The classical theory of computation usually does not refer to physics. Pioneers such as Turing, Church, Post and Goedel managed to capture the correct classical theory by intuition alone and, as a result, it is often falsely assumed that its foundations are self-evident and purely abstract. They are not! Computers are physical objects and computation is a physical process. Hence when we improve our knowledge about physical reality, we may also gain new means of improving our knowledge of computation. From this perspective it should not be very surprising that the discovery of quantum mechanics has changed our understanding of the nature of computation. In this series of lectures you will learn how inherently quantum phenomena, such as quantum interference and quantum entanglement, can make information processing more efficient and more secure, even in the presence of noise.

**Prerequisites**

The course material should be of interest to physicists, mathematicians, computer scientists, and engineers. The interdsciplinary nature of this course and your diverse backgrounds means that some of you may find some lectures easy while others find them difficult. The following will be assumed as prerequisites for this course:

- elementary probability, complex numbers, vectors and matrices;
- Dirac bra-ket notation;
- a basic knowledge of quantum mechanics especially in the simple context of finite dimensional state spaces (state vectors, composite systems, unitary matrices, Born rule for quantum measurements);
- basic ideas of classical theoretical computer science (complexity theory) would be helpful but are not essential.

Lectures 2018

Some of the lectures are summarised in my handwritten notes that have not yet been typeset.

- Prerequisite Material (updated 21 Feb 2018) Linear algebra in Dirac notation plus various trivia you should really know about. It would be desirable for you to look through these notes slightly before the start of the course, or early into the course, and attempt all the exercises. But hey, I am not here to tell you how to live your life, use the notes as you see fit. Just don’t look puzzled when I talk about bras and tensor products.The notes are half-baked and may contain errors. Please do let me know if you find any.
- Lecture 1 (updated 26 Jan 2018) Basic concepts, quantum interference, “impossible” logic gates, qubits and single qubit interference (Hadamard, phase, Hadamard),
- Lecture 2 (updated 12 Feb, plus handwritten notes) Hadamard transform, quantum entanglement, entangling gates, c-not, c-U, phase kickback, quantum function evaluation.
- Lecture 3 (updated 31 Jan, only handwritten notes) Quantum function evaluation revisited, three algorithms: Bernstein-Vazirani, Grover, Simon.

**Classes 2018**

**Examinable Topics (2018) **

- fundamentals of quantum theory: addition of probability amplitudes, quantum interference, mathematical description of states and evolution of closed quantum systems (Hilbert space, unitary evolution), measurements (projectors, Born rule);
- definition of quantum entanglement (tensor product structure), Bell states, expressing Bell states in terms of stabiliser generators, modifications of Bell states by local operations on one of the two qubits, measurements on subsystems e.g. on qubits which are entangled with other qubits
- quantum gates and quantum circuits
- quantum evaluation of Boolean functions, phase “kick-back” induced by quantum Boolean function evaluation;
- Hadamard transform, phase “kick-back” induced by controlled-U,
- no-cloning theorem; quantum teleportation;
- quantum algorithms: Deutsch, Bernstein-Vazirani, Simon, Grover
- geometric interpretation of the Grover search algorithm (reflections, rotations)
- very basic understanding of complexity classes (P, NP, EXP, NP complete)
- mathematical description of open quantum systems (density matrices, completely positive maps, partial trace, Born rule for density matrices);
- distinction between positive and completely positive maps
- Bloch vectors, action of quantum operations on the Bloch vector;
- quantum error correction of bit-flip and phase-flip errors,
- n-qubit Pauli group, stabilisers, stabiliser generators, stabilisers and quantum error correction

**Textbooks and reading to complement course material**

- P. Kaye, R. Laflamme and M. Mosca,
*An Introduction to Quantum Computing*. OUP, 2007. This is probably the best textbook for this particular course. - Book for the ambitious. M. Nielsen and I. Chuang,
*Quantum Computation and Quantum Information*. Cambridge University Press, 2000. It is considered the standard textbook in the field. The book was published around 2000, so its treatment of some topics is dated, but there is still no better overview of the whole field. - John Preskill's lecture notes on quantum information theory, available at http://www.theory.caltech.edu/people/preskill/ph229/#lecture
- I personally like
*Quantum Computing since Democritus*by S. Aaronson; idiosyncratic, informative and fun to read (http://www.scottaaronson.com/democritus/)

Supplementary Material

- Beyond the Quantum Horizon by D. Deutsch and A. Ekert, Scientific American, Sep 2012.
- Less reality more security by A. Ekert, Physics World, Sep 2009.
- The Limits of Quantum Computers, by S. Aaronson, Scientific American, Mar 2008.
- A Do-It-Yourself Quantum Eraser by R. Hillmer and P. Kwiat, Scientific American, May 2007.
- Quantum Seeing in the Dark by P. Kwiat et al, Scientific American, Nov 1996
- Physical Limits of Computation by C.H. Bennett and R. Landauer, Scientific American, Jul 1985.
- I always found it an interesting coincidence that the two basic ingredients of modern quantum theory, namely probability and complex numbers, were discovered by the same person, an extra-ordinary man of many talents, a gambling scholar by the name of Girolamo Cardano (1501–1576). Start with this essay