The CONTROLLED SWAP (C-SWAP) gate takes as input 3 qubits and swaps the second and third if and only if the first qubit is a 1.
(a) Show that each of the NOT, CNOT, and C-SWAP gates are their own inverses.
(b) Show how to implement an AND gate using a C-SWAP gate, i.e., what inputs a, b, c would you give to a C-SWAP gate so that one of the outputs is a ˄b?
(c) How would you achieve fanout using just these three gates? That is, on input a and 0, output a and a.
(d) Conclude therefore that for any classical circuit C there is an equivalent quantum circuit Q using just NOT and C-SWAP gates in the following sense: if C outputs y on input x, then Q outputs |x,y,z〉 on input |x,0,0〉. (Here z is some set of junk bits that are generated during this computation.)
(e) Now show that that there is a quantum circuit Q−1 that outputs |x,0,0〉 on input |x,y,z〉.
(f) Show that there is a quantum circuit Q' made up of NOT, CNOT, and C-SWAP gates that outputs |x,y,0〉 on input |x,0,0〉.
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