MATH31020: Quantum Computing

• Analysis in Many Variables and Mathematical Physics

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• To provide an introduction to the application of quantum systems to processing information, specifically in terms of communication and computing. To study the concept of quantum entanglement and demonstrate that quantum systems have properties that are fundamentally different from those of classical systems.

• Quantum Mechanics Introduction. Review of wave mechanics, introduction of Dirac notation and the density matrix.
• Quantum Information. The qubit, Bloch sphere, bipartite systems and concept of pure and mixed states.
• Quantum properties and applications. Superdense coding, teleportation, quantum key distribution, EPR paradox, Hidden variable theories and Bell inequalities.
• Information, entropy and entanglement. Brief introduction to classical information theory including Shannon information and entanglement. Quantum entropy measures, von Neumann entropy, relative entropy and conditional entropy.
• Classical computing. Universal gates/circuit models, very brief discussion of computational complexity.
• Quantum computing. Quantum circuit model and universal gates, example algorithms (e.g. Grover's and Shor's), brief discussion of quantum computational complexity and comparison to classical examples (e.g. Shor's algorithm in context of RSA cryptography.)
• Quantum error correction. Contrast to classical use of redundancy, examples of single qubit errors, use of entanglement to correct errors, example of Shor code. Discussion of error correction in quantum computing, including fault tolerant gates.

Subject-specific Knowledge:

• By the end of the module students will: be able to solve novel and/or complex problems in Quantum Information.
• have a systematic and coherent understanding of theoretical mathematics in the field of Quantum Information.
• have acquired coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
• understand concepts of pure and mixed states and bipartite systems
• Hidden variable theory and the EPR paradox
• Classical and quantum entropy measures
• Classical and Quantum computing
• Quantum error correction

Subject-specific Skills:

• In addition students will have specialised mathematical skills in the following areas which can be used in minimal guidance: Modelling.

Key Skills:

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

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