The rapidly growing interest in quantum computing also increases the importance of securing these computers from various physical attacks. Constantly increasing qubit counts and improvements to the fidelity of the quantum computers hold great promise for the ability of these computers to run novel algorithms with highly sensitive intellectual property. However, in today’s cloud-based quantum computer setting, users lack physical control over the computers. Physical attacks, such as those perpetrated by malicious insiders in data centers, could be used to extract sensitive information about the circuits being executed on these computers. This work shows the first exploration and study of power-based side-channel attacks in quantum computers. The explored attacks could be used to recover information about the control pulses sent to these computers. By analyzing these control pulses, attackers can reverse-engineer the equivalent gate-level description of the circuits, and the algorithms being run, or data hard-coded into the circuits. This work introduces five new types of attacks, and evaluates them using control pulse information available from cloud-based quantum computers. This work demonstrates how and what circuits could be recovered, and then in turn how to defend from the newly demonstrated side-channel attacks on quantum computing systems.