To address the rapidly growing demand for cloud-based quantum computing, various researchers are proposing shifting from the existing single-tenant model to a multi-tenant model that expands resource utilization and improves accessibility. However, while multi-tenancy enables multiple users to access the same quantum computer, it introduces potential for security and reliability vulnerabilities. It therefore becomes important to investigate these vulnerabilities, especially considering realistic attackers who operate without elevated privileges relative to ordinary users. To address this research need, this paper presents and evaluates QubitHammer, the first attack to demonstrate that an adversary can remotely induce unauthorized changes to a victim's quantum circuit's qubit's state within a multi-tenant model by using custom qubit control pulses that are generated within constraints of the public interfaces and without elevated privileges. Through extensive evaluation on real-world superconducting devices from IBM and Rigetti, this work demonstrates that QubitHammer allows an adversary to significantly change the output distribution of a victim quantum circuit. In the experimentation, variational distance is used to evaluate the magnitude of the changes, and variational distance as high as 0.938 is observed. Cross-platform analysis of QubitHammer on a number of quantum computing devices exposes a fundamental susceptibility in superconducting hardware. Further, QubitHammer was also found to evade all currently proposed defenses aimed at ensuring reliable execution in multi-tenant superconducting quantum systems.