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By Rosinger E.

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1) E E Rosinger (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) −→ −→ −→ −→ −→ −→ −→ −→ (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 1) (1, 1, 0) It is easy to see that applying twice the Toffoli gate gives the identity. Thus the Toffoli gate is invertible, being its own inverse. Consequently, the operation of the Toffoli gate is indeed reversible. It is important to note that the redundancy in the output of the Toffoli gate which reproduces identically the bits a and b is the way to avoid erasing information, which according to Landauer, is a necessary condition for allowing for reversibility.

4). 2)     1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0      α β γ δ          =  α β δ γ      assuming that | ψ > = α | 0 > + β | 1 >, | χ > = γ | 0 > + δ | 1 >. It is easy to check that the above matrix is indeed unitary. 6. This result about quantum gates corresponds to the classical result according to which every logical gate operating on bits can be obtained from the composition of NAND gates. Here we recall that a NAND gate operates on two classical bits a, b according to a NAND b = NOT (a AND b).

The precise details are as follows. Let us take any n ≥ 1 fixed. 5) Ai (| ψ1 >, . . , | ψn >) = = (| ψ1 >, . . , | ψi−1 , A| ψi >, | ψi+1 , . . , | ψn >) where | ψ1 >, . . , | ψn > ∈ C2 . In other words, Ai leaves all the qubits the same, except for | ψi >, on which it operates according to 36 E E Rosinger the one qubit gate A. 6) n CNOTi,j : C2 n −→ C2 which when applied to an arbitrary n-qubit (| ψ1 >, . . 1. It is easy to check that both Ai and CNOTi,j defined above are unitary operators.

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