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Quantum gates with multiple inputs and outputs in .NET Maker Data Matrix ECC200 in .NET Quantum gates with multiple inputs and outputs

16.3 Quantum gates with multiple inputs and outputs use .net data matrix ecc200 development todraw data matrix barcode on .net Visal Basic .NET Table 16.2 Transformation of input qubit a . x of coordinate visual .net Data Matrix s (u1 , u2 , u3 , u4 ) in computational basis {. 0 . 0 , . 0 . 1 , . 1 . 0 , . 1 . 1 } into output qubit a . x = A CNOT a . x . u1 1 0 0 0 1 0 0 u2 0 1 0 0 1 0 0 u3 0 0 1 0 0 1 0 u4 0 0 0 1 0 1 0 a . x . 0 . 0 . 0 . 1 . 1 . 0 . 1 . 1 1 (. 0 . 0 + . 0 . 1 ) = . 0 . + 2 1 (. 1 . 0 + . 1 . 1 ) = . 1 . + 2 . 0 . 0 + . 0 . 1 = . 0 ( . 0 + . 1 ) . 1 . 0 + . 1 . 1 = . 1 ( . 0 + . 1 ) . a . x . 0 . 0 . 0 . 1 . 0 . 1 . 1 . 0 1 (. 0 . 0 + . 0 . 1 ) = . 0 . + 2 1 (. 1 . 0 + . 1 . 1 ) = . 1 . + 2 . 0 . 0 + . 0 . 1 = . 0 ( . 0 + . 1 ) . 1 . 0 + . 1 . 1 = . 1 ( . 0 + . 1 ) Observation 2d Data Matrix barcode for .NET s Invariant Invariant xqubit ipped xqubit ipped Invariant Invariant Invariant x qubit amplitudes swapped. {. a . x } = {. 0 . 0 , . 0 . 1 , . 1 . 0 , . 1 . 1 }, the matrix takes the form: 1 0 0 0 0 1 0 0 I 0 ACNOT = 0 0 0 1 . 0 X 0 0 1 0 (16.29). The 2 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 ipped (sub-m Data Matrix for .NET atrix 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. ( VS .

NET Data Matrix barcode 16.28) and (16.29), we obtain: u1 1 0 0 0 u1 u 2 0 1 0 0 u 2 .

a . x = ACNOT a . x = = u 4 0 0 0 1 u 3 (16.30) u3 u4 0 0 1 0 = u 1 0 . 0 + u 2 0 . 1 + u 4 1 . 0 + u 3 1 . 1 . The right-h VS .NET data matrix barcodes and 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 e ither a pure state ( rst 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 = . 0 + . 1 , the amplitu des ( , ) are either conserved (. a = . 0 ) or swapped (. a = . 1 ). In the spe ci c case = = 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 A CNOT B CNOT A CNOT Figure 16.3 (a) gs1 datamatrix barcode for .NET 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 input qubits a . x , the CNOT ga gs1 datamatrix barcode for .NET te can generate any of the four EPR or Bell states, as de ned in Eq. (16.

2). It is easily veri ed that the matrix ACNOT is unitary and that its inverse matrix is A 1 = ACNOT , or A2 CNOT = I (I = 4 4 identity matrix). This last result is expected, CNOT since the repeated action of CNOT (with same control qubit, by inherent circuit construction) must leave the target qubit invariant.

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