Introduction to Quantum Computing

Lectures:
1. Fundamental Properties of a Qubit and the Bloch Sphere: Classical bit versus quantum bit; qubit states; superposition; geometric representation using the Bloch sphere; examples of basic states (∣0⟩, ∣1⟩, and ∣+⟩).

2. Qubits and Their States, Dirac Notation: Fundamentals of linear algebra in quantum information science; Dirac notation; tensor products; description of multi-qubit states; separability and quantum entanglement.

3. Reversible Operations on Qubits and Qubit Measurement: Unitary operations; Pauli matrices; the Hadamard gate; measurement in the computational basis; wave-function collapse; probabilistic nature of measurement outcomes.

4. Quantum Entanglement: Formal definition; Bell states; the role of quantum entanglement in quantum algorithms and quantum communication.

5.Deutsch–Jozsa and Grover’s Algorithms: The Deutsch–Jozsa algorithm as an elementary demonstration of quantum speedup; comparison of classical and quantum solutions. Principles of quantum search; oracle and diffusion operator; quadratic speedup over classical search.

6.Quantum Phase Estimation and the HHL Algorithm: Phase kickback as a building block of quantum phase estimation (QPE); extraction of the eigenvalues of unitary operators from their eigenstates; application of QPE in the HHL algorithm for solving systems of linear equations.

7. Quantum Fourier Transform and Shor’s Algorithm: Mathematical foundations of the quantum Fourier transform (QFT); its efficient implementation; Shor’s factorisation algorithm and its significance for cryptography.

8. RSA and Cryptography: Principles of classical cryptography and RSA; application of quantum factorisation to breaking RSA. Quantum cryptography: quantum key distribution and applications of quantum communication.

9. Variational Quantum Algorithms: Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimisation Algorithm (QAOA); hybrid approaches combining classical and quantum computational resources.

10. Quantum Machine Learning: Quantum kernel methods; data encoding for quantum algorithms; quantum neural networks. (Suitable literature needs to be added.)

11. Introduction to Quantum Error Correction: Noise and decoherence in quantum computers; an example of a simple repetition code.

12. Error Diagnosis and Quantum Error-Correcting Codes: Syndrome measurements; principles of stabiliser codes; examples of quantum error models and their correction.

Practical Sessions:

1. Installation and First Steps: Installation of Qiskit and access to the IBM Quantum Platform; construction of the first quantum circuit and its execution on a simulator.

2. Tensor Algebra and Interpretation of Quantum Circuits: State-vector visualisation; working with simple two-qubit states.

3. Reversible Operations and Measurement: Implementation of Pauli and Hadamard gates; simulation of measurements; probability distributions of measurement outcomes; single-qubit resonator.

4. Quantum Entanglement in Practice: Generation of Bell pairs and verification of entanglement; experiments with multi-qubit states; quantum teleportation.

5. Deutsch–Jozsa and Grover’s Algorithms: Implementation of the Deutsch–Jozsa algorithm and comparison with its classical counterpart; implementation of Grover’s algorithm for searching a small database; comparison with classical search.

6. Quantum Phase Estimation: Implementation of the phase-kickback effect and the quantum phase estimation algorithm.

7. Quantum Fourier Transform: Implementation of the QFT in Qiskit; analysis of its complexity and outputs.

8. Shor’s Algorithm: Simulation of the factorisation of small integers; limitations of current quantum hardware.

8. Variational Quantum Algorithms: Construction of a simple variational quantum circuit; implementation of VQE and QAOA; hybrid approaches combining classical and quantum computational resources.

10. Data Encoding and Quantum Machine Learning: Data-encoding methods for quantum algorithms; construction of a neural network with a quantum layer.

11. Quantum Error Correction and Cryptography: Implementation of a simple repetition code; demonstration of the BB84 protocol on a simulator.

12. Tests and Discussion: Two class periods are reserved for continuous assessment tests and discussion of their solutions.

Project:

Individual assignment involving the implementation of a quantum algorithm—such as Grover’s, Shor’s, or Simon’s algorithm—or a quantum cryptographic protocol on an ideal quantum simulator and either a real quantum computer or a noisy simulator. The required outputs are the source code, an experimental report, and a discussion of the results.