ation of input qubit |a) |X) of coordinates (U1, U2, U3, U4) in computational basis {|0>|0), |0>|1), |1)|0), |1)|1>} into output qubit |a') 'X') = ACNOT|2)|X). |a'>|x') Observations |0)|0> |0>|0) Invariant 0 1 0 0 |0)|1) |0)|1) Invariant 0 0 1 0 |1)|0> |0)|1) xqubit flipped |1)|1) |1)|0) xqubit flipped Invariant |1)|+> Invariant Invariant = 10)(a/0)+ B|1)) = 10)(a|0) + B|1)) B|1)|0) + a|1)|1) x qubit amplitudes = |1)(B|0) -a|1)) swapped matrix takes the form: (16.29) The 2 x 2 reduced form of the above gate matrix shows that states of the form |0>|x) are invariant (sub-matrix I), while states of the form |1) |x) have the target qubit /x) flipped (sub-matrix X). Although somewhat tedious, it is useful to verify now the above result by applying the gate matrix ACNOT to the input state |a) /x). From Eqs. (16.28) and (16.29), we obtain: (16.30) The right-hand side of Eq. (16.30) can now be developed according to different input possibilities for |a)|x), i.e., concerning the control qubit |a) and the target qubit /x). Table 16.2 shows the result with the target qubit |x) as being in either a pure state (first four lines) or a superposition of states (last four lines). As expected, the table illustrates that the CNOT gate leaves the target qubit /x) = |0) or |1) invariant when the control qubit is set to |a) = |0). If the target qubit is a superposition |x) = a|0) + B|1), the amplitudes (a, B) are either conserved (|a) = |0)) or swapped (|a) = |1)). In the specific case a = B = 1, the target qubit remains invariant regardless of the control qubit |a), as expected. It is left as an exercise to analyze the action of the CNOT gate Quantum bits and quantum gates ACNOT BCNOT ACNOT Figure 16.3 (a) Quantum gate circuit based on the concatenation of three CNOT gates, with corresponding matrices ACNOT (control qubit at top) and BCNOT (control qubit at bottom). (b) Equivalent circuit representation (CROSSOVER or SWAP). with the control qubit |a) being now in a superposition of states, and show that for certain combinations of in