, stabilizer codes, and CSS codes.

Hybrid quantum error prevention, reduction, and correction methods
Daniel Lidar

These two talks will 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. The talks will cover both the theoretical background and the experimental state of the art.

Quantum Key Distribution: Theory and Practice
Norbert Lutkenhaus

Quantum Key Distribution solves a problem in the field of cryptography. In this lecture I will outline the problem of key distribution and show how quantum mechanics is used solve it. Although usually formulated abstractly with qubits, I show how these protocols can be implemented securely with simple optical devices such as laser sources and threshold photodetectors.

Quantum Computer Algorithms
Michele Mosca

Quantum computer algorithms are able to solve some problems more efficiently than the best known classical algorithms. For some "black-box" problems, the quantum improvement is provable. Feynman's original idea (in the early 1980's) was to use quantum computers to simulate quantum mechanical systems exponentially more efficiently than the best known classical algorithms. Shor's algorithms solve the integer factorization problem and the discrete logarithm problem, which are at the core of all the widely used public-key cryptosystems. Grover's quantum search algorithm solves a black-box searching problem with quadratically fewer queries than is possible with a classical algorithm. Many more quantum algorithms and algorithmic tools have been developed since these seminal results in the mid-1990's.

I will introduce some of the basic principles and tools behind quantum algorithms, survey some of the main known algorithms, and discuss future directions.

Implementing quantum information with light
Barry C. Sanders

We will see how quantum information can be encoded into light: through polarization, path, timing, frequency, or a combination thereof. Quantum key distribution is experimentally successful and provides an excellent example of how photons are excellent carriers of quantum information so we will study this example in detail. In addition we will learn how a universal set of optical quantum gates can be constructed so photonic quantum computers can be built.

Implementing quantum information in silicon
Barry C. Sanders

Of all the candidates for quantum information media, silicon (Si) is the most appealing. Si-based quantum information processing could exploit the enormous investment in Si chips made by the computer industry. A Si-based quantum chip may be able to talk easily to standard computer chips. Furthermore, quantum information can be encoded into nuclear or electron spins of dopants (e.g., Phosphorous = P) or as charge qubits whereby one excess electron is shared between two dopants. We will study Kane's original proposal for Si:P quantum computing, wherein quantum information is encoded into the nuclear spin of P. Then we will study the sophisticated Australian "update" of Kane's scheme. This scheme aims to build a Si-based quantum computer wherein quantum information is encoded into the spin of P's outermost electron.

Guifre Vidal

The basic concepts required to describe entanglement will be introduced: Schmidt decomposition, LOCC, entanglement monotones, measures of entanglement, ...

Entanglement in quantum many-body systems
Guifre Vidal

Using the concepts introduced in the previous section, I will explain what makes entanglement in many-body systems (e.g. ground states of typical local Hamiltonians) very special. Namely, many-body systems are typically only weakly entangled. The content of this lecture may serve as a basic introduction to the ideas discussed by Matthew Hastings in his second and third Distinguished Lectures.

Quantum Computational Complexity
John Watrous

In 1994 Peter Shor discovered efficient quantum computer algorithms for factoring integers and computing discrete logarithms -- problems that are conjectured not to be efficiently solvable using ordinary classical computers. These algorithms are the most well-known among several examples that suggest that quantum information has a profound effect on the general notion of computational difficulty. Quantum computational complexity studies this topic in a variety of settings that model not only quantum computations, but also the verification of quantum proofs, quantum interactions of various types, and their relationships to analogous classical concepts.

In this talk, which will be divided into two parts, I will introduce the basic notions of quantum computational complexity and survey some of the main results in this area. The talk will focus on three fundamental notions in quantum computational complexity: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems.

Asymptotic theory of quantum communication
Jon Yard

I will lecture on the asymptotic theory of quantum communication. The goal is to determine how much information can, in principle, be encoded into a collection of noisy quantum systems so that it can be retrieved with negligible error as the number of systems grows. The noise is modeled by a completely positive, trace preserving linear map on density matrices, otherwise known as a quantum channel, and we will be interested in the case where the same channel acts independently on each of the systems. I will primarily focus on the quantum capacity of a given channel, which measures the number of qubits per transmission that can be reliably protected. Main topics to be covered are:

1. Computing the quantum capacity. For certain classes of channels, we know how to write down an easily computable, closed-form formula for the quantum capacity. In other cases, the best we have is an open-form "regularized" expression, which is too unwieldy to be more than formally useful. I will outline what is known here and show that quantum capacity has some surprising behavior as well, such being a non-additive of the channel.

2. I will outline a proof of the LSD (Lloyd, Shor, Devetak) coding theorem, which shows that asymptotically good codes can be constructed by selecting a random subspace of the inputs of the channels, provided the communication rate is less than the quantum capacity. Two subtopics that will be necessary to cover are:

2.1 Approximate error correction. The usual theory of quantum error correction focuses on perfectly correcting some set (actually, a
subspace) of quantum errors. When the noise is modeled by a quantum channel, one can speak of approximately correcting encoded information in a meaningful and quantitative way.

2.2 The method of types. A basic tool coming from classical information theory, statistics and large deviation theory aiding the analysis of sequences of i.i.d. (independent and identically
distributed) random variables. It is an indispensable tool for studying channel capacities.

3. Time permitting, I will also discuss capacities for transmitting classical and private information, and how they are related to the quantum capacity.

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