Theory of quantum system certification: a tutorial

The precise control of complex quantum systems promises numerous technological applications including digital quantum computing. The complexity of such devices renders the certification of their correct functioning a challenge. To address this challenge, numerous methods were developed in the last decade. In this tutorial, we explain prominent protocols for certifying the physical layer of quantum devices described by quantum states and processes. Such protocols are particularly important in the development of near-term devices. Specifically, we discuss methods of direct quantum state certification, direct fidelity estimation, shadow fidelity estimation, direct quantum process certification, randomized benchmarking and cross-entropy benchmarking. Moreover, we provide an introduction to powerful mathematical methods, which are widely used in quantum information theory, in order to derive theoretical guarantees for the protocols.
Comment: 47 pages, 4 figures, 62 Theorems, Propositions, Lemma, Protocols, Definitions and Examples; v2: published version with numerous small improvements; v3: the argument leading to the diamond norm normalization of quantum channels has been corrected