Supervisor: Simon Ross
Project's research area: Mathematical Physics
Description
Quantum mechanics provides a fundamentally different model of reality to classical mechanics: the fundamental description of the world is through the wave function, a vector in a Hilbert space. Observable quantities are represented by operators acting on the wave function. Quantum systems can be in a superposition of eigenstates of an observable, and measurements probabilistically select one state.
Differences between quantum and classical systems are strikingly illustrated in the context of computation. The state of a classical computer can be represented by a string of bits - zeros and ones. For a quantum computer, we can formulate the analogue of a bit, called a qubit, by considering a two-dimensional Hilbert space, with two basis states labelled by 0 and 1. The state of a qubit is in general some superposition of these basis states. When we have several qubits, the Hilbert space is the tensor product of the Hilbert space of the individual qubits. The most general state is a sum of products of states of the individual qubits: we say it is entangled.
These properties of superposition and entanglement lead to a much richer set of possible operations in a quantum computer. This has been exploited to devise quantum algorithms, which would allow us to solve certain problems much more efficiently on a quantum computer than we can classically. This has motivated an intense effort to build quantum computers.
The goal of the project is to study the theory underlying quantum computers. The project will focus on an abstract description of quantum computers and the conceptual issues rather than practical details. Studying this provides both a road into an active area of current technological development, and a hands-on understanding of "quantum weirdness", getting to grips with the differences between the quantum and classical worlds.
Group project
The group project will involve understanding the use of quantum operations on a Hilbert space to perform computations. By the end of the group project we would have learned:- Basic quantum operations, entanglement and measurement
- Quantum circuit notation
- Simple examples of quantum algorithms
- Shor's algorithm and factoring
Mode of Operation and Evidence of Learning for the group project
The project will revolve around learning through reading with focus on the underlying theory and the development of conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Individual project
The individual project will build on the knowledge we have gained in the group project and will explore additional advanced topics, either in quantum computing or a related area of quantum information. Potential directions include:
- Noise and quantum error correction
- Quantum simulation
- Quantum complexity theory
- Entanglement, mixed states and von Neumann entropy
Mode of Operation and Evidence of Learning for the individual project
The project will revolve around learning through reading with focus on the underlying theory and the development of conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Pre-requisites and Co-requisites
- Pre-requisite: Linear Algebra I
- Co-requisite: Quantum Mechanics III. I will not strictly require this, as we will develop the basics of quantum mechanics that we need during the project, but I expect it will be very helpful to take this alongside the project.