Quantum Algorithms in Open Quantum Systems

This book offers a comprehensive introduction to quantum computation in the presence of noise: how noise arises in real quantum hardware, how it limits computation, and the strategies developed to correct and mitigate it. Designed to be accessible, it requires only a basic knowledge of quantum mechanics. The book begins by laying the foundations of quantum computation — qubits, gates, universality, and computational complexity — and works through the canonical quantum algorithms, from Deutsch–Jozsa and Grover to the quantum Fourier transform, phase estimation, Shor's factoring algorithm, and quantum key distribution. It then turns to variational quantum algorithms as a practical, near-term application of noisy devices. Next, it characterises the main sources of noise affecting quantum systems and develops the central ideas of quantum error correction, building from the repetition codes to the stabiliser formalism, the surface code, and the principles of fault-tolerant and measurement-based quantum computation. While fully fault-tolerant quantum computation remains a future goal, the book also explores techniques to suppress and mitigate noise on present-day hardware, such as dynamical decoupling and quantum error mitigation, framed within the theory of open quantum systems. Throughout the text, worked examples and exercises are provided.