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.

Matteo Carlesso earned his PhD in Theoretical Physics from the University of Trieste in 2018, working under the supervision of Prof. Angelo Bassi on open quantum system dynamics, decoherence, and collapse models. He then held postdoctoral positions at the University of Trieste, working on tests of the foundations of quantum mechanics, and, from 2021, at Queen's University Belfast, where his research expanded to quantum thermodynamics.

Since 2023 he has been a Research Associate at the University of Trieste. His research spans open quantum systems, the foundations of quantum mechanics, tests of the limits of quantum theory, quantum optomechanics, quantum thermodynamics, and quantum control. Since spring 2024 he has taught the course Quantum Algorithms in Open Quantum Systems — on which this book is based — to master's students in Theoretical Physics and in Quantum Science and Technology.