The left-hand gate runs first

A circuit is a drawing of one or more wires, with each gate drawn as a box on a wire, and you read that drawing from the left so the left-hand gate runs first. A measurement, if the circuit has one, is usually the last mark on the wire, because the click replaces the amplitudes and returns one bit.

Each puzzle is a one-wire circuit, with the start state at the left-hand end and a named pattern of amplitudes as the target at the right-hand end. The keys print the gates the puzzle will allow. Two sequences made from those same keys can finish on different patterns, so the order has to be copied down with the letters.

Take a qubit that is certain to read 0. H first rewrites it as two equal amplitudes of the same sign, about plus 0.71 and plus 0.71, and each fresh copy then reads 0 or 1 with equal chance. X after that swaps the amplitude for 0 with the amplitude for 1. Write plus 0.71 beside 0 and plus 0.71 beside 1, swap the two columns, and the sheet still reads as it did, which is why the chances and the matching signs stay where H put them.

X first makes that qubit certain to read 1. H on that state writes two equal amplitudes with opposite signs, about plus 0.71 and minus 0.71, so the chances of 0 and 1 are equal at this finish as well. A later gate can turn that minus into a reading that comes out one way every time, which is why a target has to name the signs and not only the chances.

Keith had put both sequences in a folder he called a split followed by a flip. Malcolm ruled two rows on one sheet and made him file that sheet in place of the folder label. One row ends at matching signs and the other ends at opposite signs, and nothing printed on the two keys records which row was run.

H, then Z, then H already sits on the bench in the note about cancellation, where the three boxes finish certain to read 1. Put those boxes down in another order and you have drawn another circuit.