The rapid growth of interest in quantum computing has brought about the need to secure these emerging computers against a range of security attacks. Among the potential security threats are physical attacks, including those orchestrated by malicious insiders within data centers where the quantum computers are located, which could compromise the integrity of computations and resulting data. To help in the understanding of emerging fault injection attacks on quantum computers, this paper presents an in-depth exploration of quantum computer fault injection attacks. This work introduces a classification of fault injection attacks and strategies, including an analysis of the domain of fault injection attacks, the fault targets, and the fault manifestations in quantum computers. The resulting classification highlights the landscape of the potential threats, and presents a road map for researchers and industry for developing security protection mechanisms against fault injection attacks for the emerging quantum computing systems.