This chapter presents an introduction to the theory of quantum fault tolerance and quantum error correction, which provide a collection of techniques to deal with imperfect operations and unavoidable noise afflicting the physical hardware, at the expense of moderately increased resource overheads.

This chapter provides an overview of how to perform quantum error correction using the surface code, which is the most well-studied quantum error correcting code for practical quantum computation. We provide formulas for the code distance—which determines the resource overhead when using the surface code—as a function of the desired logical error rate and underlying physical error rate. We discuss several decoders for the surface code and the possibility of experiencing the backlog problem if the decoder is too slow.

This chapter provides an overview of how to perform a universal set of logical gates on qubits encoded with the surface code, via a procedure called lattice surgery. This is the most well-studied approach for practical fault-tolerant quantum computation. We perform a back-of-the-envelope end-to-end resource estimation for the number of physical qubits and total runtime required to run a quantum algorithm in this paradigm. This provides a method for converting logical resource estimates for quantum algorithms into physical resource estimates.

This chapter explores the origin, key components, and essential concepts of quantum computing. It begins by charting the series of discoveries by various scientists that crystallized into the idea of quantum computing. The text then examines how certain applications have driven the evolution of quantum computing from a theoretical concept to an international endeavour. Additionally, the text clarifies the distinctions between quantum and classical computers, highlighting the DiVincenzo criteria, which are the five criteria for quantum computing. It also introduces the circuit model as the foundational paradigm for quantum computation. Lastly, the chapter sheds light on the reasons for the belief that quantum computers are more powerful than classical ones (touching on quantum computational complexity) and physically realizable (touching on quantum error correction).