qubit 1q) = a|0) + B|1) into the superposition 19) = al+) + B1-), where 1+), |-) represent a new pure states basis. As we shall see later, the two pure-state bases |0), |1) and 1+), |-) play a fundamental role in quantum computing. The action of the above 2 x 2 quantum gates on input qubits is summarized in Table 16.1. 7 With the following definition (see http://en.wikipedia.org/wiki/Levi-Civita_symbol) or 2), is (1, (3, j 2, 2, or 3), 1), j (1, (2, 3, 3, 2) 1) (3, (2, 1, 1, otherwise: i = = k or k = i. 16.3 Quantum gates with multiple inputs and outputs Table 16.1 Action of the elementary 2 X 2 quantum gates on input qubit |q> = a|0) + B11). Gate U Output |q')=U|q) Action Identity, a|0) +B|1) Invariant X,01, Ox B|0) +a|1) Swaps amplitudes Y,02,0y Swaps amplitudes and IT shift between amplitudes Z,03, Oz a|0) +eir B|1) IT shift between amplitudes Hadamard, H al+) + B|-> Switches to 1+),1-> basis For illustration purposes, consider two examples showing the action of cascades of 2 X 2 quantum gates: U=XZX- U/q) = XZX(x|0) + B|1)) = XZ(B|0) + a 11) (16.26) = e(||) U=HX-> Uq)=HX(a|0)+ = ,B|1)) (16.27) = H(B|0)+a|1)) Finally, we must note that the 2 x 2 Pauli and Hadamard matrix gates correspond to reversible 1-qubit computations. Indeed, any input qubit can be retrieved through the double operations X2= = Y2 = More generally, any gate cor- responding to a unitary matrix or operator U is reversible by the application of the Hermitian conjugate U+, since, by definition, U+U = = I. This property also applies to gate cascades UVW of unitary operators, which have for their Hermitian conjugate (UVW...)+ ..W+V+U+, thus, (UVW...)-UVW W+V+U+UVW = W+V+VW=...W+W...I. Quantum gates with multiple qubit inputs and outputs With the background from the previous section on 2 x 2 quantum gates and that from reversible logic gates introduced in Chapter 15, we can now describe the matrix representation and operation of various higher-order quantum gates, which have two or more 1-qubit inputs and outputs. The elementary gates of this typ