Chapter 10 Superoperators and its Properties

Vasily Evgenevich Tarasov

Vasily Evgenevich Tarasov

Wolfgang Mauerer

Abstract As implementations of quantum computers grow in size and maturity, the question of how to program this new class of machines is attracting increasing attention in the software engineering domain. Yet, many questions from how to design expressible quantum languages augmented with formal semantics via implementing appropriate optimizing compilers to abstracting details of machine properties in software systems remain challenging. Performing research at this intersection of quantum computing and software engineering requires sufficient knowledge of the physical processes underlying quantum computations, and how to model these. In this chapter, we review a superoperator-based approach to quantum dynamics, as it can provide means that are sufficiently abstract, yet concrete enough to be useful in quantum software and systems engineering, and outline how it is used in several important applications in the field.

J. Rae

L.T. Muus

Frank Gaitan

Having surveyed the necessary background in Chapter 1, this chapter begins developing the theory of quantum error correcting codes and fault-tolerant quantum computing. Section 2.1 introduces the dynamical framework of quantum operations. In this approach a quantum computer is treated as an open quantum system and a superoperator describes the effective interaction it has with its environment. The operator-sum representation for a quantum operation is derived, as are some of its essential properties. This section also shows how the noise model introduced in Section 1.4.1 can be represented as a quantum operation, and briefly describes other important noise models. Sec­ tion 2.2 introduces a number of fundamental notions associated with quantum error correcting codes, and the seven-qubit Calderbank-Shor-Steane code is used in Section 2.3 to show how these notions manifest in a specific quantum error correcting code.