10th Canadian Summer School on Quantum Information
http://hdl.handle.net/2429/29055
2015-05-03T16:35:33ZTopological Quantum Computation (Issued:2010-07-27)
http://hdl.handle.net/2429/30258
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.
2010-07-27T00:00:00ZMeasurement-based quantum computation (Issued:2010-07-29)
http://hdl.handle.net/2429/30257
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.
2010-07-29T00:00:00ZGraph theory in quantum information (Issued:2010-07-19)
http://hdl.handle.net/2429/30256
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.
2010-07-19T00:00:00ZClassical Simulation of Quantum Systems (Issued:2010-07-20)
http://hdl.handle.net/2429/30253
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.
2010-07-20T00:00:00Z