The rapid growth of interest in quantum computing has brought about the need to secure these powerful machines against a range of physical attacks. As qubit counts increase and quantum computers achieve higher levels of fidelity, their potential to execute novel algorithms and generate sensitive intellectual property becomes more promising. However, there is a significant gap in our understanding of the vulnerabilities these computers face in terms of security and privacy attacks. Among the potential 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. This paper presents an exploration of fault-injection attacks as one class of physical attacks on quantum computers. This work first introduces a classification of fault-injection attacks and strategies, including the domain of fault-injection attacks, the fault targets, and fault manifestations in quantum computers. The resulting classification highlights the potential threats that exist. By shedding light on the vulnerabilities of quantum computers to fault-injection attacks, this work contributes to the development of robust security measures for this emerging technology.