The Quantum Circuit model of computation is the primary model used by theoretical computer scientists to study quantum computers (in the same way that the Turing Machine is used to study classical computers). It's definitely important if you're studying computation, but can also be useful for understanding aspects of quantum mechanics more generally.

My experience was that quantum circuits were mostly dealt with algebraically. I had a hard time visualizing the quantum states I was dealing with, partly because they exist in a high-dimensional space and partly because they involve complex numbers. I've attempted to tame them a little bit by building a quantum circuit simulator, using a circular display for the complex amplitudes. For example, these are the representations for the numbers 1, 0, and (i-1)/2:

Combine that with the ability to move time forward and backward through the circuit, and the result is something that for me goes a lot further toward conveying the effects of the various quantum gates than the standard algebraic approach.

I've posted it here. It includes a circuit editor, permalinks for created circuits, and transition diagrams for individual gates. Unfortunately I don't have time at the moment for a full explanation of quantum computation, but hopefully for those unfamiliar with it the simulator can be a helpful supplement to other explanations.

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