Lecturer	Andreas Doering
Degrees	Schedule C1 — Computer Science Schedule C1 — Mathematics and Computer Science Schedule C — MSc in Advanced Computer Science MSc in Mathematics and Foundations of Computer Science
Term	Hilary Term 2012  (24 lectures)

Overview

Both physics and computer science have been very dominant scientific and technological disciplines in the previous century. Quantum Computer Science aims at combining both and may come to play a similarly important role in this century. Combining the existing expertise in both fields proves to be a non-trivial but very exciting interdisciplinary journey. Besides the actual issue of building a quantum computer or realising quantum protocols it involves a fascinating encounter of concepts and formal tools which arose in distinct disciplines.

Remark: Students who intend to write their MSc thesis in the Quantum Group at Comlab should also take the Categories, Proofs and Processes course in Michaelmas term.

This course provides an interdisciplinary introduction to the emerging field of quantum computer science, explaining basic quantum mechanics (including finite dimensional Hilbert spaces and their tensor products), quantum entanglement, its structure and its physical consequences (e.g. non-locality, no-cloning principle), and introduces qubits. We give detailed discussions of some key algorithms and protocols such as Grover's search algorithm and Shor's factorisation algorithm, quantum teleportation and quantum key exchange, and analyse the challenges their significance for computer science, mathematics etc. We also provide a more conceptual semantic analysis of some of the above. Other important issues such as quantum information theory (including mixed states) will also be covered, although not in great detail. We mainly discuss the circuit model and briefly mention alternative computational paradigms like measurement-based quantum computing, we argue the need for high-level methods, provide some recent results concerning a graphical language and categorical semantics for quantum informatics and delineate the remaining scientific challenges for the future.

Learning outcomes

The student will know by the end of the course what quantum computing and quantum protocols are about, why they matter, and what the scientific prospects of the field are. This includes a structural understanding of some basic quantum mechanics, knowledge of important algorithms such as Grover's and Shor's algorithm and important protocols such as quantum teleportation. The student will also know where to find more details and will be able to access these. Hence this course also offers computer science and mathematics students a first stepping-stone for research in the field, with a particular focus on the newly developing field of quantum computer science semantics, to which Oxford University Computing Laboratory has provided pioneering contributions.

Prerequisites

We do not assume any prior knowledge of quantum mechanics. However, a solid understanding of basic linear algebra (finite-dimensional vector spaces, matrices, eigenvectors and eigenvalues, linear maps etc.) is required as a pre-requisite. The course notes and the slides contain an overview of this material, so we advise students with a limited background in linear algebra to consult the course notes before the course starts.

Synopsis

1. Some background

• Classical physics
• Quantum physics
• Quantum computer science

    
2. Hilbert spaces and operators

• Inner product and orthonormal bases
• Adjoints
• Unitary, self-adjoint and projection operators
• The spectral theorem

3. Quantum theory and qubits

• Von Neumann formalism
• Dirac notation
• Qubits
• The Bloch sphere

4. Entanglement

• Direct sum and tensor product of Hilbert spaces
• Composite systems and entanglement
• Bell nonlocality
• No-cloning theorem

5. Protocols from entanglement

• Bell basis
• Teleportation
• Entanglement swapping

6. The structure of entanglement and graphical calculus

• Graphical calculus for quantum theory
• Map-state duality and compositionality
• The logic of bipartite entanglement

7. Quantum algorithms

• Gates and circuits
• Mixed states, trace and partial trace
• The concept of quantum computation
• Grover's search algorithm
• Shor's factoring algorithm
• Quantum key distribution

8. Quantum logic

• The lattice of projections
• Gleason's theorem
• Kochen-Specker theorem

9. More on mixed and entangled states

• Majorisation
• POVMs
• LOCC transformations
• Measuring bipartite entanglement

10. Semantics for quantum computing

• Quantum computing with relations
• Quantum computing with categories

Syllabus

Finite dimensional vector space, inner product, complex numbers, linear adjoints, unitary maps, self-adjoint operators, projectors, trace, tensor product of Hilbert spaces, Dirac notation, bit, qubit, entanglement, map-state duality, no-cloning, quantum circuits, quantum gates, Shor's algorithm, Grover's algorithm, quantum teleportation, quantum key-exchange, teleportation and measurement based quantum computing, decoherence, mixed states, quantum information, quantum logic, quantum categorical semantics.

Reading list

Reading list

Lecture notes which cover the whole course and which provide detailed pointers to additional reading will be made available. Moreover, there will be slides. The following book can serve as a standard reference:

•     Mermin, N.D. (2007), Quantum Computer Science, Cambridge University Press

Other standard books on the subject that might be of use are:

•     Gruska, J. (1999), Quantum Computing, McGraw-Hill
•     Nielsen, M. and Chuang, I. L. (2000), Quantum Computation and Quantum Information, Cambridge University Press
•     Kitaev, A. Yu., Shen, A. H. and Vyalyi, M. N. (2001), Classical and Quantum Computing, Graduate Studies in Mathematics 47, American Mathematical Society

On-line available courses elsewhere which can be consulted are:

•     By Braunstein: http://www.weizmann.ac.il/chemphys/schmuel/comp/comp.html
•     By Pub http://www.theory.caltech.edu/people/preskill/ph229/
•     By Preskil http://www.theory.caltech.edu/people/preskill/ph229/

A different angle which is also very much reflected in the course is available at:

•     http://arxiv.org/abs/quant-ph/0510032
•     http://arxiv.org/abs/quant-ph/0506132

Taking our courses

This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses

Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.