With the anticipated end of Moore’s law for integrated circuits fast approaching and continued advances in low-power electronics, interest in quantum computing has increased. This shifts the focus from deterministic logical circuits to potentially more powerful circuits based on controllable quantum systems. In this column, we present a mathematical tour of the quantum circuit model, beginning with reversible logic circuits and expanding to quantum circuits, gates, and measurement. We highlight quantum mechanical phenomena such as superposition, entanglement, and measurement, review the Gottesman-Knill theorem, which states that some subclasses of quantum operations can be simulated efficiently on a classical computer, and describe sets of quantum gates that are universal for quantum computation.